Variations, Geometry and Physics: In Honour of Demeter Krupka's Sixty-fifth Birthday

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VARIATIONS , G EOMETRY AND P HYSICS No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

VARIATIONS , G EOMETRY AND P HYSICS IN HONOUR OF

D EMETER K RUPKA’ S

SIXTY- FIFTH BIRTHDAY

O LGA K RUPKOV A´ AND

DAVID S AUNDERS

Nova Science Publishers, Inc. New York

c 2009 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter cover herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal, medical or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Library of Congress Cataloging-in-Publication Data Variations, geometry & physics / Olga Krupkova and David Saunders (editor). p. cm. ISBN 978-1-61324-186-8 (eBook) 1. Calculus of variations. 2. Global analysis (Mathematics) 3. Mathematical physics. I. Krupkov, Olga, 1960- II. Saunders, D. J., 1964- III. Title: Variations, geometry, and physics. QA315.V38 2008 510–dc22 2008024454

Published by Nova Science Publishers, Inc. ✜ New York

Contents vii

Preface

PART I

VARIATIONAL P RINCIPLES ON J ET B UNDLES

1

Chapter 1

Lepage Forms in Variational Theories: From Lepage’s Idea to the Variational Sequence Jana Musilová and Michal Lenc

3

Chapter 2

Lepage Forms in the Calculus of Variations Olga Krupková

Chapter 3

On a Generalization of the Poincaré-Cartan Form in Higher-Order Field Theory D. R. Grigore

57

Krupka’s Fundamental Lepage Equivalent and the Excess Function of Wilkins D. J. Saunders

77

Chapter 4

27

Chapter 5

Lepage Congruences in Discrete Mechanics Antonio Fernández and Pedro L. Garc´ıa

85

Chapter 6

Finite Order Variational Sequences: a Short Review Raffaele Vitolo

99

Chapter 7

Concatenating Variational Principles and the Kinetic Stress-Energy-Momentum Tensor M. C. López, M. J. Gotay and J. E. Marsden

117

A Geometric Hamilton-Jacobi Theory for Classical Field Theories M. de León, J. C. Marrero and D. M. de Diego

129

Chapter 8

vi

Contents

PART II

N ATURAL B UNDLES AND D IFFERENTIAL I NVARIANTS

141

Chapter 9

Natural Lagrangian Structures Josef Janyˇska

143

Chapter 10

Connections on Higher Order Frame Bundles and Their Gauge Analogies Ivan Kolárˇ

Chapter 11

Natural Lifts in Riemannian Geometry Oldˇrich Kowalski and Masami Sekizawa

Chapter 12

Invariant Variational Problems and Invariant Flows via Moving Frames Peter J. Olver

Chapter 13

Differential Invariants of the Motion Group Actions Boris Kruglikov and Valentin Lychagin

PART III

D IFFERENTIAL E QUATIONS S TRUCTURES

Chapter 14

AND

167

189

209

237

G EOMETRICAL

Remarks on the History of the Notion of Lie Differentiation Andrzej Trautman

Chapter 15

Second-Order Differential Equation Fields with Symmetry M. Crampin and T. Mestdag

Chapter 16

Dimensional Reduction of Curvature-Dependent Central Potentials on Spaces of Constant Curvature J. F. Cariñena, M. F. Rañada and M. Santander

253

255

261

277

Chapter 17

Direct Geometrical Method in Finsler Geometry L. Tamássy

293

Chapter 18

Linear Connections Along the Tangent Bundle Projection W. Sarlet

315

Chapter 19

On the Inverse Problem for Autoparallels G. E. Prince

341

PART IV

A PPENDIX

353

Demeter Krupka: List of Publications

355

Index

363

Preface This book is a commemorative volume to celebrate the sixty-fifth birthday of Demeter Krupka, an internationally recognized researcher in several fields of mathematics and mathematical physics. Demeter was born on 22 January 1942 in Levoˇca, Slovakia. He studied first at Masaryk University, Brno and in 1976 he received a Ph.D. degree from Charles University, Prague; during his studies he also spent a year at Warsaw University under the supervision of Professor Andrzej Trautman. In 1981 he was awarded the degree of Doctor of Sciences in geometry and topology by the Czechoslovak Academy of Sciences, Prague. Demeter has held posts at Masaryk University and at the Silesian University in Opava, and is currently Professor in the Department of Algebra and Geometry at Palacký University, Olomouc. He also holds an honorary position as an IAS Distinguished Fellow at the Institute for Advanced Study, La Trobe University, Melbourne. His main research areas are global variational analysis, differential geometry, tensor algebra and mathematical physics. Among his recognized achievments we should mention the development of a global theory of variational functionals in fibred spaces in the early 1970s, and its higher-order generalisations in the early 1980s, based on his concept of a Lepage form and the Euler–Lagrange form. This approach provides a completely intrinsic setting for Euler–Lagrange theory and the invariance of variational functionals, including Noether theorem and its generalisations. It also facilitates the study of regular variational functionals, casting new light on the concept of regularity in the calculus of variations, Hamilton theory and Hamilton–Jacobi theory. He also was among the first to find necessary and sufficient conditions for a system of higher-order PDEs in covariant form to be variational, and he solved the local problem of the structure of higher-order null Lagrangians. In 1990 Demeter published his key paper on variational sequences, a finite order alternative to Vinogradov’s C -spectral sequence and the variational bicomplex introduced by, for example, Dedecker and Tulczyjew. Another important impact of his work concerns the geometric foundations of theory of differential invariants developed in the 1970s and 1980s, based on the concept of a higher order differential group, and the creation of a rigorous setting for natural variational principles. While solving technical problems appearing in the higher order variational calculus, he also contributed to linear algebra by studying and

viii

Preface

solving the trace decomposition problem in general tensor spaces over the real numbers. During his career, Demeter has written numerous articles for publication (a list is given in the Appendix to this volume) and taught a wide variety of topics in mathematics and mathematical physics; other highlights include being a founding editor of two major journals (Journal of Geometry and Physics and Differential Geometry and its Applications) and holding the post of Rector at the Silesian University in Opava. Most recently he has been joint editor of a major reference work (Handbook of Global Analysis). To celebrate Demeter’s sixty-fifth birthday, a Colloquium was held in Olomouc on 25th and 26th August, 2007 with twenty-one invited speakers, all with established reputations in their field. Many of the speakers have prepared versions of their talks for the present volume, and we have also been able to include some articles from authors who were not able to attend. We have divided the articles into three groups, covering three major areas of Demeter’s research interests: Variational principles on jet bundles, Natural bundles and differential invariants and Differential equations and geometric structures. We are grateful to all the authors who have provided these articles, and we are sure that they will be of great interest to the community.

Olga Krupková, David Saunders (Editors) Olomouc, January 2008

Part I

VARIATIONAL P RINCIPLES ON J ET B UNDLES

1

In: Variations, Geometry and Physics Editors: O. Krupková and D. Saunders, pp. 3-26

ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc.

Chapter 1

L EPAGE F ORMS IN VARIATIONAL T HEORIES : F ROM L EPAGE ’ S I DEA TO THE VARIATIONAL S EQUENCE∗ Jana Musilová† and Michal Lenc‡ Institute of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotláˇrská 2, 611 37 Brno, Czech Republic

Abstract In the presented paper the development of the concept of Lepage forms is followed from the initial idea given by Lepage in 1930’s to its important role in the contemporary geometrical analysis and variational theories in physics. The attention is paid especially to the role of the concept of Lepage forms for the main formula of calculus of variations (called the first variational formula) as well as to their meaning as geometrical objects resulting by a natural way from the theory of variational sequences on finite order prolongations of fibered manifolds. As a possible application Lepage forms are studied in the context of our results concerning the representation of the variational sequence by differential forms on the one hand, and in concrete physical examples on the other. For the illustration we show that the concept of Lepage forms can be effectively used in such physical situations in variational field theories, as are e.g. variational problems in the classical string and brane theories.

1.

Introduction

The contemporary general concept of Lepage forms is closely related to fundamentals of local and global aspects of the modern calculus of variations. It represents differential forms of a special type defined on finite order jet prolongations of fibered manifolds. These manifolds are chosen as appropriate underlying geometrical structures for the majority of ∗ This paper is dedicated to 65th birthday of our colleague and friend professor Demeter Krupka, whose contribution to the development of the concept of Lepage forms as the key geometrical objects of modern calculus of variations and its application in physical theories is undoubtedly important. † E-mail address: [email protected] ‡ E-mail address: [email protected]

4

Jana Musilová and Michal Lenc

physical theories. Theories developed on fibered manifolds with one-dimensional bases are called “mechanics”, while for n-dimensional bases of underlying fibered manifolds we speak about “field theories”. There are specific geometrical (i.e. coordinate free) objects defined on a fibered manifold and its jet prolongations, closely connected with the fibered structure of the manifold: projectable and vertical vector fields, horizontal and contact differential forms. These objects play the key role in formulation of the fundamental variational problem which lies in searching such sections of the underlying fibered manifold, e.g. “trajectories” of studied physical systems, representing critical (stationary) points of a functional appropriately chosen with respect to a given theory. Such a functional is, for a theory formulated on a fibered manifold (Y, π, X) with the n-dimensional base X, the (m + n)-dimensional total space Y , and the projection π, defined by the integral of a horizontal n-form λ = L ω0 , called r-th order Lagrangian, where L is a function on the r-th prolongation (J r Y, πr , X) of the fibered manifold and ω0 is the volume element on the base, in local coordinates ω0 = dx1 ∧ . . . ∧ dxn . The general questions most intensively studied in last decades are especially: ⊲ the variational equivalence problem leading to the concept of Lepage equivalents of a given Lagrangian, i.e. differential forms giving the same variational integral as the corresponding Lagrangian, and in general to the concept of Lepage n-forms, ⊲ the structure of the Euler-Lagrange mapping assigning to a Lagrangian its EulerLagrange form representing the equations of motion; the study of this structure enables us to find trivial Lagrangians, leading to identically zero Euler-Lagrange forms, ⊲ the structure of Helmholtz-Sonin mapping; this enables us to solve the inverse problem of calculus of variations, i.e. stating whether given equations of motion arise from a variational problem, and leads to the concept of Lepage (n + 1)-forms and their equivalence with respect to given equations of motion, ⊲ the theory of the finite-order variational sequences and the problem of their representation by differential forms, both inspired by the above exposed equivalence problems, namely by the close relation between the Euler-Lagrange mapping and the exterior derivative operator, ⊲ a cohomological structure of the variational sequence in which both Euler-Lagrange and Helmholtz-Sonin mappings are included, as an effective tool for describing global aspects the theory, ⊲ an idea of the quite general definition of Lepage forms arising naturally from the structure of the variational sequence itself. Both primary ideas leading to the concept of Lepage forms (the importance of the exterior derivative operator and the idea of equivalence of forms in variational problems) were ´ Cartan and Th. de Donder, formulated by Lepage in [35] and [36], and presented by Elie respectively. They were later developed by Paul Dedecker in [6]. In Sec. 2 we present the brief review of considerations concerning these “historical” ideas and their consequences. The concept of a Lepage equivalent of a Lagrangian λ = L ω0 as a form leading to the same variational problem, strictly based on the geometrical structure of fibered manifolds

Lepage Forms in Variational Theories

5

and their jet prolongations, was introduced by Krupka (see e.g. his key paper [16]), and then developed up to a coherent theory. His studies of the Euler-Lagrange mapping in 1970’s and 1980’s inspired the later considerations concerning the variational sequence. (See e.g. in [17], [18], [19], [20], [21], [22], [24], a complete review with contemporary results see in [26].) The concept of a Lepage (n + 1)-form and Lepage class of a dynamical (n + 1)form E = Eσ ω σ ∧ ω0 , important from the point of view of equations of motion Eσ ◦ J s γ (variational or non-variational) for trajectories γ : X → Y , U ⊂ X, π ◦ γ = IdU of a dynamical system, was introduced by Krupková for mechanics, i.e. n = 1 (see [30], [31], [32]). It was well elaborated to a coherent theory by the same author (see the key book [33]). The concept was later extended to non-holonomic constrained mechanical systems in [34]. Main ideas and results obtained in studies of Lepage forms made by Krupka and Krupka’s coworkers are summarized in Sec. 3. The above mentioned special type of a Lepage form, Lepage equivalent of a Lagrangian, is important namely for so called first variation formula. This formula represents an appropriate decomposition of the basic object of the calculus of variations – the variational functional (the integral first variation formula), or the decomposition of the integrand itself with respect to the fibered structure of the underlying manifold (the infinitesimal first variation formula). Moreover, the first variational formula gives information concerning some properties of symmetry transformations of variational problems. Thus, the application of Lepage equivalents of Lagrangians for formulation of variational physical theories is straightforward. In Sec. 4 we give some examples of such an application in modern variational field theories in physics – string and brane theories. We present there the Lepage equivalents in three physical situations: a relativistic particle (first order mechanics), the classical variational functional for a boson string (first order theory, see e.g. [5]), the Polyakov functional for D-dimensional space and its special case for Minkowski space (first order field theory). With help of results following from the first variational formula we use these Lepage equivalents for obtaining quantities important for conservation laws. The quite general definition of Lepage forms arises by a natural way from the structure of the variational sequence on finite order jet prolongations of fibered manifolds. Especially, it is closely related to the problem of representation of the variational sequence by forms. The idea of the finite order variational sequence was exposed by Krupka in 1989 (see [23]), who studied the structure of morphisms of the sequence important for calculus of variations, as well as the cohomology aspects important for global questions. Recall that the infiniteorder counterpart of Krupka’s theory is the theory of variational bicomplex, formulated in 1970’s and 1980’s by Anderson and Duchamp in [3], Dedecker and Tulczyjew in [7], [43], Takens in [42], and Vinogradov and his school in [44], [45], [46], [4], [11]. Both theories, the variational sequence as well as variational bicomplex, are based on the analysis of the exterior derivative operator d. (Recall that the r-th order variational sequence is the quotient of the well-known De Rham sequence by its exact subsequence of forms of certain kind of contactness or “variational irrelevance”, the quotient being exact as well.) However, there are some differences resulting from the mentioned different choice of basic structures as domains of definition, finite order jet prolongations J r Y of a fibered manifold Y and infinite jets J ∞ Y . The main difference lies in the fact that the finite order theory keeps the order of all sequence operation fixed (contrary to the case of bicomplex, where the order can be increased arbitrarily). This gives a deeper understand-

6


ing of the role of the order, of the analytic structure of fundamental variational concepts. Moreover, it is more appropriate for application in physical situations, always described as finite order problems. The instructive comparison of both theories is presented in [25] and [48]. The finite order variational sequence was intensively studied from 1990’s to the present by Krupka and coauthors or coworkers, and by other authors, mainly Grigore [9], [10] and Vitolo [47], [48], [49], [50]. (The practically complete bibliography related to both the variational bicomplex and mainly to the variational sequence is presented in [50].) A specific problem connected with the finite order variational sequence is its representation by differential forms. The idea lies in the requirement to represent every class [̺] of differential forms in the r-th order variational sequence by an appropriately chosen form with specific properties: R-linearity, coordinate invariance, exactness of the sequence of representatives, projection property of the operator assigning to a class its representative. The representation problem was also studied namely by Krupka’s school and important partial results were obtained (except for cited Krupka’s works see also e.g. [37], [12], [13], [14], [28], [38], [39], [29], [41].) A complete solution of the representation problem in r-th order field theory was inspired by works of Anderson [1], [2]. The complete solution of the representation problem for finite order variational sequence in field theories is presented in [15]. It was inspired by the Euler representation operator introduced by Anderson for the variational bicomplex: the Anderson’s concept of this operator was adapted to the finite order case and the required properties of the obtained representation were proved by taking into account specific features of the finite-order problem. It appeared that the general definition of Lepage forms arises immediately from the structure of the variational sequence representation. Note that the general concept of Lepage forms was introduced by an alternative way in [29] for the case of mechanics. Some aspects of the representation of the variational sequence in the context with the general concept of Lepage forms are discussed in Sec. 5.

2.

Lepage’s Idea – a Brief Historical Review

The famous mathematician Théophile Lepage gained the degree of Doctor of Science de ´ Cartan. He had 11 docl’Université de Liege in 1924. He was probably the student of Elie toral students and 168 descendants in the period 1937–1973. He was the dean of the Faculté des Sciences de l’Université Libre de Bruxelles in 1953–1955. His bibliography amounts to 19 scientific papers published in 1929–1955, mostly devoted to geometry and calculus of variations. As a certain curiosity we can mention that Théophile Lepage had introduced a symplectic analog of Hodge theory before the Hodge theory itself was presented. His paper [36] notifying the concept of future Lepage forms was published in 1936. However, in 1933 some Lepage’s ideas was published in Comptes Rendus des Séances de ´ l’ Académie des Sciences, being presented on the seance 18-th December 1933 by Elie Cartan (see [35]). In the presentation called Sur certaines formes différentielles extérieures et la variation des intégrales doubles (corresponds to the field theory on a fibered manifold with the two-dimensional base) a predecessor of a contact 1-form is introduced and some factorization of 2-forms with respect to contact forms is made. Let us now present some results from the Lepage’s key paper [36] important for the concept of Lepage forms and discuss them from the point of the contemporary theory. In [36] a double variational integral


7

defined by the formula I(zi ) =

Z Z

∂zn ∂z1 ∂zn ∂z1 ,..., ; ,..., f x, y; z1 , . . . , zn ; ∂x ∂x ∂y ∂y

dx dy

(2.1)

is studied. There x and y are independent variables, zi (x, y) are declared as unknown functions. It is evident that this corresponds to a first order variational problem on a fibered manifold (Y, π, X), where dim X = 2, dim Y = n + 2, dim J 1 Y = 2 + n + 2n = 3n + 2. A 2-form is defined (in the Lepage’s terminology called the symbolic quadratic form) ω = f (x, y; z1 , . . . , zn ; p1 , . . . , pn ; q1 , . . . , qn ) dx dy,

(2.2)

where function f is differentiable. (The exterior product “∧” is not explicitly written but it is evidently taken into account.) This form corresponds to a Lagrangian defined on J 1 Y . Thus, the integral (2.1) is a variational functional of the type Z

J 1 γ ∗ ω,

Ω

γ : (x, y) → (z1 γ(x, y), . . . , zn γ(x, y))

and our aim is to find its stationary paths γ. The forms ωi = dzi − pi dx − qi dy,

1 ≤ i ≤ n,

(2.3)

are introduced as Pfaffian forms, being the predecessor of today’s contact forms on J 1 Y . Moreover, congruences (i.e. equivalence classes) are introduced as follows: Denote θi = Xi dx + Yi dy + Aij ωj ,

Aij + Aji = 0,

1 ≤ i, j ≤ n,

Xi , Yi , Aij being differentiable functions of (x, y, zj , pj , qj ). The relation Ω = f dx dy + θi ωi ,

Ω ≡ ω (mod ω1 , . . . , ωn )

(2.4)

defines the equivalence relation of forms giving the same variational integral (2.1). Then there is requirement to calculate all forms Ω defining congruences dΩ ≡ 0

(mod ω1 , . . . , ωn ).

(2.5)

The result is as follows: Ω = f dx dy +

∂f ∂f dx − dy ∂qi ∂pi

ωi + Aij ωi ωj .

(2.6)

It can be easily verified that from the point of view of the contemporary theory the form Θω given by first two summands of (2.6), i.e. Θω = f dx dy +

∂f ∂f dx − dy ∂qi ∂pi

ωi ,

is exactly the principal component of the Lepage equivalent of the Lagrangian ω (compare with the relation (3.5) for n = 2, s = 1), while the form Aij ωi ω2 is 2-contact in the

8


present terminology. (Note that the aim of the paper is to study so called geodesic vector fields relative to the form Ω. These fields given by functions v : (x, y, z1 , . . . , zn ) −→ (pi (x, y, z1 , . . . , zn ), qi (x, y, z1 , . . . , zn )) are defined by the condition d[Ω] = 0, where [Ω] = Ω ◦ v and [ωi ] = ωi ◦ v.) Finally let us mention an important contribution of Paul Dedecker to the concept of Lepage forms. It is presented in the paper [6]. The notation used therein is close to that used at present. The variational integral in mechanics is introduced by I1 =

Z c

L(t, q i , q˙i ) dt,

1 ≤ i ≤ n,

(2.7)

where c is “an arc in the space W with local coordinates (t, q i , q˙i ) satisfying the equations ω i = 0”. In fact, this means that c can be interpreted as a first prolongation J 1 γ of a section γ of a fibered manifold with one-dimensional base. The so called Pfaffian form is defined by ∂L ω = L dt + i ω i , ω i = dq i − q˙i dt. (2.8) ∂ q˙ We can see that from the point of view of the today’s theory the form (2.8) is exactly the Lepage equivalent of the first order Lagrangian in mechanics. The semibasic forms are then introduced as θ = L dt mod ω i and it is stated and proved that the the relation (2.8) gives the unique semibasic form such that dθ ≡ 0 mod ω i . The procedure is then generalized for the field theory and a version of the first variational formula is derived.

3.

Lepage Forms on Fibered Manifolds

The concept of a Lepage form and the Lepage equivalent of a Lagrangian on fibered manifolds and their jet prolongations was introduced by Krupka in one of his basic works [16] in the connection of properties of the variational integral as well as the so called first variational formula. This formula is closely related to the fibered structure of the underlying manifold. The definition and properties of both a Lepage form and a Lepage equivalent of a Lagrangian were then studied by Krupka in [22]. In this section we follow the development and improvement of the concept of Lepage forms culminating in Krupka’s latest work on this topic [26]. First let us briefly expose basic geometrical structures for the variational integral, variational principle and the first variation formula. We use the standard definitions and standard notations given e.g. in [26]. Moreover, we often present definitions of geometrical objects by their chart expressions, which is more appropriate for their practical use.


3.1.

9

Lagrange Structures, Variational Functionals

Let (Y, π, X), in shortened notation Y or π, be a fibered manifold with the n-dimensional base X, (m + n)-dimensional total space Y , m-dimensional fibers π −1 ({x}), x ∈ X and the projection π : Y → X (surjective submersion). Note that for n = 1 or n > 1 this structure represents the geometrical background for mechanics or field theories, respectively. Denote by (J r Y, πr , X), r ≥ 0, the r-th jet prolongation of (Y, π, X), where we put J 0 Y = Y , π0 = π. Canonical projections are denoted by πs,r : J s Y → J r Y for s > r ≥ 0. Let (V, ψ) be a fibered chart on Y , V being an open subset of Y , ψ = (xi , y σ ), 1 ≤ i ≤ n, 1 ≤ σ ≤ m. The pair (U, ϕ), U = π(V ), ϕ = (xi ), is the associated chart on X. A smooth mapping γ : U → Y such that π ◦ γ = Id U is called a section of π defined on U . Denote by ΓU a set of all such sections. The pair (Vr , ψr ) where Vr = πr−1 (V ), ψr = (xi , y σ , yiσ1 , . . . , yiσ1 ...ir ),

yiσ1 ...ik =

∂ k yσ γ , ∂xi1 . . . ∂xik

1 ≤ k ≤ r, 1 ≤ i1 , . . . , ik ≤ n, is the associated fibered chart on (J r Y, πr , X). A vector field field ξ on Y is called π-projectable, if there exists a vector field ξ0 on X such that T π · ξ = ξ0 ◦ π. It is called π-vertical, if ξ0 = 0. A π-projectable vector field is expressed as ∂ ∂ ξ = ξ i (xj ) i + Ξσ (xj , y ν ) σ . ∂x ∂y For π-vertical vector field ξ we have ξ i = 0, 1 ≤ i ≤ n. Definitions of πs -projectable or πs,r -projectable vector fields, and definitions of πs -vertical or πs,r -vertical vector fields on J s Y , s > r ≥ 0, are quite analogous. A vector field on J s Y is of the form ξ = ξi

s X ∂ ∂ σ ∂ + Ξ + Ξσi1 ...ik σ , i σ ∂x ∂y ∂yi1 ...ik k=1

where the components of ξ are functions on J s Y , in general. Let ξ be a π-projectable vector field in Y . Then its s-jet prolongation is the vector field on J s Y given by the chart expression J s ξ = ξ i (xj )

s X ∂ ∂ ∂ σ j ν + Ξ y ) + Ξσi1 ...ik (xj , y ν , . . . , yjν1 ...jk ) σ (x , , i σ ∂x ∂y ∂yi1 ...ik k=1

Ξσi1 ...ik = where dj f =

dΞσi1 ...ik−1 dxik

s X df ∂f = + dxj ∂xj k=0

− yiσ1 ...ik−1 j

X

1≤i1 ≤...≤ik

dξ j , dxik

∂f yiσ1 ...ik j σ ∂y i1 ...ik ≤n

is the total derivative of a function f on J s Y with respect to xj . A differential form ω on J s Y is called πr -horizontal if iξ ω = 0 for every πs -vertical vector field ξ. A form ω on J s Y is called πs,r -horizontal if iξ ω = 0 for every πs,r -vertical

10


vector field ξ. A form ω on J s Y is called contact if J s γ ∗ ω = 0 for every section γ of π. Differential forms ω σ = dy σ − yiσ dxi , ωiσ1 = dyiσ1 − yiσ1 j dxj ,

ωiσ1 ...is−1 = dyiσ1 ...is−1 − yiσ1 ...is−1 j dxj , dyiσ1 ...is ,

(3.1)

where 1 ≤ σ ≤ m, 1 ≤ i1 , . . . , is ≤ n, form the basis of 1-forms called the basis adapted to the contact structure. Note that these forms are contact with the exception of dyiσ1 ...is . ∗ A form ω on J s Y is called k-contact if every summand of the chart expression of πs+1,s ω contains the exterior product of exactly k 1-contact factors of the type (3.1). Let ω be a differential q-form on J s Y . Then there exists a unique decomposition ∗ πs+1,s ω = hω + pω =

q X

pq ω,

p0 ω = hω,

(3.2)

k=0

where h ω is the horizontal form called the horizontal or 0-contact component of ω and p ω is the contact form called the contact component of ω. The contact component p ω is uniquely decomposed into its k-contact components, 1 ≤ k ≤ q. Let (V, ψ) be a fibered chart on Y . We denote by Ωs0 V the ring of functions on Vs , and by Ωsq V a Ωs0 V -module of q-forms on Vs . Note that the sets Ωsq,X V of πs -horizontal q-forms and Ωsq,Y V of πs,0 -horizontal q-forms on Vs are submodules of Ωsq V . Now, let us introduce the concept of variational functional. Let W ⊂ Y be an open set. −1 A horizontal n-form Λ defined on Wr = πr,0 (W ) is called r-th order Lagrangian. It holds Λ = L(xi , y σ , yiσ1 , . . . , yiσ1 ...ir ) ω0 ,

ω0 = dx1 ∧ . . . ∧ dxn .

The pair (π, Λ) is so called Lagrange structure. Let U = π(W ). Let Ω ⊂ U be a compact n-dimensional submanifold of X with the boundary ∂Ω. Denote by ΓΩ,W a set of all sections γ ∈ ΓU such that supp γ = Ω and γ(Ω) ⊂ W . The mapping ΛΩ (γ) : ΓΩ,W ∋ γ −→ ΛΩ (γ) =

Z

J r γ∗Λ

(3.3)

Ω

is called the variational functional or variational integral over Ω. In physics it is often called the action function. For a π-projectable vector field ξ on Y the expression (∂J r ξ Λ)Ω (γ) =

Z

J r γ ∗ ∂J r ξ Λ

(3.4)

Ω

represents the variation of the variational functional, called its variational derivative or the first variation induces by the vector field ξ. A section γ ∈ ΓΩ,W is called the extremal of the variational functional (3.3) if it is its stationary point, i.e. the variational derivative of ΛΩ vanishes for γ. The aim of the calculus of variations is to find equations for extremals (in physics equations of motion of physical systems – particles and fields) and in general to study properties of variational functionals.


3.2.

11

Lepage Equivalents of Lagrangians

The definition of the Lepage equivalent of a Lagrangian as one of concepts of coherent theory on fibered manifolds was introduced by Krupka in [16]. It was based on primary ideas of Lepage and Dedecker and on some generalizations of Sniatycki [40]. In the above cited Krupka’s paper horizontal and contact differential forms on jet prolongations of fibered manifolds are introduced practically in the today’s form, being called horizontal and pseudovertical forms. Especially, for a differential p-form ̺ on J r Y a form h(̺) on J r+1 Y is defined (the notation of the time inclusive) by the relation hh(̺)(jxr+1 γ), ξ1 × · · · × ξp i

= h̺(jxr γ), Tx j r γ · T πr+1 · ξ1 × · · · × Tx j r γ · T πr+1 · ξp i, for r-jets of all sections γ of π at x, and every collection of vector arguments ξ1 , . . . , ξp on J r+1 Y . By this relation the form h(̺) is defined. A form with the property that the above value is zero whenever one of vector arguments is vertical, i.e. T πr+1 · ξi = 0, is ∗ called horizontal. Inspired by the relation j r+1 γ ∗ p(̺) = 0, where p(̺) = πr+1,r ̺ − h(̺) r r ∗ a pseudovertical p-form η on J Y , is defined by the property j γ η = 0 for any section γ of π. Then, the properties of the mapping h : ̺ → h(̺) for n-forms are studied. The definition of the horizontalization mapping is extended for (n + 1)-form by the relation ∗ h(i(ξ) πr+1,r ̺) = i(ξ) ̺

for every πr+1 -vertical vector field ξ on J r+1 Y , proving the existence and uniqueness of ̺. The symbol i(ξ) η denotes the contraction of the form η by the vector field ξ. Finally, a ˜ : ̺ → h(̺) ˜ mapping h = ̺ is defined. A Lepage n-form is then defined as follows: Let ̺ be an n-form on J r Y . ̺ is called ˜ Lepagian form, if the (n+1)-form h(d̺) is horizontal with respect to πr+1,0 . The properties of Lepagian forms (in today’s terminology Lepage forms) are studied and explicit formula ˜ for h(d̺) is derived for the second order r = 2 and for a n-form ̺ on J 1 Y . The relation of ˜ to Lepage congruencies [36] is mentioned and the connections of Lepage mappings h and h forms with equations of extremals of Lagrangians is studied, giving rise to Euler-Lagrange mapping by the formula ΩnX (J 1 Y ) ∋ λ → E(λ) ∈ Ω1Y (J 2 Y ),

˜ h(d̺) = E(h(̺)) ∧ ω,

where ω = F ω0 is the volume element on X and F > 0 is a function. Note that here E ∈ Ω1Y (J 2 Y ) is a uniquely defined pseudovertical 1-form. The Lepage equivalent of the Lagrangian λ is in fact introduced by an example, but it is not explicitly defined including the terminology. (Note that ΩqX (J s Y ) or Ω1Y (J s Y ) in the previous definition of the mapping λ → E(λ) means Ωsq,X V or Ωs1,Y V in the today’s notation.) Let us now present the contemporary definition of Lepage equivalents of Lagrangians, as presented in [26]. The concept of Lepage forms is introduced by the following lemma. Lemma. Let W ⊂ Y be an open set, and let ̺ ∈ Ωsn W . The following conditions are equivalent: (1) The (n + 1)-form p1 d̺ is πs+1,0 horizontal.

12

Jana Musilová and Michal Lenc −1 (2) For every πs,0 -vertical vector field ξ on W s = πs,0 (W ) it holds h iξ d̺ = 0, ∗ (3) The form πs+1,s ̺ has the chart expression ∗ πs+1,s ̺ = f0 ω0 +

s X

k=0

fσi,j1 ...jk ωjσ1 ...jk ∧ ωi + η,

ωi = i

∂ ∂xi

ω0 ,

∂f0 j ,j ...j − dp fσp,j1 ...jk − fσk 1 k−1 = 0, sym(j1 . . . jk ), 1 ≤ k ≤ s, ∂yjσ1 ...jk ∂f0 − fσjs+1 ,j1 ...js = 0, sym(j1 . . . js+1 ). ∂yjσ1 ...js+1 Any form satisfying the Lemma is called the Lepage form. The structure of Lepage forms is given by the following explicit formulas. Theorem. Let W ⊂ Y be an open set. A form ̺ ∈ Ωsn W is a Lepage form if and only ∗ if for every fibered chart (V, ψ), ψ = (xi , y σ ), on Y such that V ⊂ W , πs+1,s ̺ has an expression ∗ πs+1,s ̺ = Θ + dµ + η,

Θ = f0 ω0 +

s X

k=0

s−k X

∂f0

ℓ

(−1) dp1 . . . dpℓ

ℓ=0

where

∂yjσ1 ...jk p1 ...pℓ i

!

ωjσ1 ...jk ∧ ωi ,

where f0 is a function, defined in coordinates by h̺ = f0 ω0 , µ is a contact (n − 1)-form, and η is a form of contactness ≥ 2. The Lepage equivalent of a Lagrangian Λ = L ω0 is defined as such a Lepage form ̺ for which h̺ = Λ up to a possible projection. This leads to the conclusion that the function f0 in the preceding relation is equal to L, i.e. ΘΛ = L ω0 +

s X

k=0

s−k X

∂L

ℓ

(−1) dp1 . . . dpℓ

ℓ=0

∂yjσ1 ...jk p1 ...pℓ i

!

ωjσ1 ...jk ∧ ωi .

(3.5)

For mechanics (n=1) and Λ of order r we have r−1 X

dk ΘΛ = L dτ + (−1) dτ k k=0 k

∂L σ ∂qk+1

!

ωkσ .

(3.6)

The Euler-Lagrange mapping assigning to every Lagrangian Λ its Euler-Lagrange form defined by the standard way as a dynamical form σ

EΛ = Eσ (L) ω ∧ ω0 =

r X

∂L (−1) dj1 . . . djℓ σ ∂yj1 ...jℓ ℓ=0 ℓ

!

ω σ ∧ ω0

is closely related to the concept of Lepage equivalents. For every Lepage equivalent of a Lagrangian Λ it holds p1 d̺ = EΛ , and, moreover, the following theorem can be proved:


13

Theorem. Let Λ ∈ Ωrn,X W be a Lagrangian and let ̺ ∈ Ωsn W be its Lepage equivalent. Then the section γ ∈ ΓΩ,W is the extremal of the corresponding variational functional (3.3) if and only if for every π-vertical vector field ξ defined on W such that the support supp (ξ ◦ γ) ⊂ Ω it holds J s γ ∗ iJ s ξ d̺ = 0, or equivalently, the Euler-Lagrange form of Λ vanishes along J 2 γ. Another important result lies in the first variational formula resulting directly from properties of the Lepage equivalent ̺ of the Lagrangian Λ, the expression for the variational derivative (3.4) of the variational functional and the formula for Lie derivative ∂Ξ η = iΞ dη + diΞ η. Let γ be a section of π, ξ a π-projectable vector field on Y . Then it holds J r γ ∗ ∂J r ξ Λ = J s γ ∗ iJ s ξ d̺ + dJ s γ ∗ iJ s ξ ̺, (3.7) and using the Stokes theorem Z

J r γ ∗ ∂J r ξ Λ =

Ω

Z

J s γ ∗ iJ s ξ d̺ +

Ω

Z

J s γ ∗ iJ s ξ ̺,

(3.8)

∂Ω

The relations (3.7) and (3.8) give the infinitesimal first variational formula and the integral first variational formula, respectively. There are very important consequences resulting from the first variational formula for applications in variational physical theories. π-projectable vector field ξ on V is called the generator of invariance transformations of Lagrangian Λ ∈ Ωrn,X V , if ∂J r ξ Λ = 0. (3.9) For such a vector field the left hand side of the integral first variational formula (3.8) vanishes. Let γ be an extremal of the functional ΛΩ , Ω ⊂ π(V ), i.e. J s γ ∗ iJ s ξ d̺ = 0. Then the first variational formula gives Z

J s γ ∗ iJ s ξ ̺ = 0.

(3.10)

∂Ω

On the other hand, the integral (3.10) represents the flow of the quantity corresponding to the integrand through the boundary ∂Ω of Ω. It holds ∗ J s γ ∗ iJ s ξ ̺ = J s+1 γ ∗ πs+1,s iJ s ξ ̺ = h iJ s ξ ̺.

So, we can interpret the expression Ψ = h iJ s ξ ̺ ◦ J s γ

(3.11)

as a quantity obeying a conservation law along the extremal (so called elementary flow). At the end of this section let us note that a concept of Lepage forms introduced primarily for n-forms was later extended by Krupková (see e.g. [33]) for (n + 1)-forms. However, in this paper we don’t discuss the properties of Lepage (n + 1)-forms separately. On the other hand in Sec. 5 we mention the quite general definition of a Lepage form as resulting from the structure of the variational sequence.

14

4.


Application – Lepage Equivalents of Lagrangians for String Theories

Relativistic particles and strings are physical systems typically described by variational theories. So, their equations of motion are derived from conditions for stationary sections of a variational functional. These two situations are appropriate for application of the concept of Lepage equivalents of Lagrangians, representing the case of first order mechanics and first order field theory, respectively. There are of course physical arguments for the choice of variational integral (for strings see [5]). Apart from these arguments, the obtained results enables us to formulate these variational functionals as variational integrals for a certain Lagrange structure of an appropriately chosen fibered manifold. In this section we give corresponding Lagrange structures (π, Λ) and calculate Lepage equivalents ̺. Moreover, using the infinitesimal first variational formula we can study invariance transformations of Lagrangians. Let the underlying fibered manifold be again (Y, π, X). We denote fibered charts on Y , associated charts on X and associated fibered chart on J 1 Y by the standard way (V, ψ), (U, ϕ), (V1 , ψ1 ). We will use the standard relativistic notation. Recall that considering the infinitesimal first variational formula in the form (3.7) we can see that quantities obeying conservation laws along the prolongations of extremals of the Lagrangian are (n − 1)-forms given by (3.11), where ξ is a invariance transformation of Λ. From the physical point of view it is suitable to take into account the “full” Lepage equivalent, i.e. including a trivial Lagrangian of the same order as the Lagrangian given by the physical situation itself. The last assumption is induced by the standard form of boundary conditions of the variational problem. As a good example we can mention the Lagrangian L = (1/2)mx˙ 2 +U (x) of a non-relativistic particle with one degree of freedom (the motion along x-axis). For non-constant potential energy U (x) only the time translation is the invariance transformation of the Lagrangian. However, adding an appropriately chosen trivial Lagrangian we can show that for linear potential energy U (x) = Kx + C (a homogeneous field) the translation along x is invariance transformation as well.

4.1.

Relativistic Particle

In the case of relativistic mechanics the base X of the underlying manifold is one-dimensional (n=1), the unique coordinate being a “non-physical parameter”. Every fiber is a time-space with metric g = gαβ dxα ⊗ dxβ . Thus the fiber dimension is m = 4. Thus, we have dim Y = 5. So on (Y, π, X) we have coordinates ϕ = (τ ),

ψ = (τ, xα ),

ψ1 = (τ, xα , x˙ α ),

0 ≤ α ≤ 3.

The Lagrange structure (π, Λ) with “classical” Lagrangian is Λ1 = L1 dτ,

q

L1 = −mc gαβ x˙ α x˙ β .

The Lagrange structure appropriate for both the zero and nonzero mass is given by the Lagrangian ! 1 gαβ x˙ α x˙ β 2 2 + λ(τ )m c , Λ2 = L2 dτ, L2 = − 2 λ(τ )


15

where λ(τ ) is a function of the parameter. Both Lagrangians are reparametrizable. The unique Lepage equivalent of this Lagrangian given by the general expression (3.6) is of the form ∂L ∂L ∂L ̺ = L dτ + ν ω ν = L − ν x˙ ν dτ + ν dxν . ∂ x˙ ∂ x˙ ∂ x˙ By the simple calculation we finally obtain gµν x˙ ν

̺1 = −mc q

gαβ x˙ α x˙ β

1 ̺2 = 2

dxµ ,

(4.1)

!

gαβ x˙ α x˙ β gαβ x˙ β α − λ(τ )m2 c2 dτ − dx . λ(τ ) λ(τ )

(4.2)

Now, let us study conservation laws resulting from invariance transformations of “full” Lagrangians (Λ + Λ0 ) for both cases, Λ1 and Λ2 , Λ0 being a trivial Lagrangian. For our situation it holds r = s = 1, and thus J 1ξ = ξ

dξ µ dξ ξ˜µ = − x˙ µ . dτ dτ

∂ ∂ ∂ + ξ µ µ + ξ˜µ µ , ∂τ ∂x ∂ x˙

Every trivial Lagrangian of the first order has the form h df for a function f = f (τ, xµ ). Using Lepage equivalents (4.1) and (4.2) we obtain for “full” Lagrangians the resulting general expressions for quantities given by (3.11) Ψ1 = and 1 Ψ2 = 2

4.2.





x˙ ν

∂f ∂f mc gµν ξ − q − µ  ξµ, ∂τ ∂x α β gαβ x˙ x˙

gαβ x˙ α x˙ β ∂f − λ(τ )m2 c2 + λ(τ ) ∂τ

!

ξ−

gαβ x˙ β ∂f − α λ(τ ) ∂x

(4.3)

!

ξα.

(4.4)

Classical Bosonic String

Strings can be studied within the field theory. The base of the underlying manifold is in case of strings two dimensional, i.e. n = 2. The base coordinates denoted by τ 0 and τ 1 have the meaning of the time evolution parameter and the position on the string, respectively. Every fiber is again time-space, i.e. m = 4. Metric is denoted again by g = gαβ dxα ⊗ dxβ . For such a structure Lagrangian has a form Λ = L(ξ i , xµ , xµi ) dτ 0 ∧ dτ 1 . The principal component of Lepage equivalent is, following (3.5) ΘΛ = L dτ 0 ∧ dτ 1 +

ΘΛ = L −

∂L µ ω ∧ i ∂ (dτ 0 ∧ dτ 1 ), ∂xµi ∂ξi

∂L µ ∂L µ ∂L ∂L x − x dτ 0 ∧ dτ 1 + µ dxµ ∧ dτ 1 − µ dxµ ∧ dτ 0 . (4.5) ∂xµ0 0 ∂xµ1 1 ∂x0 ∂x1

16

Jana Musilová and Michal Lenc For an example we consider a bosonic string. Let us denote h = hij dτ i ⊗ dτ j = gαβ xαi xβj dτ i ⊗ dτ j . The Lagrangian of a string is defined as √ Λ = L dτ 0 ∧ dτ 1 = −T −det h dτ 0 ∧ dτ 1 , L = −T

q

(gαβ xα0 xβ1 )2 − (gαλ xα0 xλ0 )(gβν xβ1 xν1 )

= −T

q

(gαβ gλν − gαλ gβν ) xα0 xλ0 xβ1 xν1 .

(4.6)

Note that in the original derivation of the variational functional of the string the NambuGoto idea formulated e.g. in [8] is based on the variational integral (action) S = −T

Z

dΣ,

Σ

where dΣ is understood as the “elementary surface area”. This means that dS is, correctly speaking, the (unique) volume element of the relativistic time-space. Thus, in this approach h can be understood as the induced metric for a mapping γ : (τ 0 , τ 1 ) −→ (x0 (τ 0 , τ 1 ), x1 (τ 0 , τ 1 ), x2 (τ 0 , τ 1 ), x3 (τ 0 , τ 1 )), h = γ ∗ g = (gαβ ◦ γ)

∂xα ∂xβ i dτ ⊗ dτ j . ∂ξ i ∂ξ j

However, in the approach based on Lagrange structures formalism on fibered manifolds and their jet prolongations, the object h is defined on first jet prolongation of (Y, π, X). Using (4.5) for Lagrangian (4.6), we obtain after some simple calculations the principal component of Lepage equivalent √ ΘΛ = T −det h dτ 0 ∧ dτ 1 T +√ (gαµ gβν − gαβ gµν ) dxµ ∧ (xα0 xβ0 xν1 dτ 0 − xα1 xβ1 xν0 dτ 1 ). (4.7) −det h

Now, let us study Lepage equivalents of trivial Lagrangians for the considered physical situation. Taking into account that every trivial Lagrangian of first order has the form h dχ (see [27]), where χ is a (n − 1)-form on J r−1 Y , we can obtain the corresponding principal component of Lepage equivalents. In coordinates

Λ0 = h dχ = di χj + xµj di Xµ dτ i ∧ dτ j , where

(4.8)

χ = χi dτ i + Xµ dxµ . Thus, we have Λ0 = L0 ω0 , where ∂χ1 ∂Xν ∂χ1 ∂χ0 + xν0 − − 0 1 ∂τ ∂τ ∂xν ∂τ 1 ∂χ0 ∂Xν ∂Xν ν µ −xν1 − + (x x − xν0 xµ1 ) . ∂xν ∂τ 0 ∂xµ 1 0 L0 =

(4.9)


17

The corresponding Lepage equivalent has the form ̺0 = Θ0 + η + dµ, where η is a 2-contact 2-form and µ is a contact 1-form. For conservation laws only the principal component Θ0 is important, because h dµ has the character of a special type of trivial Lagrangian and J s γ ∗ η = 0. Θ0 =

∂χ1 − ∂τ 0 ∂χ1 + ∂xµ ∂χ0 + ∂xµ

∂χ0 ∂Xµ µ ν + (x x − xµ1 xν0 ) dτ 0 ∧ dτ 1 ∂τ 1 ∂xν 0 1 ∂Xµ ∂Xµ ν ∂Xν − + x1 − dxµ ∧ dτ 1 ∂τ 1 ∂xµ ∂xν ∂Xµ ∂Xµ ν ∂Xν − + x0 − dxµ ∧ dτ 0 . ∂τ 0 ∂xµ ∂xν

(4.10)

Let

∂ ∂ ∂ + ξ 1 1 + Ξµ µ 0 ∂τ ∂τ ∂x be an invariance transformation of the Lagrangian (Λ + Λ0 ), where Λ0 is a trivial Lagrangian. We obtain the corresponding elementary flow ξ = ξ0

Ψ = ΨΛ + Ψ0 = h iJ 1 ξ ΘΛ + h iJ 1 ξ Θ0 ,

(4.11)

where √ T ΨΛ = −T −det h ξ 1 + √ (gαµ gβν − gαβ gµν ) · −det h

h

io

· xα0 xβ0 xν1 Ξµ − xµ0 xα0 xβ0 xν1 ξ 0 − xα1 xβ1 xν0 ξ 1 dτ 0 √ T + T −det h ξ 0 + √ (gαµ gβν − gαβ gµν ) · −det h h

· −xα1 xβ1 xν0 Ξµ − xµ1 xα0 xβ0 xν1 ξ 0 − xα1 xβ1 xν0 ξ 1 and

i

dτ 1

(4.12)

(4.13)

Ψ0 = A00 ξ 0 + A01 ξ 1 + B0µ Ξµ dτ 0 + A10 ξ 0 + A11 ξ 1 + B1µ Ξµ dτ 1 , where

A00 = A01 = B0µ = A10 = A11 = B1µ =

∂Xµ ∂χ0 ∂Xµ ∂Xν − µ + xµ0 xν0 − , ∂τ 0 ∂x ∂xν ∂xµ ∂χ0 ∂χ1 ∂Xµ ∂χ1 − + xµ0 − µ , ∂τ 1 ∂τ 0 ∂τ 1 ∂x ∂χ0 ∂Xµ ∂Xµ ν ∂Xν − + x0 − , ∂xµ ∂τ 0 ∂xµ ∂xν ∂χ1 ∂χ0 ∂Xµ ∂χ0 − + xµ1 − µ , 0 1 ∂τ ∂τ ∂τ 0 ∂x ∂χ1 ∂Xν µ ∂Xµ µ ν ∂Xµ x1 − µ + x1 x1 − , ∂τ 1 ∂x ∂xν ∂xµ ∂χ1 ∂Xµ ∂Xν ∂Xµ − + xν1 − . µ 1 µ ∂x ∂τ ∂x ∂xν

xµ0

18


The invariance transformations ξ of the Lagrangian (Λ + Λ0 ) given by (4.6) and (4.9) can be obtained from (3.9) in principle. Conservation laws can be then studied using (4.11).

4.3.

Polyakov Action for D-Dimensional Space

The Polyakov action is a variational integral arising from generalization of string Lagrangian to a D-dimensional space by the following way: Λ = L(τ i , xµ , xµi ) dτ 0 ∧ dτ 1 ,

0 ≤ i, j ≤ 1,

0 ≤ µ ≤ D − 1,

T p −det f f ij gµν xµi xνj . (4.14) 2 Here f is the metric on the base, without a relation to g. Using again relation (4.5) we can calculate the principal component of Lepage equivalent of the Lagrangian (4.14) in the following form. L=−

T p −det f f ij gαβ xαi xβj dτ 0 ∧ dτ 1 2p + T −det f gαβ xαi dxβ ∧ (f 1i dτ 0 − f 0i dτ 1 ).

ΘΛ =

(4.15)

Note that the quantities Ψ for conservation laws can be calculated for the Polyakov action by the quite analogous procedure as for a bosonic string.

5.

Lepage Forms in Variational Sequences

In previous sections a concept of a Lepage equivalent of a Lagrangian was discussed and its usefulness for application to concrete situations occurring in variational theories in physics was demonstrated. In this section we present the generalized concept of a Lepage form in the context of the variational sequence representation (for the complete solution of the representation problem in field theories see [14] and [15], for mechanics see [29]). We follow the general ideas and structures employed by Krupka in [23]. The main results are formulated in the language of the theory of sheaves. (We use sheaves of germs of C ∞ differential q-forms.)

5.1.

Variational Sequence

Let Ωrq , q ≥ 0, be the direct image of the sheaf of smooth q-forms over J r Y by the jet projection πr,0 (functions are labeled as 0-forms). Denote by Ωrq,c

=

(

ker p0 ker pq−n

for 1 ≤ q ≤ n, for n + 1 ≤ q ≤ dim J r Y,

(5.1)

where p0 and pq−n are morphisms of sheaves induced by mappings p0 and pq−n , assigning to a form ̺ its horizontal and (q − n)-contact component, respectively. The dimension of n+r r J Y is N = m n + n. We further denote Θrq = Ωrq,c + d Ωrq−1,c ,

(5.2)


19

d Ωrq−1,c is the image sheaf of Ωrq−1,c by d. Let us consider the sequence {0} → Θr1 → · · · → Θrn → Θrn+1 → Θrn+2 → · · · → ΘrP → {0},

(5.3)

with P = m n+r−1 + 2n − 1 being the maximal nontrivial degree of forms of the type n (5.2). The arrows (except the first one) are given by exterior derivatives d. It can proved that the sequence (5.2) is an exact subsequence of the de Rham sequence of forms

{0} → Ωr1 → · · · → Ωrn → Ωrn+1 → Ωrn+2 → · · · → ΩrN → {0}. The resulting quotient sequence is called the variational sequence of order r. The situation is shown in the following figure. The variational sequence is, of course, also exact. We denote the quotient mappings as follows Eqr : Ωrq /Θrq ∋ [̺] → Eqr ([̺]) = [d ̺] ∈ Ωrq+1 /Θrq+1 .

(5.4)

r The mappings Enr and En+1 correspond to the Euler-Lagrange mapping and HelmholtzSonin mapping of calculus of variations, respectively. The global properties of the variational sequence are described using the abstract de Rham theorem and can be found in [23]:

(1) Each sheaf Ωrq is soft. (2) The variational sequence (concisely {0} → RY → V) is an acyclic resolution of the constant sheaf RY over Y . (3) For every q ≥ 0 it holds H q (Γ(RY , V)) = H q (Y, R), where Γ(Y, V) : {0} → Γ(Y, RY ) → Γ(Y, Ωr0 ) → · · · → Γ(Y, ΩrN ) →{0} is the cochain complex of global sections and H q (Γ(RY , V)) denotes its q-th cohomology group.

5.2.

Representation of the Variational Sequence

In this subsection we present the representation of the r-th order variational sequence by forms. All details and proofs of theorems can be found in [15]. By the representation we mean a mapping assigning to every class [̺] of the variational sequence of the r-th order an appropriately chosen representative with specific properties mentioned in Sec. 1: Rlinearity, coordinate invariance, exactness of there sequence of representatives, projection property of the operator assigning to a class its representative. Such a mapping is given by Euler operator. Its construction was inspired by the idea of Anderson in [1], [2] for the variational bicomplex. For finite order variational sequences the construction uses the concept of the Lie derivative of differential forms with respect of vector fields along maps, and is based on the generalized integration by parts procedure, both introduced in [15]. Let us give here main results only. Integration by parts is described by the following lemma. (Note that J = (j1 . . . jℓ ), 0 ≤ ℓ ≤ r, is a multiindex and |J| = ℓ is its length.)

{0}

{0}

{0}

R

Ωr0

d0

{0}

{0}

Θr1

d1

Θr2

d2

Ωr1

d1

Ωr2

d2

···

···

{0}

dq−1

Θrq

dq

dq−1

Ωrq

dq

···

···

dP −1

ΘrP

dP

dP −1

ΩrP

dP

E0

{0}

E1

Ωr2 /Θr2

{0}

E2

···

Ωr /Θrq Eq−1 q Eq

{0}

···

Ωr /ΘrP EP −1 P

{0}

ΩrP +1

dP +1

EP +1

EP

Ωr1 /Θr1

{0}

···

dN −1 EN −1

ΩrN

dN EN

{0}


21

Lemma. Let (V, ψ), ψ = (xi , y σ ) be a fibered chart on Y and ̺ ∈ Ωrn+k V a form. Let pk ̺ be expressed as r X

pk ̺ =

|J|=0

ωJσ ∧ ησJ .

Then there exists the decomposition pk ̺ = I(̺) + pk d pk R(̺) where σ

σ

I(̺) = ω ∧ ζσ = ω ∧

r X

(5.5)

(−1)|J| dJ ησJ .

(5.6)

|J|=0

and R(̺) is a local k-contact (n + k − 1)-form. Moreover, we can write the operator I in the form I:

Ωrn+k W

∋ ̺ → I(̺) =

1 σ kω

∧

r X

|J|

(−1)

|J|=0

dJ

∂ y pk ̺ ∂yJσ

!

∈ Ω2r+1 n+k W,

where we used the obvious identity pk ̺ =

1 k

r X

ωJσ

|J|=0

∧

∂ y pk ̺ ∂yJσ

!

We can see that I is clearly R-linear. (Note that the operator “y” stands for the contraction, i.e. ξ y η = iξ η means the contraction of a form η by the vector field ξ.) It can be proved that the decomposition 5.5 is valid globally (the proof see in [15]) and following the terminology in [2] we call the operator I interior Euler operator. The following Theorem summarizes all properties of the interior Euler operator important for the representation of the variational sequence. Theorem. Let W ⊂ Y be an open set and let ̺ ∈ Ωrn+k W , 1 ≤ k ≤ N − n, be a form. Then (1) (π2r+1,r )∗ ̺ − I(̺) ∈ Θ2r+1 n+k W,

(2) (3) (4)

I(pk d pk R(̺)) = 0, 2

∗

I (̺) = (π4r+3,2r+1 ) I(̺), ker I =

Θrn+k W.

(5.7) (5.8) (5.9) (5.10)

Now let us define a family of mappings Rq : Rq : Ωrq W/Θrq W ∋ [̺] → Rq ([̺]) ∈ Ωsq W, as follows Rq ([̺]) =

   p0 ̺

I(̺)

  ̺

for 0 ≤ q ≤ n, s = r + 1 for n + 1 ≤ q ≤ P, s = 2r + 1 for P + 1 ≤ q ≤ N, s = r

(5.11)

22


Evidently, in all three cases the mapping Rq assigns to every class [̺] of q-forms on J r Y a correctly defined representative Rq ([̺]). In this sense, every class [̺] is represented by a form Rq ([̺]). Consider a sequence {0} → R0 (Ωr0 W ) → R1 (Ωr1 W/Θr1 W ) → · · · → RP (ΩrP W/ΘrP W ) →

→ RP +1 (ΩrP +1 W ) → · · · → RN (ΩrN W ), (5.12)

where arrows denote the mappings E q : Rq (Ωrq W/Θrq W ) → Rq+1 (Ωrq+1 W/Θrq+1 W )

(5.13)

induced by the commutativity of diagrams Ωrq W/Θrq W

Eqr

(5.14)

Rq+1

Rq

Rq (Ωrq W/Θrq W )

Ωrq+1 W/Θrq+1 W

Eq

Rq+1 (Ωrq+1 W/Θrq+1 W )

i.e. E q ◦Rq ([̺]) = Rq+1 ◦ Eqr ([̺]) = Rq+1 ([d ̺]). The sequence (5.12) is called the representation sequence of the r-th order variational sequence. The representation sequence is exact, as it is proved in [15]. There are practical applications od the representation of the variational sequence. Especially, the representatives Rq ([̺]) of classes of q-forms [̺] for q = n − 1, n, n + 1, n + 2 are the well-known forms on calculus of variations, i.e. r-th order currents, r-th order Lagrangians, r-th order dynamical forms and r-th order Helmholtz-Sonin forms. So, the exactness of the representation sequence (5.12) enables us e.g. to solve the problem of trivial Lagrangians and the inverse problem of calculus of variations.

5.3.

Generalized Lepage Forms

One may generalize the concept of Lepage forms to all (n + k)-forms, k ≥ 0. Such a generalization is directly induced by the structure of the representation of the variational sequence. An (n + k)-form ̺ on J r Y is called Lepage if pk+1 d ̺ = I(d ̺),

i.e.

pk+1 d ̺ = I(pk+1 d ̺).

(5.15)

The equivalent definition resulting from the decomposition (5.5) can be given in the form pk+1 d pk+1 R(d ̺) = pk+1 d R(pk+1 d ̺) = 0. A Lepage form ̺ is also called the Lepage equivalent of the class [̺]. The definition of Lepage forms as well as the general definition of Lepage equivalents was introduced also in [29] for mechanics. There the properties of Lepage forms for mechanics are studied in details and useful examples of chart expressions are presented. Note that in the terminology of [29] the Lepage equivalent of the class [̺] is called the Lepage equivalent of the source form. This is a form η ∈ [̺] such that pk η = I(η), i.e. the representative Rn+k [̺] (see (5.11)) in our notation.


6.

23

Conclusion

The aim of the paper was to show the brief history of the concept of Lepage forms from the initiating Lepage’s idea up to the most general definition resulting by the quite natural way from geometrical structures built on underlying fibered manifolds and their jet prolongations. The development of this concept supports the unquestionable meaning of Lepage forms for the calculus of variations. Moreover, numerous examples following from variational physical theories, as e.g. classical Newtonian and relativistic mechanics, theory of fields, string theories, show the usefulness of Lepage forms, especially Lepage equivalents of Lagrangians, for many physical applications, as e.g. studies of conservation laws, etc. It is evident that the relation (5.15) represents the most general possibility to define the concept of a Lepage form. It ought to be mentioned that this definition inspired primarily by Lepage is undoubtedly based on Krupka’s complex theoretical considerations on Lagrange structures and Lepage equivalents of Lagrangians and on his idea and studies of the finiteorder variational sequence.

Acknowledgements The research is supported by the grant 201/06/0922 of the Czech Grant Agency and by the grant MSM 0021622409 of the Ministry of Education, Youth and Sports of the Czech Republic.

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24


[8] T. Goto, Prog. Theor. Phys. 46 (1971) 1560; Derived also in unpublished lectures of Y. Nambu presented in Copenhagen, 1970. [9] D. R. Grigore, The variational sequence on finite order bundle extensions and the Lagrange formalism, Diff. Geom. Appl. 10 (1999) 43–77. [10] D. R. Grigore, Variationally trivial Lagrangians and locally variational equations of arbitrary order, Diff. Geom. Appl. 10 (1999) 79–105. [11] I. S. Krasilschik, V. V. Lychagin and A. M. Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations (Advanced Studies in Contemporary Mathematics 1, Gordon & Breach, New York, 1986). [12] M. Krbek, J. Musilová and J. Kaˇsparová, The variational sequence: Local and global properties, In: Proceedings of the Seminar on Differential Geometry (Math. Publications 2, Silesian University, Opava, 2000) 15–38. [13] M. Krbek, J. Musilová and J. Kaˇsparová, The representation of the variational sequence in field theories, In: Steps in Differential Geometry (Proc. Colloq. Differential Geometry, Debrecen, Hungary, 2000, (L. Kozma, P. T. Nagy and L. Tamássy, Eds.) Debrecen, 2001) 147–160. [14] M. Krbek and J. Musilová, Representation of the variational sequence, Rep. Math. Phys. 51 (2003) 251–258. [15] M. Krbek and J. Musilová, Representation of the Variational Sequence by Differential Forms, Acta Appl. Math. 88 (2005) 177–199. [16] D. Krupka, Some geometric aspects of variational problems in fibred manifolds, Folia Fac. Sci. Nat. Univ. Purk. Brunensis XIV (10) (1973) pp. 65. [17] D. Krupka, On the structure of Euler-Lagrange mapping, Arch. Math. (Brno) 10 (1974) 353–358. [18] D. Krupka, A map associated to the Lepagean forms of the calculus of variations in fibered manifolds, Czechoslovak Math. J. 27 (1977) 114–118. [19] D. Krupka, Natural Lagrangian Structures, In: Differential geometry (Semester in Differential Geometry, Banach Center, Warsaw, 1979, Banach Center Publications 12, 1984) 185–210. [20] D. Krupka, On the local structure of Euler-Lagrange mapping of the calculus of variations, In: Differential Geometry and its Applications (Proc. Conf., Nové Mˇesto na Moravˇe, Czechoslovakia, 1980, (O. Kowalski, Ed.) Charles University, Praha, 1981) 181–188. [21] D. Krupka, Regular lagrangians and lepagean forms, In: Differential Geometry and ˇ its Applications (Proc. Conf., Brno, Czechoslovakia, 1986, (D. Krupka and A. Svec, Eds.) D. Reidel, Dordrecht, 1986) 85–101.


25

[22] D. Krupka, Geometry of Lagrangean structures 2, 3, In: 14th Winter School on Abstract Analysis (Proc. Conf., Srn´ı, Czechoslovakia, 1986, Suppl. Rend. del Circ. Mat. di Palermo 14, 1987) 178–224. [23] D. Krupka, Variational sequences on finite order jet spaces, In: Differential Geometry and its Applications (Proc. Conf., Brno, Czechoslovakia, 1989, (J. Janyˇska and D. Krupka, Eds.) World Scientific, Singapore, 1990) 236–254. [24] D. Krupka, The Geometry of Lagrange Structures, Preprint Series in Global Analysis GA 7 (1997), Mathematical Institute, Silesian University, Opava 1987, pp. 82. [25] D. Krupka, Variational sequences and bicomplexes, In: Differential Geometry and its Applications (Proc. Conf., Brno, Czech Republic, 1998, (O. Kowalski, D. Krupka and J. Slovák, Eds.) Masaryk University, Brno, 1998) 525–532. [26] D. Krupka, Global variational theory in fibred spaces, In: Handbook of Global Analysis ((D. Krupka and D. Saunders, Eds.) Elsevier, Amsterdam, 2008) 773–836. [27] D. Krupka and J. Musilová, Trivial Lagrangians in Field Theory, Diff. Geom. and Appl. 9 (1998) 393–505. [28] D. Krupka and J. Musilová, Recent results in variational sequence theory, In: Steps in Differential Geometry (Proc. Colloq. Differential Geometry, Debrecen, Hungary, 2000, (L. Kozma, P. T. Nagy and L. Tamássy, Eds.) Debrecen, 2001) 161–186. ˇ enková, Variational sequences and Lepage forms, In: Differential [29] D. Krupka and J. Sedˇ Geometry and its Applications (Proc. Conf., Prague, Czech Republic, 2004, (J. Bureˇs, O. Kowalski, D. Krupka and J. Slovák, Eds.) Charles University, Prague, 2005) 617– 627. [30] O. Krupková, Lepagean 2-forms in higher order Hamiltonian mechanics. I. Regularity, Arch. Math. (Brno) 22 (1986) 97–120. [31] O. Krupková, Lepagean 2-forms in higher order Hamiltonian mechanics. II. Inverse problem, Arch. Math. (Brno) 23 (1987) 155–170. [32] O. Krupková, Variational analysis on fibered manifolds over one-dimensional bases, PhD Thesis, Silesin University, Opava and Charles University, Prague, 1992, pp. 67. [33] O. Krupková, The Geometry of Ordinary Variational Equations (Lecture Notes in Mathematics, 1678, Springer, Berlin, 1997). [34] O. Krupková, Mechanical systems with non-holonomic constraints, J. Math. Phys. 38 (1997) 5098–5126. [35] Th. H. J. Lepage, Calcul des Variations. Sur certaines formes différentielles extérieures et la variation des intégrales doubles, Comptes rendus des séances de l’Académie des sciences, Tome cent-quatre-vingt-dix-septieme (1933) 1718–1719. [36] Th. H. J. Lepage, Sur les champs géodesiques du Calcul des Variations I, II, Bull. Acad. Roy. Belg., Cl. Sci. 22 (1936) 716–729, 1036–1046.

26


[37] J. Musilová, Variational sequence in higher order mechanics, In: Differential Geometry and its Applications (Proc. Conf., Brno, Czech Republic, 1995, (J. Janyˇska, I. Koláˇr and J. Slovák, Eds.) Masaryk Unversity, Brno, 1996) 611–624. [38] J. Musilová and M. Krbek, A note to the representation of the variational sequence in mechanics, In: Differential Geometry and its Applications (Proc. Conf., Brno, Czech Republic, 1998, (I. Koláˇr, D. Krupka, O. Kowalski and J. Slovák, Eds.) Brno, 1999) 511–623. ˇ enková, On the invariant variational sequences in mechanics, In: Rend. Cont. [39] J. Sedˇ Mat. Palermo (Proc. 22-nd Winter School Geometry nad Physics, Srn´ı, Czech Republic, 2002) 185–190. [40] J. Sniatycki, On the geometric structure of classical field theory in Lagrangian formulation, Proc. Camb. Phil. Soc. 68 (1970) 475–484. ˇ anek, A representation of the variational sequence in higher order mechanics, [41] J. Stef´ In: Differential Geometry and its Applications (Proc. Conf., Brno, Czech Republic, 1995, (J. Janyˇska, I. Koláˇr and J. Slovák, Eds.) Masaryk University, Brno, 1996) 469– 478. [42] F. Takens, A global version of the inverse problem of calculus of variations, J. Diff. Geom. 14 (1979) 543–562. [43] W. M. Tulczyjew, The Euler-Lagrange resolution, In: Differential Geometric Methods in Mathematical Physics (Proc. Internat. Coll., Aix-en-Provence, France, 1979, Lecture Notes in Math. 836, Springer, Berlin, 1980) 22–48. [44] A. M. Vinogradov, On the algebro-geometric foundations of Lagrangian field theory, Soviet. Math. Dokl. 18 (1977) 1200–1204. [45] A. M. Vinogradov, A spectral sequence associated with a non-linear differential equation, and algebro-geometric foundations of Lagrangian field theory with constraints, Soviet. Math. Dokl. 18 (1978) 144–148. [46] A. M. Vinogradov, The C-spectral sequence, Lagrangian formalism and conservation laws I and II, J. Math. Anal. Appl. 100 (1) (1984) 1–129. [47] R. Vitolo, Finite order Lagrangian bicomplexes, Math. Proc. Cambridge Philos. Soc. 125 (1999) 321–333. [48] R. Vitolo, On different geometric formulations of Lagrangian formalism, Diff. Geom. Appl. 10 (3) (1999) 225–255. [49] R. Vitolo, Finite order formulation of Vinogradov’s C-spectral sequence, Acta Appl. Math. 70 (1-2) (2002) 133–154. [50] R. Vitolo, Variational sequences, In: Handbook of Global Analysis ((D. Krupka and D. Saunders, Eds.) Elsevier, Amsterdam, 2008) 1115–1163.

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupková and D. Saunders, pp. 27-55

Chapter 2

L EPAGE F ORMS IN THE C ALCULUS OF VARIATIONS∗ Olga Krupková† Department of Algebra and Geometry, Faculty of Science, Palacký University, Tomkova 40, 779 00 Olomouc, Czech Republic and Department of Mathematics, La Trobe University, Bundoora, Victoria 3086, Australia

Abstract Lepage forms represent a fundamental concept in the global calculus variations. Inspired by the work of Lepage, they were introduced by Demeter Krupka in 1973 in his seminal paper Some Geometric Aspects of Variational Problems in Fibered Manifolds, published in the journal of Brno University Folia Fac. Sci. Nat. Univ. Purk. Brunensis (see also arXiv:math-ph/0110005 for an electronic transcription). In this paper we recall Lepage forms, their recent generalisations, the role they play in the current global variational analysis, and their applications in the geometric theory of differential equations, the theory of variational sequences, and higher-order mechanics and field theory.

2000 Mathematics Subject Classification. 58-02, 70-02, 58E15, 70H50. Key words and phrases. Lagrangian, Lepage form, the first variation formula, Euler– Lagrange form, Noether theorem, conservation law, Hamilton equations, regular Lagrangian, the variational sequence, null-Lagrangian, the inverse problem of the calculus of variations, Helmholtz form.

1.

Introduction

This paper is a survey of the theory of Lepage forms, its current development and numerous applications in the calculus of variations on fibred manifolds. ∗ †

Paper dedicated to Demeter Krupka on the occasion of his 65th birthday E-mail address: [email protected]

28

Olga Krupková

The concept of Lepage form (more precisely, Lepage n-form where n is the dimension of the base manifold) was introduced in the early 1970’s by D. Krupka, inspired by a classical paper of Th. Lepage [61]. D. Krupka was among the first who realised that fibred and prolongation structures introduced by Ehresmann [14] represent an appropriate background for the investigation of first and higher order variational functionals on manifolds (cf. [7, 8, 17, 22, 36, 77]). In his seminal paper [36] published 35 years ago he introduced the fundamental concept of the Lepage equivalent of a Lagrangian, and since that time he has systematically developed techniques and concepts of a global higher order variational theory on fibred spaces where Lepage forms and variational sequences play a central role. I am grateful to Demeter that from the beginning of my university education I had the possibility of joining his seminar, to learn his theory of Lepage forms in jet bundles, and to participate in the fascinating process of discovering geometric structures in the calculus of variations.

Figure 1. Demeter Krupka in the early 1980’s lecturing on Lepage forms. This paper, reviewing the current status of the theory of Lepage forms, starts by briefly recalling the calculus of vector fields and horizontal and contact forms on jet bundles over fibred manifolds [36, 37, 41], including the fundamental Krupka’s decomposition formula of a differential form into contact components, concepts and techniques that are essential for understanding intrinsic constructions and properties of geometric objects appearing in “fibred” variational analysis. Based on papers [36, 37, 38, 39, 41, 42, 43, 45, 49], the next section is devoted to Lepage n-forms and their role in the theory of global variational functionals. Lepage n-forms are fundamental for obtaining the intrinsic first variation formula, that gives various geometric formulations of the Euler–Lagrange equations on one hand, and, combined with the concepts of invariant variational functionals, of Noether theorems on the other hand. Moreover, it is demonstrated that this setting provides a new look at Hamiltonian theory and regularity of variational problems, that has not yet been completely explored. The reader should be aware that the family of Lepage equivalents of a Lagrangian contains the well-known Cartan form [5, 78], and the Poincaré–Cartan form [11, 17, 22, 36] and its higher order generalisations [11, 15, 18, 31, 33, 41], that are very popular in vari-

Lepage Forms in the Calculus of Variations

29

ous geometric formulations of the calculus of variations (see e.g. [22, 12, 16, 18, 33, 34], or the quite recent multisymplectic formalism [13, 21, 24, 62] and others). It should be stressed, however, that “Cartan-like” forms, though often preferred, are not the only possibility; in some situations they are too restrictive (exterior differential systems for variational equations, Hamilton theory, regularity), or even apparently unavailable (higher order field theories, homogeneous Lagrangians, non-fibred structures), so that some other Lepage equivalents of a Lagrangian need to be considered (we refer to [4, 39, 68] for constructions and [8, 36, 41, 54, 58, 59, 65, 71] for discussions and applications). The last section of the present paper is concerned with generalisations of Lepage forms. Roughly speaking, a motivation for introducing Lepage forms of higher degrees is to “transfer” variational operators to the exterior derivative [8, 40]; as a first step, to introduce Lepage equivalents of Euler–Lagrange forms [50] in such a way that the following diagram should be commutative: Lagrangian   y

Euler–Lagrange mapping

−−−−−−−−−−−−−→ Euler–Lagrange form d

  y

Lepage n-form −−−−−−−−−−−−−−−→ Lepage (n + 1)-form

The idea comes from the solution of the inverse problem of the calculus of variations and of the problem of the structure of null Lagrangians: with help of Lepage n-forms these problems are transferred to the application of the Poincaré Lemma [40, 41]. Lepage equivalents of Euler–Lagrange forms introduced in [50] and further studied in [25, 28, 29, 53, 54], play a fundamental role in investigating variational equations by means of exterior differential systems methods, enabling the study of the geometry of these equations and structure of their solutions, symmetries and conservation laws, and exact methods of integration (see e.g. [25, 51, 52, 55, 56, 57]). In particular, a new approach to Hamiltonian theory and regular variational problems is achieved, associating Hamilton equations directly to the Euler–Lagrange form (not to individual Lagrangians) [50, 51, 53, 54]. Finally, it is shown how the concept of Lepage form can be generalised to arbitrary k-forms (k ≥ n), and how it fits in with Krupka’s variational sequence [35, 44, 48, 57].

2.

Horizontal and Contact Forms on Fibred Manifolds

Throughout the paper, all manifolds and mappings are smooth and summation over repeated indices is understood. We refer to [66] for the theory of jet bundles and to [36, 41] or [45, 51] for details on the calculus of vector fields and differential forms on fibred manifolds. We denote by π : Y → X a fibred manifold, dim X = n, dim Y = m + n, πr : J r Y → X the r-jet prolongation of π, (V, ψ), ψ = (xi , y σ ), 1 ≤ i ≤ n, 1 ≤ σ ≤ m, a fibred chart on Y , and (Vr , ψr ), ψr = (xi , y σ , yjσ1 , . . . , yjσ1 ...jr ) the associated chart on J r Y . If dim X = 1, we write (t, q σ ), and (t, q σ , q1σ , . . . qrσ ), respectively, (t, q σ , q˙σ , q¨σ ) if r = 2. Next we put ω0 = dx1 ∧ dx2 ∧ ... ∧ dxn ,

ωi1 ...ik = i∂/∂xik ωi1 ...ik−1 , 1 ≤ k ≤ n,

for a local volume on X and its contractions, and ∂ ∂ ∂ σ ∂ di = + yiσ σ + yji + · · · + yjσ1 ...jr i σ σ i ∂x ∂y ∂yj ∂yj1 ...jr

(2.1)

(2.2)

30

Olga Krupková

if dim X > 1, respectively, d ∂ ∂ ∂ ∂ σ = + q1σ σ + q2σ σ + · · · + qr+1 dt ∂t ∂q ∂q1 ∂qrσ

(2.3)

if dim X = 1, the operators of total derivative. A mapping γ : U → Y defined on an open subset U ⊂ X is called a section of the fibred manifold π, if π ◦ γ = idU . A section δ of πr is called holonomic if δ = J r γ, i.e., the r-jet prolongation of a section γ of π. A vector field ξ on J r Y , r ≥ 0, is called πr -projectable if there exists a vector field ξ0 on X such that T πr .ξ = ξ0 ◦ π, and πr -vertical if ξ projects onto a zero vector field on X, i.e., T πr .ξ = 0. Quite similarly one can define a πr,s -projectable or a πr,s -vertical vector field on J r Y , where r > s. Local flows of projectable vector fields transfer sections into sections; consequently, π-projectable vector fields on Y can be naturally prolonged to vector fields on J r Y as follows. Given a π-projectable vector field ξ on Y with projection ξ0 , denote by {φu } and {φ0u } the corresponding local one-parameter groups. For every u, the mapping φu is an isomorphism of the fibred manifold π, i.e. π ◦ φu = φ0u ◦ π. Then for every section γ, the composition γ¯ = φu ◦ γ ◦ φ−1 0u is a section of π and we can define the r-jet prolongation of φu by J r φu (Jxr γ) = Jφr0u (x) (φu γφ−1 0u ).

(2.4)

Then J r φ is a local flow corresponding to a vector field on J r π, denoted by J r ξ and called the r-jet-prolongation of ξ. The vector field J r ξ is both πr -projectable and πr,s -projectable for 0 ≤ s < r, and its πr -projection, (resp. πr,s -projection) is ξ0 (resp. ξ, resp. J s ξ, 1 ≤ s ≤ r − 1). A q-form η on J r Y is called contact if J r γ ∗ η = 0 for every section γ of π. The ideal of contact forms on J r Y is generated by local contact 1-forms ωjσ1 ...jk = dyjσ1 ...jk − yjσ1 ...jk l dxl ,

0 ≤ k ≤ r − 1,

(2.5)

and 2-forms dωjσ1 ...jr−1 . If dim X = 1 we write σ ωkσ = dqkσ − qk+1 dt,

0 ≤ k ≤ r − 1.

(2.6)

It is important to notice that contact 1-forms (2.5) give rise to a local basis of 1-forms on J r Y adapted to the contact structure, namely, (dxi , ω σ , . . . , ωjσ1 ...jr−1 , dyjσ1 ...jr ). In coordinate expressions and computations in fibred coordinates this basis is much more convenient than the canonical basis (dxi , dy σ , . . . , dyjσ1 ...jr ). A q-form η on J r Y is called πr -horizontal (or 0-contact) if the contraction iξ η vanishes for every πr -vertical vector field on J r Y . Similarly, η is called πr,s -horizontal if it vanishes whenever at least one of its arguments is a πr,s -vertical vector field. A contact q-form η on J r Y is called i-contact, 1 ≤ i ≤ q, if for every πr -vertical vector field ξ on J r Y , iξ η is (i − 1)-contact. By Krupka’s theorem, the module of πr+1,r -horizontal q-forms on J r+1 Y is the direct sum of the submodules of i-contact forms, 0 ≤ i ≤ q, meaning that every πr+1,r -horizontal q-form on J r+1 Y has a unique decomposition into a sum of i-contact forms, 0 ≤ i ≤ q.


31

If we denote by h and pi , 1 ≤ i ≤ q, projectors onto the submodules of the horizontal and i-contact forms, where 1 ≤ i ≤ q, we obtain the above theorem in the following form: Every q-form η on J r Y admits a unique decomposition ∗ πr+1,r η = hη + p1 η + · · · + pq η

(2.7)

into a sum of i-contact forms, 0 ≤ i ≤ q, on J r+1 Y . The operators h and pi are called the horizontalisation and i-contactisation, respectively. We also say that hη is the horizontal part of η and pi η is the i-contact part of η. Note that every q-form where q > n is contact; it is called strongly contact if pq−n η = 0.

3. 3.1.

Lepage Forms and the First Variation Variations

Let us start with Lagrangians and variations, following [36]. Consider a fibred manifold π : Y → X, where X is an orientable manifold of dimension n. A Lagrangian of order r for π is defined to be a horizontal n-form λ on J r Y . In a fibred chart (V, ψ), ψ = (xi , y σ ), on Y , λ = L ω0 , −1 where L is a real function on Vr = πr,0 (V ), i.e., L = L(xi , y σ , yjσ1 , . . . , yjσ1 j2 ...jr ). Denote by Ω a piece of X, i.e. a compact n-dimensional submanifold of X with boundary, and ΓΩ (π) the set of smooth sections of π over Ω. The real valued function

ΓΩ (π) ∋ γ →

Z

Ω

J r γ∗λ ∈ R

(3.1)

is called the variational function or action function of the Lagrangian λ over Ω. Since J r γ ∗ η = 0 for any contact n-form η on J r Y , the action function remains the same if one considers the form λ + η instead of λ. In other words, for any n form ρ such that hρ = λ, Z

J r γ∗λ =

Ω

Z

J r γ ∗ ρ.

(3.2)

Ω

Let γ : U → Y be a section defined on an open set U ⊂ X. Let ξ be a π-projectable vector field defined in a neighborhood of γ(U ), and ξ0 the projection of ξ. If αt is the local 1-parameter group of ξ and α0t is the projection of αt (i.e., α0t is the local 1-parameter −1 group of ξ0 ), then γt = αt ◦ γ ◦ α0t is a 1-parameter family of sections of π, depending smoothly on the parameter t. The family {γt } is called the variation or deformation of the section γ induced by the vector field ξ. Often the vector field ξ itself is called a variation of γ. Now, taking the action function (3.1) for a fixed section γ ∈ ΓΩ (π), and considering a variation of γ, we get the following real-valued function, defined in a neighborhood I of the point 0 in R: Z I∋t→

α0t (Ω)

J r γt∗ λ ∈ R.

32

Olga Krupková

Differentiating this function at the point t = 0 we arrive at nd Z

dt

α0t (Ω)

J r γt∗ λ

o

0

=

Z

Ω

J r γ ∗ LJ r ξ λ.

(3.3)

The real number (3.3) is called variation of the action function at the point γ, induced by the vector field ξ. The function ΓΩ (π) ∋ γ →

Z

Ω

J r γ ∗ LJ r ξ λ ∈ R

(3.4)

is then called the first variation of the action function (3.1) by the vector field ξ. Let us recall the key idea that lead Krupka to the discovery of Lepage forms: The problem is to find an intrinsic form of the first variation formula, i.e. a decomposition of the integral in (3.4) to a sum of a term characterising extremals (Euler–Lagrange term) and a boundary term (connected with conservation laws). To this end, one would like (and should) utilize the well-known Cartan formula for the decomposition of the Lie derivative. Unfortunately, the result cannot be obtained by a direct application of this formula to LJ r ξ λ, since iJ r ξ dλ depends not only upon the variation ξ itself, but also upon prolongations (“derivations”) of ξ. However, taking into account property (3.2) of the action function, one can study the following question: Given a Lagrangian λ, is it possible to add to λ a contact form ν in such a way that after the decomposition of LJ r ξ (λ + ν) in (3.4) the first term would depend upon the variation vector field ξ only? Then, for such forms, Cartan’s formula would provide the desired intrinsic first variation formula, and, consequently, a geometric form of the Euler–Lagrange equations and of conservation laws.

3.2.

Lepage n-Forms and the First Variation Formula

Recall that n = dim X, and let s ≥ 0. Formalization of the above idea leads to the following definition [36]: Definition 3.1. A n-form ρ on J s Y is called a Lepage n-form of order s if for every πs+1,0 vertical vector field ξ on J s Y hiξ dρ = 0. (3.5) Note that if ρ is a Lepage n-form on J s Y then hρ is a Lagrangian on J s+1 Y . We write hρ = λ = Lω0 .

(3.6)

From the definition we obtain three Krupka theorems on the structure of Lepage n-forms [36, 41]: Theorem 3.2. Let ρ be an n-form on J s Y . The following conditions are equivalent: (1) ρ is a Lepage form of order s. (2) The (n + 1)-form p1 dρ is πs+1,0 -horizontal. (3) dρ satisfies ∗ πs+1,s dρ = E + F,

where E is 1-contact and πs+1,0 -horizontal, and F is at least 2-contact.

(3.7)


33

∗ (4) In every fibred chart, πs+1,s ρ is expressed as follows: ∗ πs+1,s ρ = L ω0 +

s X

k=0

fσi, j1 j2 ...jk ωjσ1 j2 ...jk ∧ ωi + η,

(3.8)

where η is an arbitrary at least 2-contact n-form, and fσ(js+1 , j1 j2 ...js ) = (j , j1 j2 ...jk−1 )

fσ k

=

∂L , ∂yjσ1 j2 ...js+1 ∂L ∂yjσ1 j2 ...jk

(3.9)

− di fσi, j1 j2 ...jk ,

1 ≤ k ≤ s,

where (. . . ) means symmetrisation in the indicated indices, i.e. (i, j1 j2 . . . jp ) = sym(i, j1 j2 . . . jp ). Theorem 3.3. A form ρ on J s Y is Lepage n-form if and only if in every fibred chart ∗ πs+1,s ρ = Θλ + dν + µ,

(3.10)

where Θλ = L ω0 +

s s−k X X

(−1)l dp1 dp2 . . . dpl

k=0 l=0

∂L ∂yjσ1 j2 ...jk p1 p2 ...pl i

ωjσ1 j2 ...jk ∧ ωi ,

(3.11)

ν is a contact (n − 1)-form, and µ is an at least 2-contact form. Note that ∗ πs+1,s ρ = Θλ + dν + µ = Θλ + p1 dν + η,

(3.12)

and that in this decomposition, the forms Θλ + p1 dν and η are global (being the at most ∗ 1-contact and the at least 2-contact part of πs+1,s ρ, respectively). We stress that, on the contrary, the decomposition Θλ + p1 dν need not be invariant under the change of fibred coordinates, meaning that in general Θλ is not a global differential form. Theorem 3.4. If ρ is a Lepage n-form then the 1-contact part p1 dρ of dρ depends only on hρ = λ = Lω0 and reads p1 dρ = Eλ =

∂L

∂y σ

−

s+1 X

(−1)l+1 dp1 dp2 . . . dpl

l=1

∂L ∂ypσ1 p2 ...pl

ω σ ∧ ω0 .

(3.13)

Definition 3.5 ([36]). Given a Lagrangian λ, we say that a form ρ is a Lepage equivalent of λ if ρ is a Lepage n-form and hρ = λ. The (n + 1)-form Eλ = p1 dρ is then called the Euler–Lagrange form of λ, and its fibred-chart components are called Euler–Lagrange expressions. Theorem 3.6 (Krupka [36, 41], Marvan [63]). Every Lagrangian has a Lepage equivalent. Taking into account the above theorems we obtain:

34

Olga Krupková

Corollary 3.7. (1) Every Lagrangian λ of order r has a Lepage equivalent of order 2r − 1, that is generally nonunique. Every Lepage equivalent of λ takes the form as described by theorem 3.3 or 3.2 (4). (2) Every Lagrangian λ has a Lepage equivalent that is at most 2-contact. If λ is of order r that Lepage equivalent is of order ≤ 2r − 1, is generally nonunique, and reads Θ = Θλ + p1 dν,

(3.14)

where Θλ is given by formula (3.11) and ν is a contact (n − 1)-form. In particular, every Lagrangian of order r has (global) 2-contact Lepage equivalents that are π2r−1,r−1 horizontal; they take the form (3.14) where ν is an arbitrary 1-contact π2r−1,r−1 -horizontal (n − 1)-form. (3) To every Lagrangian λ there exists a uniquely determined Euler–Lagrange form Eλ . If λ is defined on J r Y then Eλ is generally on J 2r Y and is given by formula Eλ =

∂L

∂y σ

−

r X

(−1)l+1 dp1 dp2 . . . dpl

l=1

∂L ∂ypσ1 p2 ...pl

ω σ ∧ ω0 = p1 dρ,

(3.15)

where ρ is any Lepage equivalent of λ. It should be stressed that the splitting of Θ in (3.14) into the term Θλ that is determined by the Lagrangian, and an auxiliary term p1 dν generally is not invariant under changes of fibred coordinates. This means that the expression (3.11) may have only a local meaning, not providing a differential form on J 2r−1 Y . If Θλ happens to be a global differential form, we speak about the (higher-order) Poincaré–Cartan form associated to the Lagrangian λ, or, about the Poincaré–Cartan equivalent of λ. Equipped with Lepage n-forms, we are able to write down the intrinsic first variation formula. Theorem 3.8 (Krupka [36, 41]). Let λ be a Lagrangian of order r, ρ its Lepage equivalent, ξ a π-projectable vector field on Y . Then (i) the Lie derivative LJ r ξ λ is expressed by the formula LJ r ξ λ = hiJ 2r−1 ξ dρ + hdiJ 2r−1 ξ ρ,

(3.16)

J r γ ∗ LJ r ξ λ = J 2r−1 γ ∗ iJ 2r−1 ξ dρ + dJ 2r−1 γ ∗ iJ 2r−1 ξ ρ,

(3.17)

(ii) for any section γ of π,

(iii) for any piece Ω of X with boundary ∂Ω, and any section γ of π such that Ω ⊂ dom γ, Z

Ω

r ∗

J γ LJ r ξ λ =

Z

Ω

J

2r−1 ∗

γ iJ 2r−1 ξ dρ +

Z

∂Ω

J 2r−1 γ ∗ iJ 2r−1 ξ ρ,

(3.18)

and, in all the above formulas, the first term on the right-hand side depends on the vector field ξ only (not on its prolongations). Formula (3.16) or (3.17) is called the infinitesimal first variation formula, (3.18) is the integral first variation formula.


3.3.

35

Examples of Lepage Equivalents of Lagrangians

We have seen that to a Lagrangian one has in general many Lepage equivalents. However, a few particular cases are known, when either the Lepage equivalent is unique, or, the family of Lepage equivalents contains some interesting distinguished representatives. When discussing special cases, first of all, one has to mention classical and higher order mechanics. In the case dim X = 1 we denote local fibred coordinates on Y by (t, q σ ), and the associated higher order coordinates by (t, q σ , q1σ , . . . , qrσ ), or (t, q σ , q˙σ , q¨σ , . . . ). The σ dt, 0 ≤ j ≤ r. Taking into corresponding local contact 1-forms then read ωjσ = dqjσ − qj+1 account that now ρ is a 1-form, i.e., at most 1-contact, and ν is a contact function (i.e. = 0), we immediately obtain: Theorem 3.9. Let dim X = 1. To every Lagrangian λ on J r Y there exists a unique Lepage equivalent. In fibred coordinates where λ = Ldt, it reads r−1 ∂L d ∂L d2 ∂L r−1 d + − · · · + (−1) ωσ ∂q1σ dt ∂q2σ dt2 ∂q3σ dtr−1 ∂qrσ ∂L d ∂L σ ∂L σ − ωr−2 + σ ωr−1 . + ··· + σ σ ∂qr−1 dt ∂qr ∂qr

Θλ = L dt +

∂L

−

(3.19)

The above form Θλ is called (higher order) Cartan form. For a first order Lagrangian the formula is reduced to the well-known one, introduced to the calculus of variations and classical mechanics by Whittaker [78] and Cartan [5]: Θλ = Ldt +

∂L σ ∂L σ ∂L ω = L − q˙ dt + σ dq σ . ∂ q˙σ ∂ q˙σ ∂ q˙

(3.20)

Another important special case is dim X = n > 1 and r = 1, the so-called first order field theory. If λ is a first order Lagangian then theorem 3.3 gives all first-order Lepage equivalents of λ as follows: ρ = Θλ + p1 dν + η ∂L ik σ ij ijk σ = Lω0 + − d g k σ ω ∧ ωi − (gσ + dk gσ ) ∧ ωj ∧ ωi + η, ∂yiσ

(3.21)

where η is an arbitrary at least 2-contact n-form, and ν is an arbitrary 1-contact (n − 1)form, ν = 21 gσij ω σ ∧ ωij + 16 gσijk dω σ ∧ ωijk , where we assume the components be functions skewsymmetric in the upper indices. Obviously the part of ρ, uniquely determined by the Lagrangian, i.e. ∂L Θλ = Lω0 + σ ω σ ∧ ωi , (3.22) ∂yi is invariant with respect to changes of fibred coordinates; it is called the Poincaré–Cartan form. We can see that the form Θλ is uniquely characterised and intrinsically defined as the Lepage equivalent of λ that is π1,0 -horizontal and at most 1-contact. It is known that behind the Poincaré–Cartan form, the family (3.21) contains the following two global Lepage forms, completely determined by the Lagrangian:

36

Olga Krupková – Carathéodory form ∂L σ1 ∂L σn n ∧ · · ω · ∧ L dx + ω σ 1 Ln−1 ∂y1 ∂ynσn 1 ∂L ∂L σ1 ∂L ω ∧ ω σ 2 ∧ ωj 1 j 2 + · · · , = L ω0 + σ ω σ ∧ ωj + ∂yj 2L ∂yjσ11 ∂yjσ22

1

ρCλ =

L dx1 +

(3.23)

this form is invariant under changes of any (not only fibred) coordinates (cf. [65]), and – Krupka form [39] (later also found by Betounes [4]) ρK λ

= L ω0 +

n X

k=1

1 ∂kL ω σ1 ∧ · · · ∧ ω σk ∧ ωj1 ···jk . σ 1 (k!)2 ∂yj1 · · · ∂yjσkk

(3.24)

This form has an important property dρK λ = 0 if and only if the Euler-Lagrange form of λ vanishes identically. As pointed out in [36], an interesting situation arises when dim X = n > 1 and r = 2 (second order Lagrangians in field theory). First of all, in this case Θλ is no longer intrinsically characterised similarly as for r = 1, i.e. as the Lepage equivalent of λ that is π3,1 -horizontal and at most 1-contact: indeed, such form is no longer unique and is given by formula Θ = L ω0 +

∂L

∂yiσ

− dk

∂L

σ ∂yik

+ gσik

ω σ ∧ ωi +

∂L

σ ∂yij

− gσij ωjσ ∧ ωi ,

(3.25)

where gσjk are arbitrary functions skewsymmetric in the upper indices. However, one can check by a direct calculation that the part of Θ determined by the Lagrangian, i.e. Θλ = L ω0 +

∂L

∂yiσ

− dk

∂L σ ∂L ω ∧ ωi + σ ωjσ ∧ ωi , σ ∂yik ∂yij

(3.26)

is invariant under fibred coordinate transformations [41]. This means that for second order Lagrangians we still have a well-defined Poincaré–Cartan form. Also the question on the existence of a higher order Carathéodory form has an affirmative answer; the form is defined on J 3 Y by the following formula [68]: ρCλ =

1

∂L σ1 ∂L σ1 ω + σ σ 1 ωj 1 ∧ . . . Ln−1 ∂y1j11 ∂y1j 1 ∂L ∂L σn ∂L σn n ∧ L dx + − d j n σ n ω + σ n ωj n . ∂ynσn ∂ynjn ∂ynjn

L dx1 +

∂L

∂y1σ1

− d j1

(3.27)

Quite a different story concerns a possible Krupka form ρK λ for second order Lagrangians: yet, the question of whether such a (global) form may be found is still open, however, some arguments coming from the geometry of homogeneous Lagrangian systems support the suggestion that it does not exist [68]. For dim X = n > 1 and r > 2 (higher order field theory) the situation is even more complicated. Surprisingly, in the class Θ we have no global differential form determined by the Lagrangian, meaning that a “higher-order Poincaré–Cartan form” Θλ does not exist;


37

so we have no distinguished Lepage equivalent of λ of “Cartan type”. The non-existence of a global higher-order Poincaré–Cartan form initiated studies of geometric constructions providing a “globalization” of the expression Θλ (see e.g. ([6, 15, 18, 23, 31, 33]). From the point of view of the general theory of Lepage forms the problem of “globalization of Θλ ” is very simple: a desired form always exists and is obtained by adding to Θλ an appropriate term p1 dν where ν is 1-contact and horizontal with respect to the projection π2r−1,r−1 (cf. corollary 3.7).

3.4.

Extremals and Euler–Lagrange Equations

As a first application of Lepage forms one obtains different global expressions for the Euler– Lagrange equations [36, 42]. A section γ ∈ ΓΩ (π) is called a stable point of the action function of λ with respect to a variation ξ if Z Ω

J r γ ∗ LJ r ξ λ = 0.

(3.28)

γ is said to be an extremal of λ on Ω if (3.28) holds for all variations ξ of γ such that supp(ξ ◦ γ) ⊂ Ω; here supp(ξ ◦ γ) denotes the support of the vector field ξ ◦ γ along the section γ, defined as the closure of the set {x ∈ dom γ|(ξ ◦ γ)(x) 6= 0}. Finally, γ is called an extremal of the Lagrangian λ if it is an extremal of λ on every Ω in the domain of definition of γ. Necessary and sufficient conditions for a section be an extremal of a Lagrangian follow from the first variation formula: Theorem 3.10. Let λ be a Lagrangian on J r Y , ρ a Lepage equivalent of λ. Let γ be a section of π. The following conditions are equivalent: (1) γ is an extremal of λ. (2) For every vector field ξ on J 2r−1 Y J 2r−1 γ ∗ iξ dρ = 0 .

(3.29)

(3) For every π-projectable vector field ξ on Y J 2r−1 γ ∗ iJ 2r−1 ξ dρ = 0 .

(3.30)

(4) For every π-vertical vector field ξ on Y J 2r−1 γ ∗ iJ 2r−1 ξ dρ = 0 .

(3.31)

(5) J 2r−1 γ is an integral section of the exterior differential system generated by the system of n-forms iξ dρ, where ξ runs over all vertical vector fields on J 2r−1 Y .

(3.32)

38

Olga Krupková

(6) The Euler-Lagrange form Eλ vanishes along J 2r γ, i.e., Eλ ◦ J 2r γ = 0 .

(3.33)

(7) In every fibred chart, γ satisfies the following system of differential equations r X

k=0

k

(−1) dj1 · · · djk

∂L ∂yjσ1 ···jk

!

◦ J 2r γ = 0,

1 ≤ σ ≤ m.

(3.34)

Any of the equivalent conditions (2)–(7) above may be called the Euler-Lagrange equations of the Lagrangian λ. Note that (4) means that although in place of variations one can take arbitrary vector fields, extremals are determined only by the vertical part of variations σ , . . . ). Thus among defined on Y (they do not depend upon components at ∂/∂yjσ , ∂/∂yjk equations (2)–(4), (4) represents the most simple and most frequently used form of intrinsic Euler–Lagrange equations. We stress that for dim X = 1 (3.32) is a system of 1-forms, annihilating a distribution on J 2r−1 Y , called the Euler–Lagrange distribution [46]. Notice that for any Lepage equivalent ρ of λ, dρ = dΘ + F , where F is at least 2-contact. Hence iξ F is contact and vanishes along J 2r−1 γ. This means that for the Euler– Lagrange equations (3.31) the at most 2-contact part of a Lepage equivalent is essential, so that we can simply consider them in the form J 2r−1 γ ∗ iJ 2r−1 ξ dΘ = 0,

∀ξ.

(3.35)

Finally, recall that Lagrangians λ1 , λ2 are said to be equivalent if Eλ1 = Eλ2 (possibly up to a projection). A Lagrangian λ is called null if Eλ = 0. Obviously, λ1 and λ2 are equivalent if and only if λ1 − λ2 is null.

3.5.

Hamilton Equations, Regular Lagrangians

We have seen that Euler–Lagrange equations of a Lagrangian λ have a geometric meaning as equations for holonomic integral sections of the exterior differential system (3.32), where ρ is a Lepage equivalent of λ. However, one can study all integral section of the corresponding EDS, and explore that for different Lepage equivalents of λ different exterior differential systems are obtained. Let λ be a Lagrangian on J r Y , ρ its Lepage equivalent. We call the EDS defined by (3.32) Hamiltonian system of ρ. Equations for integral sections of a Hamiltonian system, i.e., δ ∗ iξ dρ = 0, ∀π2r−1 -vertical ξ, (3.36) are called Hamilton equations, the integral sections are then called Hamilton extremals of ρ [42]. Obviously, the set of prolongations of extremals of λ is a subset of the set of Hamilton extremals. Properties of this subset, however, depend both upon properties of the Lagrangian λ and the choice of its Lepage equivalent ρ: due to Dedecker [8] and Krupka and his collaborators, this understanding is a key to the concept of a regular variational problem [49, 53, 54, 58, 59, 70, 71].


39

In higher-order mechanics we associate to λ a unique Lepage equivalent, Θλ , and hence a unique Hamiltonian system, generated by 1-forms iξ dΘλ , where ξ runs over vertical vector fields. A well-known result says that if a Lagrangian λ on J r Y satisfies the “regularity condition” ∂2L 6= 0, (3.37) det ∂qrσ ∂qrν then every Hamilton extremal of Θλ is the (2r − 1)-th prolongation of an extremal of λ, i.e. the sets of extremals and Hamilton extremals of λ are in 1-1 correspondence; moreover, σ σ , p , . . . pr−1 ), where pi are components the mapping (t, q σ , . . . , q2r−1 ) → (t, q σ , . . . , qr−1 σ σ σ σ of Θλ at dqi , 0 ≤ i ≤ r − 1, is a local diffeomorphism (called Legendre transformation) [9, 46]. In the case dim X = n > 1, the Hamiltonian theory is richer, since the Lepage equivalent of λ is no longer unique. Going back to De Donder and Goldschmidt and Sternberg [11, 22], the most commonly considered Hamilton equations are those based on the Poincaré–Cartan form Θλ (cf. also Hamilton equations in the “multisymplectic formalism” [13, 21, 24, 62]); we call these Hamilton equations De Donder–Hamilton equations. The result now is completely analogous to that of mechanics: if a Lagrangian λ on J 1 Y satisfies the “regularity condition” ∂2L det (3.38) 6= 0, ∂yiσ ∂yjν then every Hamilton extremal of Θλ is the prolongation of an extremal of λ, meaning that De Donder–Hamilton equations and Euler–Lagrange equations are equivalent. Writing Θλ = L −

∂L σ ∂L yi ω0 + σ dy σ ∧ ωi = −Hω0 + piσ dy σ ∧ ωi , σ ∂yi ∂yi

(3.39)

we get the Hamiltonian and momenta of λ. The above regularity condition then guarantees that (xi , y σ , yjσ ) → (xi , y σ , pjσ ) is a local coordinate transformation on J 1 Y (Legendre transformation). In Legendre coordinates, De Donder–Hamilton equations read ∂y σ ∂H = j, j ∂x ∂pσ

∂pjσ ∂H = − σ. j ∂x ∂y

(3.40)

A higher-order version of this result was first considered by De Donder [11]. However, the generalisation is not so straightforward, since Θλ may be not globally well-defined, and if “globalised”, is non-unique. Saving the property of being determined completely by the Lagrangian, one has to resign on global Hamilton equations. Given a Lagrangian λ of order r, and its local Poincaré–Cartan equivalent Θλ (3.11), Hamilton equations now read δ ∗ iξ dΘλ = 0

for every π2r−1 -vertical vector field ξ on J 2r−1 Y

(3.41)

and are defined on the domain W of the coordinates (xi , y σ , yjσ1 , yjσ1 j2 , . . . , yjσ1 j2 ...jr ). Put Θλ = −Hω0 + piσ dy σ ∧ ωi + pjσ1 i dyjσ1 ∧ ωi + · · · + pjσ1 ...jr−1 i dyjσ1 ...jr−1 ∧ ωi ,

(3.42)

40

Olga Krupková

where pjσ1 ...jk i =

r−k−1 X

(−1)l dp1 dp2 . . . dpl

l=0

H = −L +

r X

∂L

, ∂yjσ1 ...jk p1 ...pl i

0 ≤ k ≤ r − 1,

(3.43)

pjσ1 ...jk yjσ1 ...jk ,

k=1

and denote by [q1 . . . qs ] the number of all different sequences arising by permuting the sequence q1 , . . . , qs . As proved by Shadwick [69], if the rank of all the matrices ∂2L 1 · σ [j1 . . . j2r−s (pr+1 . . . ps ] [p1 . . . pr )] ∂yj1 ...j2r−s (pr+1 ...ps ∂ypν1 ...pr )

!

(3.44)

is maximal, where r ≤ s ≤ 2r − 1, the σ, j1 ≤ · · · ≤ j2r−s label columns, ν, p1 ≤ · · · ≤ ps label rows, and the brackets (· · · ) denote symmetrization in the indicated indices, then every Hamilton extremal δ of Θλ passing in W is of the form π2r−1,r ◦ δ = J r γ where γ is an extremal of λ. Moreover, the system of functions xi , y σ , yjσ1 , . . . , yjσ1 ...jr−1 , pjσ1 ...jr , . . . , pjσ1 ,

j1 ≤ · · · ≤ jr

(3.45)

forms a part of a coordinate system on W (called Legendre coordinates). In Legendre coordinates De Donder–Hamilton equations read ∂yjσ1 ...jk ∂H = j1 ...jk i , i ∂x sym{j1 ...jk i} ∂pσ

∂pjσ1 ...jk l ∂H =− σ , l ∂x ∂yj1 ...jk

(3.46)

where 0 ≤ k ≤ r − 1, and in the second set of equations, summation over l takes place. We note that Shadwick’s regularity condition above can be put into a geometric form and can be expressed equivalently by means of certain bilinear foms or by a linear mapping [20, 34]. Considering in place of Θλ a global form Θ = Θλ +p1 dν we get to λ non-unique global De Donder–Hamilton equations. There arises the question of whether Shadwick’s regularity condition still can guarantee an analogous nice correspondence between extremals and Hamilton extremals, According to Krupka [43] and Gotay [24] the answer is affirmative: If Shadwick’s regularity condition is satisfied then every solution δ of De Donder–Hamilton equations of Θ is of the form π2r−1,r ◦ δ = J r γ where γ is an extremal of λ. However, note that De Donder–Hamilton equations of Θ in Legendre coordinates may have a more complicated form compared to (3.46). For other interesting aspects of the theory see e.g. [1, 10, 16, 24, 34, 42, 43, 67]. It is important to notice that for higher-order regular Lagrangians De Donder–Hamilton equations are no longer equivalent with the Euler–Lagrange equations: the subset of Hamilton extremals that is in bijective correspondence with extremals consists of sections that are holonomic up to the order r (note r = order of the Lagrangian). Within De Donder–Hamilton theory the concept of regularity can be reconsidered to give regularity conditions for higher order Lagrangians different from Shadwick’s regularity condition. The idea proposed in our joint paper [49] was that the “true order” of the De Donder–Hamilton equations in essential. Namely, for some Lagrangians of order r ≥ 2,


41

their Poincaré–Cartan form is π2r−1,s -projectable, where s < 2r − 1; in this case, it is apparently inappropriate to apply the standard procedure leading to considering Hamilton equations of order 2r − 1. The problem should instead be studied as a problem of order s. In the above mentioned paper this idea was applied to the class of second order Lagrangians with π3,1 -projectable Θλ . This concerns the following second order Lagrangians affine in the second derivatives: λ = Lω0 where i σ ν L = L0 (xi , y σ , yjσ ) + hpq ν (x , y ) ypq .

(3.47)

In [49] we called a Lagrangian regular if all its Hamilton extremals were holonomic, and we proved the following theorem: ˇ epánková). Let λ be a Lagrangian of the form (3.47). If the Theorem 3.11 (Krupka and Stˇ condition ! ∂ 2 L0 ∂hik ∂hki σ ν 6= 0 (3.48) det − − ∂yiσ ∂ykν ∂y ν ∂y σ is satisfied then λ is regular, its Euler–Lagrange and De Donder–Hamilton equations are equivalent, and the mapping (xi , y σ , yjσ ) → (xi , y σ , pjσ ),

pjσ =

∂hjk ∂L0 ∂hjk ∂hkj σ σ ν − − + yν ∂yjσ ∂xk ∂y ν ∂y σ k

(3.49)

is a local coordinate transformation on J 1 Y (“Legendre transformation”). It should be stressed that for second order Lagrangians (3.47) the momenta (3.49) and Hamiltonian ∂L0 ∂hjk H = −L0 + σ yjσ − σν yjσ ykν , (3.50) ∂yj ∂y are first order functions. De Donder–Hamilton equations expressed in Legendre coordinates (3.49) then take the “usual” form (3.40). The above results directly apply to the Einstein– Hilbert Lagrangian (scalar curvature) of the General Relativity Theory (see [49] for explicit computations). Thus, within this setting, gravity naturally appears as a first order regular theory (without constraints). Later the above ideas were applied to study also some other kinds of higher order Lagrangians with projectable Poincaré–Cartan forms by Garcia and Muñoz [19, 20]. We have seen that regularity is a property of a Lepage form, rather than of a Lagrangian itself, and it carries a geometric content that all Hamilton extremal are holonomic (for first order Hamiltonian systems), or holonomic up to a proper order (for higher order systems). However, in field theory we have for a given Lagrangian a family of Hamiltonian systems, defined by different Lepage equivalents of λ. In this way we expect to obtain regularity conditions depending on λ and some auxiliary functions coming from parts of ρ that are not uniquely determined by the Lagrangian, and we may consider the Hamiltonian theory in a completely new setting: instead of asking whether a Lagrangian is regular (that usually means regularity of its De Donder–Hamilton system) we may ask a question if the family of Hamiltonian systems associated with a Lagrangian contains a Hamiltonian system that is regular. This possibility was first observed by Dedecker [8], and recently studied for

42

Olga Krupková

Lepage equivalents of first and higher order Lagrangians in [53, 54, 58, 59, 70, 71]. In [59] a procedure of “regularisation” of conventionally singular Lagrangians was applied to important physical field Lagrangians (Dirac field and electromagetic field Lagrangian), and “corrected” Hamiltonians and momenta were found, providing Hamilton equations without constraints.

3.6.

Symmetries and Conservation Laws, Noether Theorem

Lepage forms play an important role also in the theory of invariant variational functionals developed by Krupka in the early 1970’s [36, 38]. Let λ be a Lagrangian on J r Y , Eλ its Euler-Lagrange form. An isomorphism φ of the fibred manifold π : Y → X is called an invariance transformation of λ, respectively, generalised invariance transformation of λ if J r φ∗ λ = 0,

respectively,

J 2r φ∗ Eλ = 0.

(3.51)

If ξ is a π-projectable vector field on Y and {φu } is a local one-parameter group of ξ such that φu is an invariance transformation, respectively, generalised invariance transformation, for every u, we get the following infinitesimal version of the above invariance conditions: LJ r ξ λ = 0,

respectively,

LJ 2r ξ Eλ = 0.

(3.52)

The above conditions are called the Noether equation and the Noether–Bessel-Hagen equation, respectively. Noether and Noether-Bessel-Hagen equations can be used to find the group of invariance or generalised invariance transformations of a given Lagrangian, or conversely, to find all Lagrangians (Euler–Lagrange forms) invariant with respect to a given group of transformations of Y ; of course, solving the Noether–Bessel-Hagen equation in this case we obtain invariant differential equations that need not come from a Lagrangian: variationality is an additional property to be satisfied. We have an important theorem due to Krupka [38]: Theorem 3.12. Let λ be a Lagrangian on J r Y and Eλ its Euler–Lagrange form. Given an isomorphism φ of the fibred manifold π : Y → X, respectively, a π-projectable vector field ξ on Y , it follows that J 2r φ∗ Eλ = EJ r φ∗ λ ,

LJ 2r ξ Eλ = ELJ r ξ λ .

(3.53)

Corollary 3.13. Every invariance transformation of a Lagrangian λ is its generalised invariance transformation. Corollary 3.14. If LJ 2r ξ Eλ = 0 then LJ r ξ λ is a null Lagrangian. As proved in [26, 47] (see also the next section), the above condition means that around every point in J r−1 Y there is an (n − 1)-form η such that LJ r ξ λ = hdη.

Equipped with the concept of invariant Lagrangian, we obtain a classical result of Emmy Noether [64] and its generalisation, that in the setting of Lepage forms appear as an easy consequence of the first variation formula (Krupka [38]):


43

Theorem 3.15 (Noether Theorem). Let λ be a Lagrangian on J r Y , ρ its Lepage equivalent. If a π-projectable vector field ξ on Y generates invariance transformations of λ, and if γ is an extremal then d(J 2r−1 γ ∗ iJ 2r−1 ξ ρ) = 0. (3.54) Theorem 3.16 (Generalised Noether Theorem). Let λ be a Lagrangian on J r Y , ρ its Lepage equivalent. If a π-projectable vector field ξ on Y generates generalised invariance transformations of λ, and if γ is an extremal then d(J 2r−1 γ ∗ (iJ 2r−1 ξ ρ − η)) = 0,

(3.55)

where η is a (local) (n − 1)-form such that LJ r ξ λ = hdη. Equation (3.54), respectively (3.55), is called a conservation law, and the (n − 1)form iJ 2r−1 ξ ρ respectively, iJ 2r−1 ξ ρ − η, that is closed along prolongations of extremals, is called a conserved current. Using the Poincaré lemma we get around every point x ∈ X an (n − 2)-form ϕ for which dϕ = f i ωi is a conserved current on X, so that the corresponding conservation law d(f i ωi ) = 0 takes a “divergence form” div f = 0. In mechanics (dim X = 1) the situation is simpler: a “conserved current” is a function, F , and a conservation law reads F ◦ J 2r−1 γ = const. Therefore, F is called constant of the motion. Since in field theory the Lepage equivalent of λ is not unique, there arises a question on how a conserved current depends upon a choice of a Lepage equivalent ρ of λ. Using formulas (3.10) and (3.14), i.e. ρ = Θ + µ ¯ where µ ¯ is an at least 2-contact form, and taking into account that contraction of µ ¯ gives a contact form that vanishes along prolongations of sections of π, we can see that the conservation law corresponding to ρ depends merely upon the at most 2-contact part of ρ; hence, for any Lepage equivalent ρ of an invariant Lagrangian λ, the conserved current is the (n − 1)-form iJ 2r−1 ξ Θ. Of course, in higher order field theory we must take into account the non-uniqueness of Θ and the non-existence of a global Poincaré–Cartan form Θλ , discussed in the previous sections.

4. 4.1.

Lepage Forms and Differential Equations The Inverse Problem of the Calculus of Variations

By a dynamical form of order s we understand a 1-contact and πs,0 -horizontal (n + 1)form E on J s Y [51]. This means that in every fibred chart (V, ψ), ψ = (xi , y σ ) on Y , −1 E = Eσ ω σ ∧ ω0 , where the components Eσ are functions on Vs = πs,0 (V ). A section γ of π satisfying E ◦ J sγ = 0 (4.1)

is called a path of E. Equation (4.1) in fibred coordinates takes the form Eσ ◦ J s γ = 0, 1 ≤ σ ≤ m, that is a system of m partial, respectively ordinary, differential equations of order s if dim X > 1, respectively if dim X = 1. Let E be a dynamical form on J s Y . We have the following important definitions [40]: E is called locally variational if to every point in J s Y there exists a neighbourhood U , an integer r ≤ s, and a Lagrangian λ defined on πs,r (U ) such that E|U = Eλ . E is called

44

Olga Krupková

globally variational if λ is defined on J r Y , i.e., if E is locally variational and has a global Lagrangian. Paths of a locally variational form E are called extremals of E. Note that extremals of a globally variational form coincide with extremals of any of its global Lagrangians. On the other hand, if E is locally but not globally variational, the family of extremals of E contains, but is not equal to, the family of extremals of any individual Lagrangian of E. Remark 4.1. We have seen that a dynamical form on J s Y determines a system of m sth order differential equations (4.1). Conversely, given a system of m sth-order differential equations Eσ ◦ J s γ = 0, 1 ≤ σ ≤ m, for graphs of mappings from Rn to Rm (n ≥ 1) we can represent it by a dynamical form E defined on an open subset of J s (Rn × Rm ) setting E = Eσ ω σ ∧ dt if dim X = 1, respectively, E = Eσ ω σ ∧ ω0 if dim X > 1. Such equations are called variational if the associated dynamical form E is locally variational. There is a close connection between locally variational forms and closed forms: Theorem 4.2 (Krupka [40, 41]). Let E be a dynamical form on J s Y . E is locally variational if and only if to every point x ∈ J s Y there exists a neighbourhood W and an at least 2-contact form FW on W such that the form αW = E + FW is closed. Since dαW = 0, we have by the Poincaré lemma to every point z ∈ W a neighborhood U and a form ρ on U such that dρ = αU . However, the one-contact part p1 dρ of dρ equals the dynamical form E (restricted to U ), i.e. p1 dρ is horizontal with respect to the projection onto Y . This means that ρ is a Lepage n-form. Now, λ = hρ is a Lagrangian such that Eλ = E|U . In this way we can obtain a local Lagrangian for E simply by means of the Poincaré lemma. Moreover, the geometric structure of J s Y enables one to introduce a modified homotopy operator, A, adapted to the contact structure [41] with the following properties: (1) Adα + dAα = dα, (2) if α is k-contact then Aα is (k − 1)-contact. Then, however, λ = hAα = Ap1 α = AE,

(4.2)

in fibered coordinates L = qσ

Z

1

Eσ (t, uq ν , uq1ν , . . . , uqsν )du

(4.3)

Eσ (xi , uy ν , uyjν , . . . , uyjν1 ...js )du

(4.4)

0

if dim X = 1, and L=y

σ

Z

0

1

if dim X > 1, respectively. The formula for L was obtained in [74] and [75], and is called Tonti–Vainberg Lagrangian. Necessary and sufficient conditions for a dynamical form be locally variational were first studied by Helmholtz for the case of second order ordinary differential equations [30].


45

A generalisation to ordinary differential equations of an arbitrary order is due to Vanderbauwhede [76], and to higher-order partial differential equations to Anderson and Duchamp [3], and Krupka [40, 41]. The celebrated conditions read as follows: Theorem 4.3. A dynamical form E on J s Y is locally variational if and only if its components in every fibred chart satisfy the following conditions: dim X = 1 :

s X k ∂Eσ l ∂Eν − (−1) − (−1)k ∂qlν ∂qlσ l k=l+1

!

dk−l ∂Eν = 0, dtk−l ∂qkσ

(4.5)

dim X = n : s X (4.6) ∂Eσ ∂Eν k ∂Eν l − (−1) − (−1)k = 0, djl+1 djl+2 . . . djk σ ∂yjν1 j2 ...jl ∂yjσ1 j2 ...jl k=l+1 l ∂yj1 j2 ...jk

!

for all σ, ν, and 0 ≤ l ≤ s. To a locally variational form a local Lagrangian is given by formula (4.3), respectively, (4.4).

4.2.

Lepage Equivalents of Locally Variational Forms

Now we are able to extend the concept of Lepage n-form to (n + 1)-forms as proposed in [50, 53]. Theorem 4.2 shows that for a dynamical form be locally variational it is essential that it can be completed (at least locally) to a closed form. This property was a motivation for the following definitions: Definition 4.4. Let s ≥ 0. A closed (n + 1)-form α on J s Y is called a Lepage (n + 1)form if p1 α is a dynamical form. A Lepage (n + 1)-form α is called Lepage equivalent of a dynamical form E if p1 α = E. With the help of the Poincaré lemma, and taking account of the definition of a Lepage n-form, we immediately obtain relations between Lepage (n + 1)-forms and n-forms: Theorem 4.5. (1) If α is a Lepage (n + 1)-form then locally α = dρ where ρ is a Lepage n-form. (2) If α is a Lepage (n + 1)-form then the dynamical form E = p1 α is locally variational. (3) Every Lepage equivalent α of a locally variational form E locally equals to dρ where ρ is a Lepage equivalent of a Lagrangian λ for E. Conversely, if ρ is a Lepage equivalent of a Lagrangian λ then dρ is a Lepage equivalent of the Euler–Lagrange form Eλ . Note that we can say that Lepage (n + 1)-forms are closed counterparts of variational equations. With help of Lepage (n + 1)-forms we realise that the difference between local and global variationality is the same as the difference between local and global exactness of (n+1)-forms. In the following theorem we denote by H n+1 (Y ) the (n+1)-st cohomology group of Y . We also note that due to the structure of the fibres of the bundle J s Y → Y , the cohomology groups of J s Y coincide with the corresponding cohomology groups of Y .

46

Olga Krupková

Theorem 4.6. (1) A locally variational form E is globally variational if and only if it has an exact Lepage equivalent. (2) If H n+1 (Y ) = {0} then every locally variational form on J s Y is globally variational. The question of the existence of Lepage equivalents of locally variational forms for general n and s is still open. However, it is known that if n ≥ 1 and the order of E is ≥ 2, a Lepage equivalent, if exists, is nonunique [53, 54], and splits into the sum α = αE + dη,

(4.7)

where αE is determined uniquely by the components Eσ of E. On the other hand, for n = 1, s arbitrary, and n > 1, s = 1 every locally variational form has a unique minimal-order Lepage equivalent [50] (cf. also [32]), and [29]: If n = 1, s ≥ 1, α = Eσ ω σ ∧ dt +

s−1 X

j,k=0

jk σ ωj ∧ ωkν , Fσν

(4.8)

where jk Fσν

1 = 2

s−j−k−1 X l=0

j+l

(−1)

j+l l

!

dl ∂Eσ , ν dtl ∂qj+k+l+1

(4.9)

jk = 0 otherwise. Note that the Lepage equivalent of a whenever 0 ≤ j + k ≤ s − 1, and Fσν dynamical form of order s is projectable onto J s−1 Y . If n > 1, s = 1, the unique minimal order Lepage equivalent of E is defined on Y and reads

α = E σ ω σ ∧ ω0 +

n X

k=1

1 ∂ k Eσ ω σ ∧ ω ν1 ∧ · · · ∧ ω νk ∧ ωj1 ···jk . (4.10) k!(k + 1)! ∂yjν11 · · · ∂yjνkk

We refer the reader to the article [27] by Grigore (in this volume), where the structure of closed (n + 1)-forms that are counterparts of variational equations (on higher order Grassmann bundles) is clarified.

4.3.

Lepage (n + 1)-Forms in the Geometry of Variational Equations

Lepage (n + 1)-forms are useful not only in the study of the inverse variational problem. They play an even more important role for investigations of the structure and symmetries of variational equations, and are essential for the formulation of exact integration methods. In this section we just mention very briefly a few important applications in the geometric theory of differential equations. However, for more details we refer to [29, 51, 52, 53, 54, 55, 56] and references therein. Let E be a locally variational form on J s Y (s ≥ 1). Given a Lepage equivalent α of E on J r Y , we define a Hamiltonian system of α to be an exterior differential system on J r Y , locally generated by n-forms iξ α, where ξ runs over all πr -vertical vector fields on J r Y .


47

Due to the nonuniqueness of α we usually have many Hamiltonian systems associated with a given locally variational form (Euler–Lagrange equations), however, all have “the same” holonomic integral sections. Moreover, for any Lepage equivalent α of E, holonomic integral sections of the Hamiltonian system of α coincide with (prolongations of) extremals of E. Therefore equations for holonomic integral sections of a Hamiltonian system, i.e., J r γ ∗ iξ α = 0, ∀πr -vertical vector fields ξ, (4.11) are Euler–Lagrange equations, and equations for (all) its integral sections, i.e., δ ∗ iξ α = 0, ∀πr -vertical vector fields ξ,

(4.12)

are called Hamilton equations. We stress that every Hamiltonian system of a locally variational form provides a complete family of extremals (global solutions of the Euler–Lagrange equations) and a complete family of Hamilton extremals (global solutions of Hamilton equations (even if the locally variational form is not globally variational). Now one can study Euler–Lagrange and Hamilton equations and their solutions from a geometric point of view by investigating properties of the corresponding exterior differential systems. Let us mention at least regularity (more generally classification of equations with respect to geometric properties of their solutions), and symmetries and conservations laws, including corresponding integration methods. Note that the above geometric representation by means of exterior differential systems demonstrates a deep difference between ordinary and partial differential equations: For n = 1 the Lepage equivalent of E is unique, hence to ordinary variational equations we have unique Hamilton equations, determined by E. This means that, in particular, we get a Hamiltonian and momenta determined by the Euler–Lagrange expressions (not by an individual Lagrangian). Moreover, the Hamiltonian EDS is locally generated by 1-forms, this means that Hamilton extremals are integral sections of a distribution, called the Euler– Lagrange distribution, uniquely determined by the Euler–Lagrange form E, and defined on J s−1 Y , where s is the order of E. We obtain a natural concept of a regular variational system: a locally variational form E is called regular if the rank of its Euler–Lagrange distribution = 1. Hence, we get the regularity condition expressed by means of the Euler– Lagrange expressions. For n > 1 Hamilton extremals are no longer described by a distribution, moreover, we have a family of Hamilton equations associated with an Euler–Lagrange form. Hamilton equations (Hamiltonians, momenta) are (similarly as in the case of dim X = 1) associated directly to the Euler–Lagrange form, not to individual Lagrangians. They do, however, depend upon a choice of a Lepage equivalent of E. In this case, regularity is understood as a property of a Hamiltonian EDS, i.e., again it does not depend upon a choice of an individual Lagrangian for E, but depends however upon a choice of α for E. Thus, in field theory, we can look for a regularisation of given variational equations: the aim is to find a Hamiltonian system for E that is regular. The EDS description also helps us look for symmetries and conservation laws for Euler– Lagrange and Hamilton equations. In particular, we can easily study symmetries of Lepage n and (n + 1)-forms (i.e., vector fields along which the Lie derivative of ρ, or α vanishes).

48

Olga Krupková

Moreover, every symmetry of a Lepage n-form ρ (resp. (n + 1)-form α) generates a conserved current (i.e. a closed n-form, belonging to the Hamiltonian EDS of dρ (resp. α)). For n = 1 this gives functions constant along extremals of E.

5.

Lepage Forms in the Variational Sequence

Within the theory of varational sequences developed by Krupka in [44], the concept of Lepage equivalent can be generalised to arbitrary k-forms, k ≥ n. Denote by Ωrq the sheaf q-forms on J r Y , Ωr0,c = {0}, and Ωrq,c the sheaf of contact q-forms if q ≤ n, respectively strongly contact q-forms if q > n, on J r Y . Set Θrq = Ωrq,c + dΩrq−1,c where dΩrq−1,c is the image sheaf of Ωrq−1,c by the exterior derivative d. There arises an exact sequence of soft sheaves 0 → Θr1 → Θr2 → Θr3 → · · · , where the morphisms are the exterior derivative, called contact sequence. It is a subsequence of De Rham sequence 0 → R → Ωr0 → Ωr1 → Ωr2 → Ωr3 → · · · . The quotient sequence 0 → R → Ωr0 → Ωr1 /Θr1 → Ωr2 /Θr2 → Ωr3 /Θr3 → · · · which is also exact, is called the r-th order variational sequence on π. It is important to stress that elements of the quotient sheaf Ωrq /Θrq are not forms, but classes of (local) qforms of order r. We denote by [ρ]rv an element of Ωrq /Θrq , that is the (variational) class of ρ ∈ Ωrq . The quotient mappings are denoted by Eqr : Ωrq /Θrq → Ωrq+1 /Θrq+1 . As proved by Krupka in [44], the variational sequence is an acyclic resolution of the constant sheaf R over Y . Due to the Abstract De Rham theorem, the cohomology groups of the cochain complex of global sections of the variational sequence are identified with the De Rham cohomology groups H q Y of Y . Quotient sheaves Ωrq /Θrq are determined up to natural isomorphisms of Abelian groups. In this way classes in Ωrq /Θrq admit representations by differential forms. Source forms [73] for the quotient sheaf Ωrq /Θrq arise by applying to q-forms the so-called interior Euler– Lagrange operator I [2, 35, 48]. Source forms for q = n and q = n + 1 are called Lagrangians and dynamical forms, respectively. In the source forms representation, the quotient mapping Enr : Ωrn /Θrn → Ωrn+1 /Θrn+1 coincides with the Euler-Lagrange mapping E : λ → Eλ . The next mapping is called Helmholtz mapping: if E is a dynamical form representing a class in Ωrn+1 /Θrn+1 then the image of E is an (n + 2)-form HE , called the Helmholtz form of E [44] (components of the Helmhotz form are the left-hand-sides of variationality conditions (4.6)). The variational sequence helps us to learn the structure of the Euler–Lagrange mapping and of null Lagrangians as follows: Condition HE = 0 for elements of the quotient sheaves r ([η]r ) = 0, and by exactness of the variational sequence means that there exists reads En+1 v a class [ρ]rv ∈ Ωrn /Θrn such that [η]rv = [dρ]rv . The source form λ = hρ for [ρ]rv is then a local Lagrangian for E. If H n+1 Y = {0}, ρ may be chosen globally defined on J r Y , so


49

we get a global Lagrangian for E. Similarly, if Eλ = 0, i.e., Enr ([η]rv ) = 0, there exists a class [ρ]rv ∈ Ωrn−1 /Θrn−1 such that [η]rv = [dρ]rv , for source forms, λ = hη = hdρ. If H n (Y ) = {0} then every null Lagrangian is globally null (there is a ρ defined on J r−1 Y ). The motivation for the generalisation of the concept of Lepage form is to obtain a representation of classes in the variational sequence such that the sequence morphisms would be the exterior derivatives [48, 57]: A q-form ρ, q ≥ n, is called a Lepage form if pq−n+1 dρ = Idρ (i.e. is a source form). If σ is a source q-form, we say that ρ is a Lepage equivalent of σ if ρ is a Lepage q-form and pq−n ρ = σ. Note that for q = n, and q = n + 1 and σ locally variational, we get definitions introduced in the previous sections. Summarising, we can express the source form representation and the Lepage form representation of the variational sequence as follows: En−1

En+1

E

En+2

· · · −−−−→ Ωrn /Θrn −−−n−→ Ωrn+1 /Θrn+1 −−−−→ Ωrn+2 /Θrn+2 −−−−→ · · · · · · −−−−→ d

· · · −−−−→

 

Iy

Ωsn

  Lepy

Ωkn

E

−−−−→ d

−−−−→

 

Iy

Ω2s n+1   Lepy

Ωkn+1

H

−−−−→ d

−−−−→

 

Iy

Ω4s n+2   Lepy

Ωkn+2

−−−−→ · · · d

−−−−→ · · ·

Generalisations of Lepage forms are studied also within the theory of non-holonomic systems [60, 72].

Acknowledgments ˇ 201/06/0922 of the Czech Science Foundation, and Research supported by grants GACR MSM 6198959214 of the Czech Ministry of Education, Youth and Sports. The author also highly appreciates support and hospitality of the Mathematics Department and the Institute for Advanced Study at La Trobe University, Melbourne, Australia.

References [1] V. Aldaya and J. de Azcárraga, Higher order Hamiltonian formalism in field theory, J. Phys. A: Math. Gen. 13 (1982) 2545–2551. [2] I. Anderson, The Variational Bicomplex (Utah State University, Technical Report, 1989). [3] I. Anderson and T. Duchamp, On the existence of global variational principles, Am. J. Math. 102 (1980) 781–867. [4] D. E. Betounes, Extension of the classical Cartan form, Phys. Rev. D 29 (1984) 599– 606. ´ Cartan, Leçons sur les invariants intégraux (Hermann, Paris, 1922). [5] E.

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Chapter 3

O N A G ENERALIZATION ´ -C ARTAN F ORM OF THE P OINCAR E IN H IGHER -O RDER F IELD T HEORY D.R. Grigore∗ Dept. of Theor. Phys., National Inst. Phys. Nucl. Engeneering “Horia Hulubei”, Bucharest-M˘agurele, P. O. Box MG 6, România

Abstract We present here a possible generalization of the Poincaré-Cartan form in classical field theory to the most general case: arbitrary dimension, arbitrary order of the theory and the absence of a fiber bundle structure. We use for the kinematical description of the system the (r, n)-Grassmann manifold associated to a given manifold X, i.e. the manifold of r-contact elements of n-dimensional submanifolds of X. The idea is to define globally a n + 1 form on this Grassmann manifold, more precisely its class with respect to a certain subspace and to write it locally as the exterior derivative of a n form which is a kind of Poincaré-Cartan form in the higher-order and non-fibred situation.

2000 Mathematics Subject Classification. 53A55, 77S25, 58A20. Key words and phrases. Grassmann bundles, Lagrangian Formalism.

1.

Introduction

It is widely accepted that the variational principles should be given in a coordinate independent formulation. This idea was first realized for a dynamical system with a finite number of degrees of freedom (i.e. particle mechanics), using a differential 1-form instead of the Lagrangian, by Poincaré and Cartan [43], [8]. There are a number of generalizations of this idea for classical field theory [27], [5], [6], [44], [45], [24], [11], [10], [12], [46], [42]. A related concept is that of Lepage equivalent of a Lagrangian form (see for instance [28], [29], ∗

E-mail address: [email protected], [email protected]

58

D.R. Grigore

[32]). All these generalizations use as geometric framework for classical field theory the jet bundle formalism (more explicitly the space-time and field variables are local coordinates on a fiber bundle X over a “space-time manifold” M ) and the derivative of the fields, up to order s, are variables in the s-th order jet bundle extension J s X of X. It was later [26], [47] suggested that it is more convenient to work with the exterior differential of the above Poincaré-Cartan form. For the case of finite number of degrees of freedom this 2-form is in general presymplectic and was used by Souriau and others [47], [23] to obtain the phase space as a symplectic manifold in a deductive way. The idea is to consider that the fundamental mathematical object for a Lagrangian system must be this 2-form and not the Lagrangian function or the Poincaré-Cartan 1-form. This point of view leads to the main features of the Lagrangian and the Hamiltonian formalism and to a natural definition of the Noetherian symmetries. For higher-order mechanics this approach has been developed in [33], [34], [35] (see also [38]). One can generalize this Lagrange-Souriau form without using the fibration hypothesis mentioned above in two particular but important cases: for classical field theory of first order [13] and for systems with a finite number of degrees of freedom and of arbitrary order [14]. Moreover, this Lagrange-Souriau form can be locally written as the exterior differential of a Poincaré-Cartan form related to some chosen local chart. This Poincaré-Cartan form is the same as that given by Krupka [27], Betounes [5]-[6] and Rund [44]. The Lagrangian is locally determined up to a variationally trivial Lagrangian, i.e. a Lagrangian giving trivial Euler-Lagrange equations. As a consequence, one can define in a geometrically nice way the Noetherian symmetries using the Lagrange-Souriau form. In this paper we give a generalization of the Lagrange-Souriau and of the PoincaréCartan forms in the most general case used in higher order field theory. We consider an arbitrary manifold X without a fiber bundle structure over some space-time manifold so instead of the s-th order jet bundle extension one must use the s-th order Grassmann bundle Pns X associated to X which was recently considered in the literature [20]. In the next Section we will summarize the main features of this construction. Next, in Section 3, we will be able to define globally a n + 1 differential form, but one will be able to see that, in general, one cannot determine this form uniquely. Fortunately one can consider the equivalence class of this form to a certain globally defined subspace of differential forms. This equivalence class is the “physical” object we are looking for. It is interesting to note that this subspace of differential forms is in fact identically zero exactly in the two particular cases mentioned above (s = 2, n arbitrary and n = 1, s arbitrary). Some combinatorial tricks introduced in [16] must be used to simplify the analysis of some tensorial identities. In Section 4 we locally exhibit the n + 1 differential form as the exterior derivative of a locally defined Poincaré-Cartan n-form and in this way the (local) Lagrangian function appears also. Some functions called hyper-Jacobians [7], [41] emerge naturally in this context and can be used to provide the most general expression for a variationally trivial Lagrangian of arbitrary order already obtained in [17] by a different method. Finally, in Section 5 we present two particular but very important cases, namely n = 1 s arbitrary and s = 2 n arbitrary. Some ideas related to the ones from this paper also appear in [31]. In the fibered situation, similar (n + 1)-forms were studied in [39], [40] and [22]. We will skip some proofs which are similar to the proofs from [16]-[19] and can be found in the web version of this paper [15].

On a Generalization of the Poincaré-Cartan Form

2. 2.1.

59

Grassmann Manifolds The Basic Constructions of the Grassmann Manifolds

In this Section we present the basic construction of Grassmannian manifolds following [20] and [18]. We will skip all the proofs. We consider N , n ≥ 1 and r ≥ 0 integers such that n ≤ N , and let X be a smooth manifold of dimension N which is the mathematical model for the kinematical degrees of freedom of a certain classical field theory. Let U ⊂ Rn be a neighborhood of the point 0 ∈ Rn , x ∈ X and let Γ(0,x) be the set of smooth maps γ : U → X such that γ(0) = x. On Γ(0,x) one has the the equivalence relationship “γ ∼ δ” iff there exists a chart (V, ψ) ψ = (xA ), A = 1, . . . , N on X such that the functions ψ ◦ γ, ψ ◦ δ : Rn → RN have the same partial derivatives up to order r in the point 0. The equivalence class of γ will be denoted by j0r γ and it is called a r (r, n)-velocity. The set of (r, n)-velocities at x is denoted by T(0,x) (Rn , Y ) ≡ Γ(0,x) / ∼. We denote [ r Tnr X = T(0,x) (Rn , X), x∈X

and define surjective mappings τnr,s : Tnr X → Tns X, where 0 < s ≤ r, by τnr,s (j0r γ) = j0s γ and τnr,0 : Tnr X → X, where 1 ≤ r, by τnr,0 (j0r γ) = γ(0). If (V, ψ), ψ = (xA ), is a chart on X we define the couple (Vnr , ψnr ) where Vnr = r,0 A (πn )−1 (V ), ψnr = (xA , xA j , · · · , xj1 ,j2 ,...,jr ) 1 ≤ j1 ≤ j2 ≤ · · · ≤ jr ≤ n, and r xA j1 ,...,jk (j0 γ)

∂k A x ◦ γ ≡ j1 , ∂t . . . ∂tjk 0

0 ≤ k ≤ r.

(2.1)

r The expressions xA j1 ,···jk (j0 γ) are defined for all indices j1 , . . . , jr in the set {1, . . . , n} but because of the symmetry property r A r xA jP (1) ,...,jP (k) (j0 γ) = xj1 ,...,jk (j0 γ)

(k = 2, ..., n)

(2.2)

for all permutations P ∈ Pk of the numbers 1, . . . , k we consider only the independent components given by the restrictions 1 ≤ j1 ≤ j2 ≤ · · · ≤ jr ≤ n. This allows one to use A multi-index notations i.e. ψnr = (xA |J| = 0, ..., r where by definition xA J ), ∅ ≡ x . The same comment is true for the partial derivatives ∂xA ∂ . j1 ,...,jk

The couple (Vnr , ψnr ) is a chart on Tnr X called the associated chart of the chart (V, ψ) and the system of charts give a smooth structure on this set; moreover Tnr X is a fiber bundle over X with the canonical projection τnr,0 . The set Tnr Y endowed with the smooth structure defined by the associated charts defined above is called the manifold of (r, n)-velocities over X. In the chart (Vnr , ψnr ) one introduces the following differential operators: ∆jA1 ,...,jk ≡

r1 ! . . . rn ! ∂ , k! ∂xA j1 ,...,jk

j1 , . . . , jk ∈ {1, . . . , n}

where rk is the number of times the index k shows up in the sequence j1 , . . . jk .

(2.3)

60

D.R. Grigore The combinatorial factors are such that the following relation is true: ∆iA1 ,...,ik xB j1 ,...,jl

=

ik i1 BS+ δA j1 ,...,jk δj1 . . . δjk 0

(

if k = l if k 6= l.

(2.4)

Here we use the notations from [14], namely Sj±1 ,...,jk are the symmetrization (for the sign +) and respectively the antisymmetrization (for the sign −) projector operators defined by 1 X Sj±1 ,...,jk fj1 ,...,jk ≡ ǫ± (P )fjP (1) ,...,jP (k) (2.5) k! P ∈P k

where the sum runs over the permutation group Pk of the numbers 1, . . . , k and ǫ+ (P ) ≡ 1,

ǫ− (P ) ≡ (−1)|P | ,

∀P ∈ Pk ;

here |P | is the signature of the permutation P . In this way one takes care of overcounting the indices. More precisely, for any smooth function on V r , the following formula is true: df =

r X

(∆jA1 ,...,jk f )dxA j1 ,...,jk =

k=0

X

(∆IA f )dxA I

(2.6)

|I|≤r

where we have also used the convenient multi-index notation. The formal derivatives are: Dir ≡

r−1 X

j1 ,...,jk xA = i,j1 ,...,jk ∆A

k=0

X

J xA iJ ∆A .

(2.7)

|J|≤r−1

The last expression uses the multi-index notation; if I and J are two such multi-indices we mean by IJ the juxtaposition of the two sets I, J. When no danger of confusion exists we simplify the notation putting simply Di = Dir . The formal derivatives give a conveniently expression for the change of charts on the velocity manifold induced by a change of charts on X. By definition the differential group of order r is the set r Lrn ≡ {j0r α ∈ J0,0 (Rn , Rn )|α ∈ Dif f (Rn )}

(2.8)

i.e. the group of invertible r-jets with source and target at 0 ∈ Rn . The group multiplication in Lrn is defined by the jet composition Lrn × Lrn ∋ (j0r α, j0r β) 7→ j0r (α ◦ β) ∈ Lrn . The canonical (global) coordinates on Lrn are defined by aij1 ,...,jk (j0r α)

∂ k αi = j1 , ∂t . . . ∂tjk 0

j1 ≤ j2 ≤ · · · ≤ jk ,

where αi are the components of a representative α of j0r α. We denote a ≡ (aij , aij1 ,j2 , . . . , aij1 ,...,jk ) = (aiJ )1≤|J|≤r

k = 0, ..., r

(2.9)


61

and notice that one has det(aij ) 6= 0.

(2.10)

A r (x · a)A I ≡ xI (j0 (γ ◦ α))

(2.11)

The group Lrn is a Lie group. The manifolds of (r, n)-velocities Tnr Y admits a (natural) smooth right action of the differential group Lrn , defined by the jet composition

i where the connection between xA I and γ is given by (2.1) and the connection between aI and α is given by (2.9). The group Lrn has a natural smooth left action on the set of smooth real functions defined on Tnr X , namely for any such function f we have:

(a · f )(x) ≡ f (x · a).

(2.12)

We say that a (r, n)-velocity j0r γ ∈ Tnr X is regular, if γ (or any other representative) is an immersion. We have the central result: Theorem 2.1. The set Pnr X ≡ ImmTnr X/Lrn has a unique differential manifold structure such that the canonical projection ρrn is a submersion. The group action defines on ImmTnr X the structure of a right principal Lrn -bundle. A point of Pnr X containing a regular (r, n)-velocity j0r γ is called an (r, n)-contact element, or an r-contact element of an n-dimensional submanifold of X, and is denoted by [j0r γ]. As in the case of r-jets, the point 0 ∈ Rn (resp. γ(0) ∈ X) is called the source (resp. the target) of [j0r γ]. The manifold Pnr is called the (r, n)-Grassmannian bundle, or simply a higher order Grassmannian bundle over X. Besides the quotient projection ρrn : ImmTnr X → Pnr we have for every 1 ≤ s ≤ r, the r s canonical projection of Pnr X onto Pns X defined by ρr,s n ([j0 γ]) = [j0 γ] and the canonical r r r projection of Pn X onto X defined by ρn ([j0 γ]) = γ(0). On Pnr X there are total differential operators; as expected, in the chart ρrn (W I,r ) they have the expression: ∂ r1 ! . . . rn ! (2.13) ∂σj1 ,...jk ≡ σ k! ∂yj1 ,...,jk We note for further use the following formula: ∂σi1 ,...,ik y¯jν1 ,...,jk = Sj+1 ,...,jk Qij11 . . . Qijkk Qνσ ,

k = 1, ..., r;

(2.14)

here we have defined: Qσν ≡ ∂ν y¯σ − y¯iσ (∂ν x ¯i )

(2.15)

and Q is the inverse of the matrix: Pji ≡ dj x ¯i ,

Pji Qjl = δli .

(2.16)

Next we define the total derivative operators on the Grassmann manifold: di ≡

r−1 X X ∂ ∂ σ σ J + yi,j ∂ j1 ,...,jk = + yiJ ∂σ . 1 ,...,jk σ i i ∂x ∂x k=0 |J|≤r−1

(2.17)

62

D.R. Grigore We note that: (ρrn )∗ (zij Dj ) = di .

(2.18)

In particular, we have for any smooth function f on ρrn (W r ) the following formula: Di (f ◦ ρrn ) = xji (dj f ) ◦ ρrn .

(2.19)

The formula for the chart change on Pnr X. can be written with this operators: let us consider two overlapping charts: (ρrn (V r ), (xi , yIσ )) and respectively (ρrn (V¯r ), (¯ xi , y¯Iσ )); then we have on the overlap: σ y¯iI = Qji dj y¯Iσ . |I| ≤ r − 1

(2.20)

Pij d¯j = di .

(2.21)

We also note that:

2.2.

Contact Forms on Grassmann Manifolds

By a contact form on Pnr X we mean any form ρ ∈ Ωrq (P X) verifying [j r γ]∗ ρ = 0

(2.22)

for any immersion γ : Rn → X. We denote by Ωrq(c) (P X) the set of contact forms of degree q ≤ n. Here [j r γ] : Rn → Pnr is given by: [j r γ] (t) ≡ [jtr γ] . We mention some of properties verified by these forms. If one considers only the contact forms on an open set ρrn (V r ) ⊂ Pnr X then we emphasize this by writing Ωrq(c) (V ). The ideal of all contact forms is denoted by C(Ωr ). By elementary computations one finds out that, as in the case of a fiber bundle, for any chart (V, ψ) on X, every element of the set Ωr1(c) (V ) is a linear combination of the following expressions: σ ωjσ1 ,...,jk ≡ dyjσ1 ,...,jk − yi,j dxi , k = 0, ..., r − 1 (2.23) 1 ,...,jk or, in multi-index notations: σ ωJσ ≡ dyJσ − yiJ dxi ,

|J| ≤ r − 1.

(2.24)

σ dωJσ = −ωJi ∧ dxi ,

|J| ≤ r − 2.

(2.25)

We have the formula

Any form ρ ∈ Ωrq (P X), q = 2, ..., n is contact iff it is generated by ωJσ , |J| ≥ r−1 and dωIσ , |I| = r − 1. In the end we present the transformation formula relevant for change of charts. ¯ two overlapping charts on X and let (W r , Φr ), Proposition 2.2. Let (V, ψ) and (V¯ , ψ) r i σ i r r r ¯ ,Φ ¯ ), Φ¯ = (¯ Φ = (x , yI , xI ) and (W xi , y¯Iσ , x ¯iI ) the corresponding charts on Tnr X. ¯ r ) ⊂ P r X: Then the following formula is true on ρrn (W r ∩ W n ω ¯ Iσ =

|I| X

|J|=1

(∂νJ y¯Iσ )ωJν − QσI,ν ω ν ,

1 ≤ |I| ≤ r − 1.

(2.26)


63

where we have defined: σ QσI,ν ≡ ∂ν y¯Iσ − y¯jI (∂ν x ¯j ),

0 ≤ |I| ≤ r − 1

(2.27)

and ω ¯ σ = Qσν ω ν

(2.28)

where Qσν is given by the formula (2.15). As a consequence we have: Corollary 2.3. If for a q-form has the expression X

ρ=

X

X

p+s=k |J1 |,...,|Jp |≤r−1 |I1 |=...=|Is |=r−1 σ ωJσ11 · · · ∧ ωJpp ∧ dωIν11

p ,I1 ,...,Is · · · ∧ dωIνss ∧ ΦJσ11,...,J ,...,σp ,ν1 ,...,νs ,

k ≤ q (2.29)

is valid in one chart, then it is valid in any other chart. This corollary allows us to define for any q = 1, dots, dim(J r Y ) = m n+r a contact n form with order of contactness k to be any ρ ∈ Ωrq such that it has in one chart (thereafter in any other chart) the expression above. We denote these forms by Ωrq,k .

3.

A Lagrange-Souriau Form on a Grassmann Manifold

3.1.

Some Invariant Conditions

As in the preceding Section we consider a differential manifold X and the associated (s, n)Grassmann manifold Pns X. We start we the following general result: Proposition 3.1. Let α ∈ Ωsq (X) a q-differential form on Pns X verifying: iξ α = 0

(3.1)

for any ρs,s−1 -vertical vector field ξ on Pns X (i.e. (ρs,s−1 )∗ ξ = 0). Then this form has the n n local expression: α=

q X

X

k=0 |I1 |,...,|Ik |≤s−1

,...,Ik TσI11,...,σ ω σ1 ∧ · · · ∧ ωIσkk ∧ dxik+1 ∧ · · · ∧ dxiq k ,ik+1 ,...,iq I1

(3.2)

,...,Ik where TσI11,...,σ are smooth functions depending on the variables k ,ik+1 ,...,iq σ i σ σ (x , y , yj , . . . , yj1 ,...,js ) and are antisymmetric in the couples (Ip , σp ) p = 1, . . . k and in the indices ik+1 , . . . , iq .

We denote the space of these forms by Ωsq,ξ (X). By elementary computations from (2.26) and (2.28) we can obtain the transformation formulas for the coefficients ,...,Ik TσI11,...,σ in the overlap of two charts. As a consequence of these transformation k ,ik+1 ,...,iq

64

D.R. Grigore

,...,Ik formulas we can prove that some constraints on the coefficients TσI11,...,σ of the k ,ik+1 ,...,iq form α are in fact globally defined. These relations are: ,...,Ik TσI11,...,σ = 0, k ,ik+1 ,...,iq

|I1 | + · · · + |Ik | ≥ t

TσI11,i2 ,...,iq = 0,

∀t ∈ N,

∀I1 6= ∅.

Ti1 ,...,iq = 0.

(3.3) (3.4) (3.5)

and the tracelessness condition ,...,Ik TσlI11,...,σ =0 k ,l,ik+2 ,...,iq

k = 1, . . . , q.

(3.6)

Some notations will be usefull. We denote the subset of the forms verifying (3.3) by Ωs,t q,ξ (X). If the condition (3.4) is fulfilled we say that the form α verifies the Lepage condition. We denote the subset of the forms verifying the conditions (3.3) and (3.4) by Ωs,t,Lep (X) and q,ξ s,t s Ωs,t q,ξ,k (X) ≡ Ωq,k (X) ∩ Ωq,ξ (X);

these are contact forms with the order of contactness equal to k and the definition is globally true. We call the subset of the forms verifying the conditions (3.3), (3.4) (i.e Lepage) and (3.6) (i.e. tracelessness) by Ωs,t,Lep q,ξ,tr (X). We finally stress again that all the spaces of the type Ω... are globally defined. ...

3.2.

The Definition of the Lagrange-Souriau Form

We need to consider one more condition on the Lepage forms, namely closeness. First we have: s,s−1 -projectable, i.e. there Proposition 3.2. Let α ∈ Ωs,s−1 q,ξ (X) be closed. Then α is ρn exists a q-form α0 ∈ Ωs−1 q (X) such that

α = (ρs,s−1 )∗ α0 . n

(3.7)

Moreover, the form α0 is closed. Proof. We exhibit the dependence of the form α on the highest-order derivatives; according to the preceding corollary these derivatives can appear in two places: in the coefficients Tσ∅,...,∅ , k = 0, . . . , q and in the contact forms ωIσ , |I| = s − 1. It is not very 1 ,...,σk ,ik+1 ,...,iq hard to write now α as follows: α=

q X

k=0 q X

X

k=1 |I1 |=s−1

Tσ∅,...,∅ ω σ1 ∧ · · · ∧ ω σk ∧ dxik+1 ∧ · · · ∧ dxiq + 1 ,...,σk ,ik+1 ,...,iq

,∅,...,∅ kTσI11,...,σ ω σ1 ∧ ω σ2 · · · ∧ ω σk ∧ dxik+1 ∧ · · · ∧ dxiq + α′ k ,ik+1 ,...,iq I1

-projectable. where the form α′ is ρs,s−1 n The expression above is, as said before, at most linear in the highest-order derivatives. One computes explicitly the coefficient of this derivatives and finds out that they are zero. This proves the first assertion. The closeness of the form α0 follows from the surjectivity . of the map ρs,s−1 n


65

s,s−1,Lep It is clear that there are strong conditions on any closed form α ∈ Ωq,ξ,tr (X). We will give a structure theorem for such a form in the case q = n + 1 which is relevant for physical applications. First we note that in this case we have ∅ Tσ,i = Tσ εi1 ,...,in 1 ,...,in

(3.8)

for some smooth functions Tσ on Pns X. Here εi1 ,...,in is the completely antisymmetric tensor. Then the general structure formula is: Theorem 3.3. Let α ∈ Ωs,s,Lep n+1,ξ,tr (X) be closed. Then α admits the following decomposition: α = T0 + dT1 (3.9) where: s,s−1 - T1 ∈ Ωs,s−2 n,ξ,2 (X) and dT1 ∈ Ωn+1,ξ,2 (X).

- T0 ∈ Ωs,s−1,1 n+1,ξ (X) has the local structure given by the formula (3.2) with the tensors Tk given by formulas of the type Tk = Pk Tσ ,

k = 2, . . . , n + 1

(3.10)

with Pk some linear differential operators which can be recursively determined. The proof relies heavily on induction on k and uses some creation and annihilation operators introduced in [16] and [17]. As a corollary of this formula we can make now the connection with the Lagrangian formalism. Namely, we have: Corollary 3.4. The expressions Tσ defined according to (3.8) verify the generalized Helmholtz equations. Proof. We write explicitly the closeness condition and select only those equations contain∅ ing the expressions Tσ,i . As a result one obtains the following set of equations 1 ,...,in ∂σI 1 Tσ2

|I|

= (−1)

X

|J|

(−1)

|J|≤s−|I|

!

|J| + |I| dJ ∂σIJ2 Tσ1 |J|

(3.11)

for |I| = s, |I| = 1, . . . , s−1 and |I| = 0 respectively. But (3.11) are exactly the Helmholtz equations (see [1], [3], [4], [9], [30]). From the theorem above we also obtain: Proposition 3.5. The decomposition (3.9) proved in the preceding theorem determines in an unique way the form T0 . Proof. It is sufficient to prove that α = T0 + dT1 = 0 =⇒ T0 = 0. Indeed, from α = 0 we have in particular Tσ,i1 ,...,in = 0 and it follows from the formula (3.10) that we have T0 = 0.

66

D.R. Grigore

As a corollary, let us denote by [α] the equivalence class of the form α ∈ Ωs,s,Lep n+1,ξ,tr (X) s,s−2 s,s−1 modulo dΩn,ξ,2 (X) ∩ Ωn+1,ξ,2 (X). Then we have [α] = 0 ⇐⇒ Tσ = 0

(3.12)

which says that the class of α is uniquely determined by the so-called Euler-Lagrange components of α: Tσ , σ = 1, . . . , m ≡ N − n. We call the globally defined class [α] of a certain form α ∈ Ωs,s,Lep n+1,ξ,tr (X) a LagrangeSouriau class. We note in closing this Section that there are two particular but important cases when the class of the form α is formed only from the form α; obviously this happens when s,s−1 dΩs,s−2 (3.13) n,ξ,2 (X) ∩ Ωn+1,ξ,2 (X) = 0. One can see that if s = 2, and n arbitrary, or n = 1, and s arbitrary, the equality above becomes an identity. So in this cases one can speak of a globally defined Lagrange-Souriau form as in [13], [21] and [14] respectively.

4.

The Associated Poincaré-Cartan n-Form

4.1.

The General Construction

Because the Euler-Lagrange expressions Tν are at most linear in the higher-order derivatives i h . yiσ1 ,...,is it is to be expected that they follow from a Lagrangian of minimal order r ≡ s+1 2 We prove this fact in this Section. The key observation is Proposition 4.1. Let α ∈ Ωs,s q+1,ξ (X) be closed. Then one can write it locally in the form α = dθ

(4.1)

where θ ∈ Ωsn (X) is a ρs,s−1 -projectable form and has the coordinates expression: n θ=

q X

X

k=0 |I1 |,...,|Ik |≤r−1

σk σ1 ik+1 k ∧ · · · ∧ dxiq LσI11,...,I ,...,σk ,ik+1 ,...,iq ωI1 ∧ · · · ∧ ωIk ∧ dx

(4.2)

k where LIσ11,...,I |I1 |, . . . , |Ik | ≤ r − 1 are smooth functions depending on the ,...,σk ,ik+1 ,...,iq , i σ σ σ variables (x , y , yj , . . . , yj1 ,...,js ) and verify the (anti)symmetry properties in the couples (Ip , σp ) p = 1, . . . k and in the indices ik+1 , . . . , iq .

Proof. Let us use the proposition 3.2 and write α ∈ Ωs,s q+1,ξ (X) in the form (3.7). It is easy to see that α0 has the following generic form: α0 =

q+1 X

X

k=0 |I1 |+···+|Ik |≤s−1

σk σ1 ik+1 k AIσ11,...,I ∧ · · · ∧ dxiq+1 ,...,σk ,ik+1 ,...,iq+1 dyI1 ∧ · · · ∧ dyIk ∧ dx

k where AIσ11,...,I |I1 |, . . . , |Ik | ≤ s − 1 are smooth functions depending on the ,...,σk ,ik+1 ,...,iq+1 , i σ σ σ variables (x , y , yj , . . . , yj1 ,...,js ) and verify the (anti)symmetry properties in the couples


67

(Ip , σp ) p = 1, . . . k and in the indices ik+1 , . . . , iq . In particular the differentials dyIσ , |I| ≥ r can appear at most once in every term of the preceding sum because if there would exists a term with at least two such differentials we would get a contradiction according to the obvious inequality 2r ≥ s. So, one can write α0 = β +

X

|I|≥r

dyIσ ∧ ασI

(4.3)

where β and ασI are (q + 1) (resp. q) forms which do not contain the differentials dyIσ , |I| ≥ r. But the form α0 is closed (see proposition 3.2) so one can write it, locally, as follows: α0 = dθ0

(4.4)

with θ0 having a structure similar to (4.3): θ0 = γ +

X

|I|≥r

dyIσ ∧ θσI ;

(4.5)

here γ and θσI are q (resp. (q − 1)) forms which do not contain the differentials dyIσ , |I| ≥ r. If we substitute (4.3) and (4.5) into (4.4) we easily obtain the consistency condition: ∂νJ θσI = ∂σI θνJ ,

∀|I|, |J| ≥ r.

Applying the usual Poincaré lemma one gets from here that θσI , following expression θσI = ∂σI λ, |I| ≥ r

where λ is a (q − 1)-form which does not contain the differentials dyIσ , substitute this into the expression (4.5) above and gets: θ0 = γ + dλ −

X

|I|≤r

dyIσ ∧ ∂σI λ − dxi ∧

|I| ≥ r have the |I| ≥ r. Now one

∂λ . ∂xi

It follows that one can take in (4.4) θ0 = γ −

X

|I|≤r

dyIσ ∧ ∂σI λ − dxi ∧

∂λ ∂xi

without affecting it. But it is clear that this form has the structure θ0 =

q X

X

k=0 |I1 |+···+|Ik |≤r−1

σk σ1 ik+1 k BσI11,...,I ∧ · · · ∧ dxiq . ,...,σk ,ik+1 ,...,iq dyI1 ∧ · · · ∧ dyIk ∧ dx

If we define θ ≡ (ρs,s−1 )∗ θ0 then we have the equality from the statement. n If q = n then we have similarly to (3.8): Li1 ,...,in = εi1 ,...,in L

(4.6)

where L is a smooth real function called the (local) Lagrangian. Now we have a result similar to theorem 2 from [32], ch. 3.2, but as we can see the proof is much simpler and do not make use of the Young diagrams technique.

68

D.R. Grigore

Theorem 4.2. Let α ∈ Ωs,s n+1,ξ (X) be closed and verifying the Lepage condition (3.4). Suppose that we have written it as in proposition 4.1. Then the following formula is true: θ = θ0 + dλ + µ

(4.7)

where θ0 ≡ n!Ldx1 ∧ · · · ∧ dxn + n also λ ∈

Ωs,r−2 n−1,ξ,1

and µ ∈

(−1)|J| dJ ∂σI11Ji1 Li1 ,...,in ωIσ11 ∧ dxi2 ∧ · · · ∧ dxin ;

X

|J|≤r−1−|I1 |

(4.8)

Ωs,s−1 n,ξ,2 .

Proof. We start from the formula (4.2) obtained before and notice that the term corresponding to k = 1 can be written as follows: LIσ11 ,i2 ,...,in ωIσ11 ∧ dxi2 ∧ · · · ∧ dxin =n

X

(−1)|J| dJ ∂σI11Ji1 Li1 ,...,in ωIσ11 ∧ dxi2 ∧ · · · ∧ dxin + dλ

|J|≤r−1−|I1 |

where λ≡

X

|I1 |≤r−2

ΛIσ11 ,i2 ,...,in−1 ωIσ11 ∧ dxi2 ∧ · · · ∧ dxin−1 .

is a (n − 1)-form with the order of contactness equal to 1. Now we define the form µ to be the sum of the terms corresponding to the contributions k ≥ 2 in the expression (4.2); this gives us a n-form with the order of contactness equal to 2. The formula from the statement follows. We prove now that the Euler-Lagrange expressions Tσ are following from a Lagrangian of order r. Proposition 4.3. In the conditions from above the following result is true: Tσ =

X

(−1)|J| dJ ∂σJ L.

(4.9)

|J|≤r

Proof. We have by direct computation ∅ Tσ,i = ∂σ Li1 ,...,in + 1 ,...,in

n 1X (−1)p dip L∅σ,i ,...,iˆ ,...,i p n 1 n p=1

(4.10)

and L∅σ1 ,i2 ,...,in = n

X

(−1)|J| dJ ∂σJi1 1 Li1 ,...,in + (δΛ)∅σ1 ,i2 ,...,in .

(4.11)

|J|≤r−1

If we substitute the second relation into the first one, we obtain the formula from the statement.


69

Let us comment this result. First we can say that because the expressions Tσ have the usual Euler-Lagrange expression, they verify the generalized Helmholtz equations, so we have an alternative proof of the corollary 3.4. Next, we notice that in fact we have a sharper result, namely the expressions Tσ follow from a Lagrangian of order r which is the minimal possible order. Indeed, if Tσ would follow from a Lagrangian of order strictly smaller than r, the the Euler-Lagrange equations would have the order strictly smaller than s which would contradict the basic stating point of our analysis. So, we can say that we have obtained above a form of the conjecture regarding the reduction to the minimal order in the higher-order Lagrangian formalism. We close this Subsection with the following result. Proposition 4.4. In the conditions of the proposition 4.1, let us suppose that q = n and k moreover that the tensors LIσ11,...,I |I1 | = · · · = |Ik | = r − 1, k = 1, . . . , n ,...,σk ,ik+1 ,...,in , are traceless. The we have the following formula: k LIσ11,...,I ,...,σk ,ik+1 ,...,in

=

!

n k (r − 1)! ∂ i1 I1 · · · ∂σikkIk Li1 ,...,in , r k (k + r − 1)! σ1

|I1 | = · · · = |Ik | = r − 1,

k = 0, . . . , n

(4.12)

The proof goes by induction and is based on the condition (3.6). We do not give the details but we only mention that the preceding formula appears also in [31].

4.2.

Hyper-Jacobians and Variationally Trivial Lagrangians

By definition, the hyper-Jacobians of order s are the following expressions: σ ,...,σ ,ik+1 ,...,in

JI11,...,Ik k

≡ εi1 ,...,in

k Y

yIσllil ,

|I1 | = · · · = |Ik | = s − 1,

l=1

k = 0, . . . , n. (4.13)

Then we have a result of combinatorial nature. Proposition 4.5. In the conditions of proposition 4.1 we have for k ≥ 2 and |I1 | + · · · + |Ik | ≥ r the following formulas: – for q ≤ n: k LIσ11,...,I ,...,σk ,ik+1 ,...,iq

X

|Ik+1 |=···=|Il |=r−1

k(q+1)

= (−1)

σ

q

!

X l 1 n εi1 ,...,in n! q − k k l=k

!

,...,σ ,jl+1 ,...,jq ,i1 ,...,ik ,iq+1 ,...,in

k+1 l l LIσ11,...,I ,...,σl ,jl+1 ,...,jq JIk+1 ,...,Il

;

(4.14)

– for q ≥ n: k LIσ11,...,I ,...,σk ,ik+1 ,...,iq

X

|Ik+1 |=···=|Il |=r−1

k(n+1)

= (−1)

!

q

X l 1 n εiq−n+1 ,...,iq n! q − k k l=k σ

,...,σ ,jl+1 ,...,jq ,iq−n+1 ,...,ik

k+1 l l LIσ11,...,I ,...,σl ,jl+1 ,...,jq JIk+1 ,...,Il

!

(4.15)

70

D.R. Grigore σ

,...,σ ,j

,...,j

q k+1 l l+1 l where JIk+1 are the hyper-Jacobians of order r and LIσ11,...,I ,...,σl ,jl+1 ,...,jq , ,...,Il 1 σ |I1 |, . . . , |Ik | ≤ r − 1 are tensors depending on the variables (x , y , . . . , yjσ1 ,...,jr−1 ) and with the symmetry properties in the couples (Ip , σp ) p = 1, . . . k and in the indices ik+1 , . . . , iq . If s = 2r − 1 then the formulas above are valid for k = 1 also.

Based on the preceding two results we can give a new proof of an important result from [17] namely that a(local) Lagrangian L of order r which is variationally trivial (i.e. the associated Euler-Lagrange expressions are identically zero) must be a linear combination of hyper-Jacobians of order r (however the coefficients are not completely arbitrary).

5.

Two Particular Cases

In this Section we present two particular case which do to have physical relevance. We will obtain, essentially, known results but we think that it is profitable to see how they follow as particularizations of the main framework developed in this paper. There will also be some refinements of these old results.

5.1.

The Case s = 2 and n Arbitrary

If we particularize in this case the main theorem of Section 3, we get: Theorem 5.1. Let X be a differential manifold of dimension N > n and let Pn2 X be the second order Grassmann manifold associated to it. Let α ∈ Ω2,2 n+1 (X) be closed and verifying the Lepage condition (3.4) and the tracelessness condition (3.6). Then α has the following local expression: α=

n X

k=1

Tσi00 ,...,σk ,ik+1 ,...,in ωiσ00 ∧ ω σ1 ∧ · · · ω σk ∧ dxik+1 ∧ · · · ∧ dxin +

n X

k=1

Tσ0 ,...,σk ,ik+1 ,...,in ω σ0 ∧ · · · ω σk ∧ dxik+1 ∧ · · · ∧ dxin ,

(5.1)

where: – the coefficients Tσi00 ,...,σk ,ik+1 ,...,in and Tσ0 ,...,σk ,ik+1 ,...,in are smooth functions of the σ ); variables (xi , y σ , yjσ , yjl – they are antisymmetric in the indices σ1 , . . . , σk (resp. in σ0 , . . . , σk ) and in the indices ik+1 , . . . , in ; – the following traceless condition is valid: Tσj0 ,...,σk ,j,ik+2 ,...,in = 0.

(5.2)

The form α is globally defined by these conditions. The proof follows directly from Theorem 3.3 if we take note that the condition (3.13) is true in this particular case. Let us remark that the previous tracelessness condition is weaker than the global condition Kα = 0 introduced in [21], [13]. One can express all the coefficients of the form α in terms of the expressions Tσ ; as a consequence we have α = 0 ⇐⇒ Tσ = 0. This is a stronger form of the relation (3.12). Finally we have the analogue of Theorem 4.1:


71

Theorem 5.2. In the conditions of the preceding theorem one can write α locally as follows: α = dθ

(5.3)

n i1 θ= ∂σ1 · · · ∂σikk Li1 ,...,in ω σ1 ∧ · · · ω σk ∧ dxik+1 ∧ · · · dxin . k! k k=0

(5.4)

where n X 1

!

Here we have Li1 ,...,in = εi1 ,...,in L

(5.5)

where L is a smooth local function depending on the variables (xi , y σ , yjσ ). It is instructive to prove all these results directly in this particular case. We remark in the end that there are no restriction on the local Lagrangian L other that the independence on the second order derivatives.

5.2.

The Case n = 1 and s Arbitrary

This case was studied in [25], [33], [35] and [14] with minor differences. As before we have: Let X be a differential manifold of dimension N = n+1 and let P1s X be the associated Grassmann manifold of order s. We will denote the local coordinates on it as follows: σ σ σ σ σ x1 7→ t and y1, . . . , 1 7→ qk i.e. we have the coordinates (t, q0 , q1 , . . . , qs ); it is natural to ωσ

| {z } k−times

put also: 1, . . . , 1 7→ ωkσ . | {z } k−times

Theorem 5.3. In the conditions described above, let α ∈ Ωs,s 2 (X) closed and verifying the tracelessness and the Lepage conditions. Then one can write it locally as follows: α = Tσ ω σ ∧ dt + with Tσ ,

X

i+j≤s−1

ij σ Tσν ωi ∧ ωjν

(5.6)

ij smooth functions depending on the variables (t, q σ , q σ , . . . , q σ ) verifying: Tσν 0 1 k ij ji Tσν = −Tνσ .

(5.7)

In this case the form α is globally defined. We define now the total derivative operator by: dt ≡

dst

s−1 X ∂ ∂ σ ≡ + qj+1 ∂t j=0 qjσ

(5.8)

and assume that ij Tσν = 0,

i + j ≥ s.

(5.9)

72

D.R. Grigore The closeness conditions for this local description is: ∂Tσ s−1,0 = 0, + 2Tνσ ∂qsν ij ∂Tσν = 0, ∂qsρ

i, j = 0, . . . , s − 1.

1 ∂Tσ j−1,0 j,0 + Tνσ + dt Tνσ = 0, 2 ∂qjν ij i−1,j i,j−1 dt Tσν + Tσν + Tσν = 0, ij ∂Tσν + ∂qkρ

jk ∂Tνρ ∂qiσ

1 2

+

ki ∂Tρσ ∂qjν

= 0,

∂Tσ ∂Tν − σ ν ∂q0 ∂q0

(5.10)

(5.11)

j = 1, . . . , s − 1,

(5.12)

i, j = 1, . . . , s − 1,

(5.13)

i, j, k = 0, . . . , s − 1,

(5.14)

00 + dt Tνσ = 0.

(5.15)

One can relax the condition (5.9). Indeed one can accept that the form α is given by the expression (5.6) with the summations restricted only to i, j ≤ s − 1. In that case one obtains from the closeness condition, beside the relations (5.10) - (5.15) above, also: s−1,j Tσν = 0,

j, . . . , s − 1.

(5.16)

Now one uses (5.12) to prove by induction that we have in fact (5.9) [35]. Let us mention two more facts. First, we have from in our particular case directly from (5.10) - (5.13) + (5.15): ij Tσν =−

1 2

s−1−i−j X

(−1)k+j

k=0

j+k ∂Tν (dt )k σ k ∂qi+j+k+1

(5.17)

so in particular we have α = 0 ⇐⇒ Tσ = 0. From this expression one can obtain the Helmholtz equations as in Corollary 3.4. They are: ! s X ∂Tσ ∂Tν k k = (−1) (dt )k−j σ , j = 0, . . . , s. (5.18) ∂qjν j ∂qk k=j We also note that the expressions (5.17) from above verify identically the system (5.10) - (5.15). Indeed, only the equation (5.14) should be investigated because the others are used completely in the induction process to obtain (5.17). But it is not very hard to prove that (5.17) verify identically (5.14) [33], [35]. Finally we give the analogue of Theorem 4.1 in this case: Theorem 5.4. In the conditions of the theorem above one can write α locally in the form α = dθ where θ=

r−1 X j=0

Ljσ ωjσ .

(5.19)

(5.20)


73

Here r = [(s + 1)/2] as before, Ljσ ≡

r−1−j X i=0

(−1)i (dt )i

∂L , σ ∂qi+j+1

j = 0, . . . , r − 1

(5.21)

and L is a smooth function depending only on the variables: (t, q σ , qjσ , . . . , qrσ ) which remains arbitrary for s = 2r and is constrained to be at most linear in qrσ for s = 2r − 1. The proof is elementary. We provide finally the expressions of the coefficients of the form α in terms of L: r X ∂L Tσ = (−1)j (dt )j σ (5.22) ∂qj j=0 (i.e. the usual Euler-Lagrange expressions) and ij Tσν =

6.

∂Ljν ∂Liσ − , σ ∂qi ∂qiν

i, j = 0, . . . , s − 1.

(5.23)

Conclusions

We first mention that one can use the formalism developed in this paper to analyse higher order Lagrangian systems with Noetherian symmetries, as in [13], [14], [35], [36],[37] and [38]. Indeed, if φ is a diffeomorphisms of the manifold X then one can see that its lift J s φ to Pns X leaves invariant the subspace of forms appearing in the left hand side of (3.13). This means that we can define a Noetherian symmetry as a map φ such that J s φ leaves the Lagrange-Souriau class invariant. It is to be expected that the computations will be much more difficult than in the two particular cases from the last Section. Next, we mention that it is not clear if in the general case studied here, the only restriction on the Euler-Lagrange expressions are given by the generalized Helmholtz equations, but it is reasonable to conjecture that this is true. Last, we remark that the formalism above could be generalized, in principle, to the case when the Euler-Lagrange expressions are not restricted by the condition of linearity in the highest order derivatives, trying for instance to relax the condition (3.3) i.e to factorize α to a smaller subspace.

References [1] H. F. Ahner and A. E. Moose, Covariant Inverse Problem of the Calculus of Variation, Journ. Math. Phys. 18 (1977) 1367–1373. [2] I. M. Anderson, The Variational Bicomplex (Utah State Univ. preprint, 1989, Academic Press, Boston, to appear). [3] I. M. Anderson and T. Duchamp, On the Existence of Global Variational Principles, American Journ. Math. 102 (1980) 781–868.

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[4] R. W. Atherton and G. M. Homsy, On the Existence and Formulation of Variational Principles for Nonlinear Differential Equations, Studies in Appl. Math. LIV (1975) 31–60. [5] D. E. Betounes, Extensions of the Classical Cartan Form, Phys. Rev. D 29 (1984) 599–606. [6] D. E. Betounes, Differential Geometric Aspects of the Cartan Form: Symmetry Theory, J. Math. Phys. 28 (1987) 2347–2353. [7] J. M. Ball, J. C. Currie and P. J. Olver, Null Lagrangians, Weak Continuity, and Variational Problems of Arbitrary Order, Journ. Functional Anal. 41 (1981) 135–174. [8] E. Cartan, Leçons sur les Invariants Integraux (Hermann, 1922). [9] A. Galindo and L. Mart´ınez Alonso, Kernels and Ranges in the Variational Formalism, Lett. Math. Phys. 2 (1978) 385–390. [10] P. L. Garcia, The Poincaré-Cartan Invariant in the Calculus of Variations, Symp. Math. XIV (1974) 219–246. [11] H. Goldschmit and S. Sternberg, The Hamilton-Cartan Formalism in the Calculus of Variations, Ann. Inst. Fourier 23 (1973) 203–267. [12] M. J. Gotay, A Multisymplectic Framework for Classical Field Theory and the Calculus of Variations. I. Covariant Hamiltonian Formalism, In: Mechanics, Analysis and Geometry: 200 Years after Lagrange ((M. Francaviglia and D. D. Holms, Eds.) North-Holland, Amsterdam, 1990) 203–235. [13] D. R. Grigore, A Generalized Lagrangian Formalism in Particle Mechanics and Classical Field Theory, Fortschr. der Phys. 41 (1993) 569–617. [14] D. R. Grigore, Higher-Order Lagrangian Theories and Noetherian Symmetries, Romanian Journ. Phys. 39 (1994) 11–35. [15] D. R. Grigore, On a Generalisation of the Poincaré-Cartan Form to Classical Field Theory; arXiv:math.DG/9801073. [16] D. R. Grigore, The Variational Sequence on Finite Jet Bundle Extensions and the Lagrangian Formalism, Diff. Geom. Appl. 10 (1999) 43–77; arXiv:dg-ga/9702016. [17] D. R. Grigore, Variationally Trivial Lagrangians and Locally Variational Differential Equations of Arbitrary Order, Diff. Geom. Appl. 10 (1999) 79–105. [18] D. R. Grigore, Higher-Order Lagrangian Formalism on Grassmann Manifolds; arXiv:dg-ga/9709005. [19] D. R. Grigore, Lagrangian Formalism on Grassmann Manifolds, In: Handbook of Global Analysis ((D. Krupka and D. Saunders, Eds.) Elsevier, 2008) 327–373.


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[20] D. R. Grigore and D. Krupka, Invariants of Velocities and Higher Order Grassmann Bundles, Journ. Geom. Phys. 24 (1997) 244–266; arXiv:dg-ga/9708013. [21] D. R. Grigore and O. T. Popp, On the Lagrange-Souriau Form in Classical Field Theory, Mathematica Bohemica 123 (1998) 73–86. [22] A. Haková and O. Krupková, Variational first-order partial differential equations, Journ. Differential Equations 191 (2003) 67–89. [23] P. Horváthy, Variational Formalism for Spinning Particles, Journ. Math. Phys. 20 (1979) 49–52. [24] I. Kijowski, A Finite-Dimensional Canonical Formalism in Classical Field Theory, Comm. Math. Phys. 30 (1973) 99–128. [25] L. Klapka, Euler-Lagrange Expressions and Closed Two-Forms in Higher Order Mechanics, In: Geometrical Methods in Physics (Conf. on Differential Geometry and Applications, Czechoslovakia, 1983, Univ. Brno, (D. Krupka, Ed.)). [26] J. Klein, Espaces Variationels et Mécanique, Ann. Inst. Fourier 12 (1962) 1–124. [27] D. Krupka, A Map Associated to the Lepagean Forms of the Calculus of Variations in Fibered Manifolds, Czech. Math. Journ. 27 (1977) 114–118. [28] D. Krupka, Lepagean Forms in Higher Order Variational Theory, In: Proceedings of the IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics (Turin, 1982, Atti della Academia delle Scienze di Torino, Suppl. al Vol. 117, 1983) 198–238. [29] D. Krupka, Geometry of Lagrangian Structures, In: Proceedings of the 14th Winter School on Abstract Analysis (Srn´ı, 1986, Suppl. ai Rendiconti del Circolo Matematico di Palermo, Serie II, no.14, 1987) 187–224. [30] D. Krupka, Variational Sequence on Finite Order Jet Spaces, In: Proceedings of the Conference Differential Geometry and its Applications (August, 1989, World Scientific, Singapore, 1990) 236–254. [31] D. Krupka, Topics in the Calculus of Variation: Finite Order Variational Sequences, In: Differential Geometry and its Applications (Proceedings Conf. Opava, 1992, Open Univ. Press) 437–495. [32] D. Krupka, The Geometry of Lagrange Structures, Preprint Series in Global Analysis, GA 7/97, Dept. of Math., Opava Univ., Czech Rep. [33] O. Krupková, Lepagean 2-forms in higher order Hamiltonian mechanics, I. Regularity, Arch. Math. (Brno) 22 (1986) 97–120. [34] O. Krupková, Lepagean 2-forms in Higher Order Hamiltonian Mechanics. II Inverse Problem, Arch. Math. (Brno) 23 (1987) 155–170.

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[35] O. Krupková, Variational Analysis on Fibered Manifolds over One-Dimensional Bases, Ph. D. Thesis, Opava Univ., 1992. [36] O. Krupková, Liouville and Jacobi theorems for vector distributions, In: Differential Geometry and Its Applications (Proc. Conf., Opava, August 1992, (O. Kowalski and D. Krupka, Eds.) Mathematical Publications 1, Silesian University, Opava, Czechoslovakia, 1993) 75–88. [37] O. Krupková, Symmetries and first integrals of time-dependent higher-order constrained systems, Journ. Geom. Phys. 18 (1996) 38–58. [38] O. Krupková, The Geometry of Ordinary Variational Equations (Lecture Notes in Mathematics 1678, Springer, Berlin, 1997). [39] O. Krupková, Hamiltonian field theory revisited: A geometric approach to regularity, In: Steps in Differential Geometry (Proc. Colloq. Diff. Geom., Debrecen, July 2000, (L. Kozma, P. T. Nagy and L. Tamássy, Eds.) Debrecen University, Debrecen, 2001) 187–207. [40] O. Krupková, Hamiltonian field theory, Journ. Geom. Phys. 43 (2002) 93–132. [41] P. J. Olver, Hyperjacobians, Determinant Ideals and the Weak Solutions to Variational Problems, Proc. Roy. Soc. Edinburgh 95A (1983) 317–340. [42] P. J. Olver, Applications of Lie Groups to Differential Equations (Springer, 1986). [43] H. Poincaré, Leçons sur les Méthodes Nouvelles de la Mécanique Céleste (GauthierVillars, Paris, 1892). [44] H. Rund, A Cartan Form for the Field Theory of Charathéodory in the Calculus of Variations of Multiple Integrals, Lect. Notes in Pure and Appl. Math. 100 (1985) 455– 469. [45] H. Rund, Integral Formulae Associated with the Euler-Lagrange Operator of Multiple Integral Problems in the Calculus of Variation, Æquationes Math. 11 (1974) 212–229. [46] D. J. Saunders, An Alternative Approach to the Cartan Form in the Lagrangian Field Theories, J. Phys. A 20 (1987) 339–349. [47] J. M. Souriau, Structure des Systemes Dynamique (Dunod, Paris, 1970).


Chapter 4

K RUPKA’ S F UNDAMENTAL L EPAGE E QUIVALENT AND THE E XCESS F UNCTION OF W ILKINS D.J. Saunders∗ Palacký University, Olomouc, Czech Republic

Abstract We recall the features of Lepage equivalents of first-order Lagrangians on jet spaces of fibred manifolds, and the corresponding structures associated with homogeneous Lagrangians. We demonstrate the correspondence between Krupka’s fundamental Lepage equivalent and a variant of the Weierstrass excess function introduced by J. E. Wilkins.

1.

Introduction

I first met Demeter Krupka in 1989 at the DGA conference; although we have not published any joint work, there are many areas of our subject where we take similar approaches, and over the years we have held many useful discussions. In this short note (which I state at the outset contains no original material) I should like to make some remarks about one of Demeter’s constructions which has fascinated me for several years: this is the Fundamental Lepage equivalent of a Lagrangian [6]. This object also appears in a paper by David Betounes [1], published a few years later but as a result of independent work. It may be described as follows. Given a Lagrangian m-form λ = Lω

(ω = dx1 ∧ · · · ∧ dxm )

in n dependent variables y α and their derivatives yjα , consider the m-form min{m,n}

X r=0

∗

1 ∂rL θα1 ∧ · · · ∧ θαr ∧ ωi1 ···ir α (r!)2 ∂yj11 · · · ∂yjαrr

E-mail address: [email protected]

78

D.J. Saunders

where θαs = dy αs − ykαs dxk are contact forms, and ωi1 ···ir = ∂/∂xir ωi1 ···ir−1 is defined recursively. This m-form has the property that it is closed precisely when the original Lagrangian m-form λ is null: that is, when the Euler-Lagrange equations of the Lagrangian vanish identically. In this note, I want to relate the circle of ideas surrounding this object — which arises, of course, in the ‘exterior differential forms’ approach to the calculus of variations — to those of an apparently different approach, associated with the Weierstrass excess function.

2.

Single-Integral Problems

Consider the variational problem with fixed endpoints given in classical notation by Z b δ L(x, y α , yxα )dx = 0 . a

If we modify the integrand L by adding to it a ‘total derivative’ the problem retains the same extremals: Z b d α α α δ L(x, y , yx ) + f (x, y ) dx = 0 , dx a because

Z b a

d α f (x, y ) dx = [f (x, y α )]ba dx

is independent of the path along which the integral is taken. We may rewrite this problem in modern notation, using the language of fibred manifolds and differential forms. Take a fibred manifold π : E → R with coordinate x on R, and fibred coordinates (x, y α ) (1 ≤ α ≤ n) on E. Given a Lagrangian 1-form λ = L dx on the first-order jet manifold J 1 π, the problem is to find local sections φ such that Z b (j 1 φ)∗ λ a

takes extreme values. For any f : E → R, adding the horizontal differential dh f = (df /dx)dx does not affect the extremals, because Z b Z b 1 ∗ (j φ) dh f = d(φ∗ f ) = [φ∗ f ]ba a

a

is independent of the section φ as the endpoints are fixed. Instead of adding a horizontal differential dh f to the Lagrangian, we could add a contact form, in other words some linear combination of the forms written locally as dy α − yxα dx where yxα are the jet coordinates. Once again, we would have (j 1 φ)∗ (L dx + gα (dy α − yxα dx)) = (j 1 φ)∗ (L dx) because (j 1 φ)∗ (dy α − yxα dx) = 0. Choosing in particular the coefficient functions g α to be gα =

∂L ∂yxα

Krupka’s Fundamental Lepage Equivalent and the Excess Function of Wilkins

79

gives the Cartan form Θλ = L dx +

∂L (dy α − yxα dx) . ∂yxα

The Cartan form Θλ is a ‘Lepage form’ [7]: that is, iZ dΘλ

is a contact form

whenever the vector field Z on J 1 π is vertical over E. Furthermore, if φ is an extremal section for λ then (j 1 φ)∗ iX 1 dλ = 0 for the prolongation X 1 of any variation field X on E; and then (j 1 φ)∗ iY dΘλ = 0 for any vector field Y on J 1 π by the Lepage property and the fixed endpoints. So j 1 φ is an extremal section for Θλ . A straightforward calculation in jet coordinates shows that the Euler-Lagrange form, ∗ dΘ : which is a 2-form on J 2 π, is the 1-contact part of π2,1 λ dΘλ =

∂L d ∂L − ∂y α dx ∂yxα

(dy α − yxα dx) ∧ dx + . . . .

Thus it is obvious that, when Θλ is closed, λ is automatically null. In this present situation, the converse also holds: we can see this because a null Lagrangian is necessarily of the form λ = L dx ehere df ∂f ∂f L= = + yxα α , dx ∂x ∂y and then Θλ =

3.

∂f ∂f + yxα α ∂x ∂y

dx +

∂f (dy α − yxα dx) = df . ∂y α

Multiple-Integral Problems

The theory for multiple-integral problems, unlike that for single-integral problems, is by no means as straightforward. Two quite distinct approaches are associated with De Donder [5] and Weyl [9], and with Carathéodory [2]. We start with the De Donder-Weyl approach. Take a fibred manifold π : E → M where i x (1 ≤ i ≤ m) are coordinates on M and (xi , y α ) (1 ≤ α ≤ n) are fibred coordinates on E; and suppose given a Lagrangian m-form λ = Lω on J 1 π, where ω = dx1 ∧ . . . ∧ dxm is a fixed volume form on the m-dimensional base manifold M (and the same symbol is used for its pull-back to J 1 π). The variational problem is to find local sections φ such that Z (j 1 φ)∗ λ C

80

D.J. Saunders

takes extreme values, where C ⊂ M is a simply-connected compact m-dimensional submanifold with boundary and φ is given on ∂C. If µ = µi ωi is a horizontal (m − 1)-form on E, adding the horizontal differential dh µ = (dµi /dxi )ω does not affect the extremals: Z Z Z 1 ∗ ∗ (j φ) dh µ = d(φ µ) = φ∗ µ C

C

∂C

is independent of the section φ as the boundary ∂C is fixed. Alternatively, instead of adding a horizontal differential dh µ, we could add a 1-contact form. Put b λ = Lω + ∂L θα ∧ ωi , Θ ∂yiα

where θα = dy α − yiα dxi are local contact forms; then

b λ = (j 1 φ)∗ (Lω) (j 1 φ)∗ Θ

b λ is often called the Cartan form, by analogy with because (j 1 φ)∗ θα = 0. The m-form Θ the single-integral case, and it is a Lepage equivalent of λ; but, unlike the single-integral case, there are other possible Lepage equivalents. b λ (it is easy to see that this will be The Euler-Lagrange form is the 1-contact part of dΘ the case for any Lepage equivalent, not merely the one we have described): ∂L d ∂L b dΘλ = − θα ∧ ω + . . . . ∂y α dxi ∂yiα b λ is closed then clearly λ is null. And for a partial converse, if a null Lagrangian is If Θ b λ is closed. But a complete converse is not available: if µi are given by λ = dh µ then Θ functions on E then i dµ λ = det ω = dh µ1 ∧ · · · ∧ dh µm dxj

b λ is not closed. is also a null Lagrangian, and now Θ A somewhat different approach was taken by Carathéodory . Again we consider local sections φ such that Z (j 1 φ)∗ λ

C

takes extreme values; but now, instead of adding a trace, we add a determinant. For any m functions µi on E, adding det dµi /dxj ω = dh µ1 ∧ · · · ∧ dh µm

does not affect the extremals: Z Z 1 ∗ i j (j φ) det dµ /dx ω = C

∂C

φ∗ µ1 dh µ2 ∧ · · · ∧ dh µm


81

is independent of the section φ as the boundary is fixed. Alternatively, instead of adding a i i horizontal form det dµ /dx ω, we could add a contact form in such a way that the result was a decomposable m-form. Put eλ = Θ

then

1 Lm−1

m ^

i=1

∂L L dx + α θα ∂yi i

;

e λ = (j 1 φ)∗ (Lω) (j 1 φ)∗ Θ

e λ is another Lepage equivalent of λ, and is called the because (j 1 φ)∗ θα = 0. The m-form Θ e λ , and Carathéodory form. As before, the Euler-Lagrange form is the 1-contact part of dΘ e λ is closed then λ is null. Furthermore, if a null Lagrangian is given by the determinant if Θ e λ is closed. λ = det dµi /dxj ω then Θ e λ is not linear, so that taking the sum of two Now, however, we have the problem that Θ determinants we obtain i i dµ2 dµ1 ω + det ω λ = det j dx dxj e λ is not closed. as another null Lagrangian, where Θ The fundamental Lepage equivalent is designed to avoid these problems; in order to do this, it is necessary to use higher derivatives of the Lagrangian function. Put min{m,n}

Θλ =

X r=0

1 ∂rL θα1 ∧ · · · ∧ θαr ∧ ωi1 ···ir ; (r!)2 ∂yjα11 · · · ∂yjαrr

then, once again, Θλ is a Lepage equivalent of λ. If λ = det dµi /dxj ω then it may be e λ . But now Θλ is linear, and dΘλ = 0 for any null lagrangian λ. shown [4] that Θλ = Θ

4.

Homogeneity

We turn now to a different, but related, type of variational problem. A ‘parametric variational problem’ is one where the submanifolds we consider are given parametrically, and are not the images of sections. The basic example of such a problem arises in Finsler geometry, where we consider 1-dimensional parametric problems: Z b δ (j 1 σ)∗ L dt = 0 a

where E is a manifold with local coordinates ua (1 ≤ a ≤ 1 + n), σ : [a, b] → E is a curve, and the Lagrangian L is a function. In order for the extremals to be independent of parametrization (but nevertheless to have a particular orientation), it is necessary and sufficient for L to be positively homogeneous of degree 1: u˙ a

∂L = L. ∂ u˙ a

82

D.J. Saunders

A homogeneous Lagrangian L gives rise to its Hilbert form ΘL =

∂L a du ; ∂ u˙ a

if σ is an extremal curve for L then j 1 σ is an extremal curve for ΘL . Any Lagrangian involving x explicitly (on a jet bundle) gives rise, in a standard way, to a homogeneous Lagrangian, and the Hilbert form of this homogeneous Lagrangian then projects to the Cartan form of the original Lagrangian. Parametric m-dimensional variational problems, on a manifold E with coordinates ua (1 ≤ a ≤ m + n) and its manifold of regular m-velocities FE with coordinates (ua , uai ) (1 ≤ i ≤ m) are considered in the same way: Z δ (j 1 σ)∗ L dm t = 0 C

where L now satisfies the homogeneity condition uaj

∂L = δji L . ∂uai

There are now m Hilbert 1-forms i θL =

∂L a du , ∂uai

and any Lagrangian involving xi explicitly (on a jet bundle) gives rise to a homogeneous Lagrangian. The Cartan form from the De Donder-Weyl theory makes no sense in this context. But we can construct m ^ i eL = 1 Θ θL Lm−1 i=1

and this projects to the Carathéodory form of the original Lagrangian [3]. If we define the tensors ∂ S i = dua ⊗ a ∂ui

then we can construct the Hilbert forms by i θL = S i dL .

We can also construct the ‘fundamental form’ ΘL =

1 1 2 S dS d · · · S m dL m!

and this projects to the fundamental Lepage equivalent of the original Lagrangian [4]. The coordinate formula for ΘL is ΘL =

1 ∂mL dua1 ∧ · · · ∧ duamm . m! ∂ua11 · · · ∂uamm 1

(1)


5.

83

The Weierstrass Excess Function

We return to inhomogeneous variational problems, and consider the classical question of whether an extremal is a genuine local minimum of the action functional; a useful reference for this is [8]. For questions like these, we need to distinguish between weak and strong local minima, with the distinction arising from the topology on the space of sections: if we regard sections as close if their values are close then we obtain a strong local minimum, whereas if we also require their derivatives to be close then we obtain a weak local minimum. In order to consider this question, one approach for a single-integral problem is to define the Weierstrass excess function: in classical notation, this is given by E(x, y α , yxα , z α ) = L(x, y α , z α ) − L(x, y α , yxα ) −

∂L α (z − yxα ) , ∂yxα

or in modern notation, with π1,0 (ψ(x)) = φ(x), we would write E(jx1 φ, ψ(x)) = L(ψ(x)) − L(jx1 φ) − dF L(ψ(x) − jx1 φ) . The Weierstrass necessary condition is then that, if φ is a strong minimum of the variational problem, then E(jx1 φ, ψ(x)) ≥ 0 for any section ψ of J 1 π → R projecting to φ. For multiple-integral problems the question is, as we might expect, more complicated. The De Donder-Weyl theory suggests that we should use b i , y α , yiα , ziα ) = L(xi , y α , ziα ) − L(xi , y α , yiα ) − ∂L (ziα − yiα ) E(x ∂yiα

as an excess function, whereas the Carathéodory theory suggests 1 ∂L α i i α α α i α α α e E(x , y , yi , zi ) = L(x , y , zi ) − p−1 det δj L + α (zj − yj ) L ∂yi

b ≥ 0 nor E e ≥ 0 is, in general, a necessary instead. It turns out, however, that neither E condition for a strong minimum. Similar difficulties arise for homogeneous problems. For these problems the De Donder-Weyl theory is not appropriate. The excess function for the Carathéodory theory becomes 1 ∂L a a a a a a e v E(u , ui , vi ) = L(u , vi ) − p−1 det L ∂uai j e ≥ 0 is not a necessary condition for a strong miniin the homogeneous case, but again E mum. Consider, for example, L = u11 u22 − u12 u21 + u31 u42 − u32 u41 :

e 6= 0. this is a null Lagrangian, but E

84

D.J. Saunders

It is, however, possible to define an excess function for homogeneous problems which does not have this problem; a suitable definition was proposed in a paper by J. E. Wilkins in 1944 [10]. This paper defines an excess function E by the formula E(ua , uai , via ) = L(ua , via ) −

1 ∂mL j1 ···jm v α1 · · · vjαmm , ε (m!)2 i1 ···im ∂uαi11 · · · ∂uαimm j1

(2)

and demonstrates that E = 0 whenever the Lagrangian is null. The structure of this formula is interesting, as it involves higher derivatives of the Lagrangian. Indeed, by comparing equations (1) and (2) it may be seen that the derivatives of the Lagrangian, and their coefficients, are combined in exactly the same arrangement here as in the fundamental form ΘL of the homogeneous Lagrangian, a form directly related to Krupka’s fundamental Lepage equivalent. It seems, therefore, that this type of structure, discovered in its two different manifestations in 1944 and 1977, is of some importance in studying the properties of null Lagrangians.

Acknowledgements The author expresses his acknowledgments to the Czech Science Foundation (grant no. 201/06/0922 for Global Analysis and its Applications).

References [1] D. E. Betounes, Extensions of the classical Cartan form, Phys. Rev. D 29 (1984), 599– 606. ¨ [2] C. Carathéodory, Uber die Variationsrechnung bei mehrfachen Integralen, Acta Szeged. Sect. Scient. Mathem. 4 (1929), 193–216. [3] M. Crampin and D. J. Saunders, The Hilbert-Carathéodory form for parametric multiple integral problems in the calculus of variations, Acta Appl. Math. 76 (2003), 37–55. [4] M. Crampin and D. J. Saunders, On null Lagrangians, Diff. Geom. Appl. 22 (2005), 131–146. [5] Th. De Donder, Théorie invariantive du calcul des variations (nouvelle e´ dit.: Paris, Gauthier-Villars, 1935). [6] D. Krupka, A map associated to the Lepagean forms in the calculus of variations, Czech Math. J. 27 (1977), 114–118. [7] Th.-H.-J. Lepage, Sur les champs géodésiques du calcul des variations, Bull. Acad. Roy. Belg. Cl. Sci. V Sér 22 (1936), 716–729, 1036–1046. [8] H. Rund, The Hamilton-Jacobi theory in the calculus of variations (London: Van Nostrand, 1966). [9] H. Weyl, Geodesic fields in the calculus of variations for multiple integrals, Ann. Math. (2nd Ser.) 36 (1935), 607–629. [10] J. E. Wilkins, Multiple integral problems in parametric form in the calculus of variations, Ann. Math. (2nd Ser.) 45 (1944), 312–334.


Chapter 5

L EPAGE C ONGRUENCES IN D ISCRETE M ECHANICS∗ 1

Antonio Fernández1† and Pedro L. Garc´ıa2‡ Dpto. de Matemática Aplicada, Universidad de Salamanca. 2 Dpto. de Matemáticas, Universidad de Salamanca

Abstract We introduce the concept of contact 1-form in Discrete Mechanics. In terms of this concept, we express the Poincaré-Cartan form of a discrete Lagrangian by two formulas that generalizes to the Discrete Mechanics the classical Lepage congruences. The requirement for these congruences to have similar properties to those in the continuous case leads to a special class of mechanical systems, which interest is illustrated with some examples.

1.

Introduction

One of the most beautiful geometrical doctrines of the last century is, without any doubt, that known in the literature from the beginning of the 70’s as the Hamilton-Cartan formalism of the Variational Calculus. This doctrine starts in the 30’s with the works of De Donder [4] and Weyl [18] and its generalization by Lepage [15, 16] some years after, its modern formulation in terms of jet fiber bundles of bundled manifolds was stablished 30 years after. Two key facts of this formulation were in its starts the identification of the Poincaré-Cartan form with the “boundary term” of the formula of variation of the integral of the action for first order variational problems and the characterization of this form from a natural globalization of the classical Lepage congruences (three basic references from this first stage are: Goldschmidt and Sternberg [10], Garc´ıa [7] and Krupka [12, 13]. The generalization of this setup to higher order problems was more problematic (Garc´ıa and Muñoz [8], ˇ epánková [14], Horák and Koláˇr [11] and refFerraris and Francaviglia [6], Krupka and Stˇ erences therein), and moreover, the recent treatment of the constrained problems and its ∗

Dedicated to Demeter Krupka E-mail address: [email protected] ‡ E-mail address: [email protected] †

86

Antonio Fernández and Pedro L. Garc´ıa

relation with the Lagrangian reduction topic (Fernandez, Garc´ıa and Rodrigo [5], Bibbona, Fatibene and Francaviglia [1], Garc´ıa and Rodrigo [9], etc.). In its simplest case –Analytical Mechanics– the setup is well known: Given a fibration p : Q˙ = Q × R → R (Q : configuration manifold of a mechanical system and R the time line), the natural space were the Lagrangian density Ldt lives is ˙ → Q˙ of the 1-jets of the local sections of p which geometry the affine bundle π : J 1 (Q) ˙ (V (Q) ˙ : bundle of the is driven by a 1-form θ with values on the vector bundle π ∗ V (Q) ˙ p-vertical vector fields of Q) defined by the rule: ˙ θjt1 s (D) = (dvert. s)(π∗ D) = π∗ D − st∗ (p∗ ◦ π∗ D) D ∈ Tjt1 s (J 1 (Q)). t ˙ In the standard set of local coordinates (t, q i , q˙i ) of J 1 (Q): X ∂ θ= dq i − q˙i dt ⊗ i . ∂q i

This differential form, known in the literature as the contact 1-form of the 1-jet bundle, allows toR obtain an intrinsic expression of the differential δs L of the integral of the action L(s) = j 1 s Ldt on a section s in the following terms: Z Z δs L = LD1 Ldt = E(s) · θ(D1 ) dt + d (j 1 s)∗ iD1 Θ , j1s

j1s

where j 1 s and D1 are the 1-jet extensions of the section s and a vector field D on Q˙ respectively, E : s → E(s) ∈ s∗ (V E)∗ is a second order differential operator (Euler-Lagrange ˙ (Poincaré-Cartan form) and · is the duality pairing. operator), Θ is a 1-form on J 1 (Q) Even more, it is possible to characterize the Poincaré-Cartan form by the conditions: Θ = p · θ + Ldt,

¯ θ, dΘ = E ∧

˙ with values on π ∗ V (Q) ˙ ∗ where p and E are, respectively, a function and a 1-form on J 1 (Q) ¯ are taken with respect to the duality pairing. and where the products · and ∧ These are in the case of the mechanics the so-called Lepage congruences, from where it is possible to recover the variation of the integral of the action by restricting to j 1 s the Lie derivative with respect to D1 of this congruences bearing in mind that θ(j 1 s) = 0 and y (j 1 s)∗ L1D θ = 0. Established the problem in these terms, in the present work we are going to start the study of this question in Discrete Mechanics, which we believe that it has not been treated up to now, perhaps because for the non existence of a clear concept of tangency to a discrete curve in this field. After a brief outline of the basic principles of the Discrete Mechanics on section 2. (see [17, 2, 3]), in section 3. we introduce a general notion of contact 1-form characterizing intrinsically one of them in terms of which the usual discretization of the “derivative” as an “incremental quotient” in two nearby instants is codified (Theorem 3). From this concept, the Poincaré-Cartan 1-form of a discrete Lagrangian is characterized by a set of conditions that extend to the discrete case the classical Lepage congruences (Theorem 4). The requirement for this congruences to have similar properties to those in the continuous case leads to a special class of mechanical systems, which interest is illustrated in section 5. with some examples.

Lepage Congruences in Discrete Mechanics

2.

87

Discrete Mechanics

Let Q˙ = R × Q be the configuration space.

Definition 1. A discrete section S d is a collection of points of Q˙ sd = {(t0 , q0 ), (t1 , q1 ), . . . , (tN , qN )} or, equivalently, a point of Q˙ N +1 . Definition 2. A discrete action is a differentiable application Sd : Q˙ N +1 → R From now onwards, we are only going to consider the discrete actions of the form d

S (t0 , q0 , t1 , q1 , . . . , tN , qN ) =

N −1 X k=0

Lk (tk , qk , tk+1 , qk+1 )(tk+1 − tk ),

(1)

where Lk : Q˙ × Q˙ → R is a differentiable function that will be called local discrete Lagrangian. Definition 3. A critical discrete section of the discrete action Sd is a discrete section sd such that dsd Sd (0, D1 , . . . , DN −1 , 0) = 0,

∀Dk ∈ T(tk ,qk ) Q˙

k = 1, . . . , N − 1.

If we calculate dSd we get: d

dS =d

NX −1 k=0

i Lk (tk , qki , tk+1 , qk+1 )(tk+1

− tk ) =

X ∂L0 ∂L0 (t1 − t0 )dq0i (t1 − t0 ) − L0 dt0 + = i ∂t0 ∂q0 i N −1 X ∂Lk−1 ∂Lk + (tk − tk−1 ) + Lk−1 + (tk+1 − tk ) − Lk dtk ∂tk ∂tk k=1 ! X ∂Lk−1 ∂Lk (tk − tk−1 ) + (tk+1 − tk ) dqki + i i ∂qk ∂qk i ∂LN −1 ∂LN −1 i + (tN − tN −1 ) + LN −1 )dtN + (tN − tN −1 dqN i ∂tN ∂qN

(2)

and, hence, the following theorem holds Theorem 1. A discrete section sd = (t0 , q0 , t1 , q1 , . . . , tN , qN ) is critical of the discrete action (1) if and only if satisfies the discrete Euler-Lagrange equations  ∂Lk−1 ∂Lk  0= (tk − tk−1 ) + (tk+1 − tk ), i = 1, . . . , n  ∂qki ∂qki k = 1, . . . , N − 1.  ∂Lk ∂Lk−1  0= (tk − tk−1 ) + Lk−1 + (tk+1 − tk ) − Lk  ∂tk ∂tk

88


If the Euler-Lagrange equations allow us to define (tk+1 , qk+1 ) as functions of (tk−1 , qk−1 ) and (tk , qk ), we can define a set of maps φk : Q˙ × Q˙ → Q˙ × Q˙

((tk−1 , qk−1 ), (tk , qk )) 7→ ((tk , qk ), (tk+1 , qk+1 )) that is called the discrete flux of the critical section sd . In terms of the discrete flux, the Euler-Lagrange equations are ∂Lk−1 ∂Lk (t − t ) =− (tk − tk−1 ), φ∗k k+1 k ∂qki ∂qki ∂Lk ∂Lk−1 ∗ φk − (tk+1 − tk ) + Lk = (tk − tk−1 ) + Lk−1 ∂tk ∂tk

(3)

and, hence, a discrete flux φk defines a critical section sd if and only if equations (3) hold. On the other hand, if we look at the first and last terms of the differential of the action (2), we can define a pair of 1-forms: X ∂Lk ∂Lk i − (t − t )dq + (t − t ) + L Θ− = − k+1 k k+1 k k dtk , k i ∂tk ∂q k i X ∂Lk ∂Lk + i Θ = (tk+1 − tk )dqk+1 + (tk+1 − tk ) + Lk dtk+1 i ∂tk+1 ∂qk+1 i

and, then, a 1-form (Θ− , Θ+ ) on Q˙ × Q˙ that will be called Poincaré-Cartan form. The two 1-forms Θ− and Θ+ are related by the formula Θ+ − Θ− = d Lk (tk , qk , tk+1 , qk+1 )(tk+1 − tk )

(4)

and, if is the discrete flux of a critical section, by (3), we have that Θ− and Θ+ in two consecutive stages are φk related, that is φ∗k Θ− = Θ+ . Finally, combining this two equations we have Theorem 2 (Discrete Cartan Equation). A discrete flux φk defines a critical section sd if and only if φ∗k Θ+ − Θ+ = d φ∗k Lk (tk+1 − tk ) . (5)

Finally, observe that if we differentiate the discrete Cartan Equation (5) we get the simplecticity of the discrete flux φk Corollary 1. If φk is the discrete flux of a critical section sd , then φ∗k dΘ+ = dΘ+ . As we have done in the continuous mechanics, we have obtained the Poincaré-Cartan form from the discrete variational principle. The next two sections of the paper will be devoted to obtain (for a particular class of Lagrangians) a characterization of the PoincaréCartan form on a stage Q˙ × Q˙ from the local discrete Lagrangian only.


3.

89

Discrete Contact 1-Forms

In order to introduce the notion of contact 1-form in the discrete realm, we need two definitions in the direct product of a manifold. Definition 4. Let M be a differentiable manifold. The conjugation of M × M is the diffeomorphism of M × M that interchanges the factors: ϕ: M × M → M × M

(x1 , x2 ) 7→ (x2 , x1 ).

The conjugation diffeomorphism acts in a natural way on the vector fields of M × M , and thus defining a 1-1 tensor on M × M , T , which local expression is the following: If (x1 , . . . , xn ) is a coordinate system in M , let (x11 , . . . , xn1 , x12 , . . . , xn2 ) be the induced coordinate system in M × M . As ϕ(x11 , . . . , xn1 , x12 , . . . , xn2 ) = (x12 , . . . , xn2 , x11 , . . . , xn1 ), it is ϕ∗ (xi2 ) = xi1 and ϕ∗ (xi1 ) = xi2 , and, hence, ∂ ∂ ∂ ∂ T = , T = . ∂xi1 ∂xi2 ∂xi2 ∂xi1 ˙ (θ− , θ+ ) verifying Definition 5. A contact 1-form on Q˙ × Q˙ is a 1-1 tensor on Q˙ × Q, 1 (θ− , θ+ ) is a projector of Q˙ × Q˙ → R × R. 2 T ∗ θ+ = θ− Local expression: Let (q 1 , . . . , q n ) be a set of local coordinates on Q, and let ˙ Then, the general (t0 , q01 , . . . , q0n , t1 , q11 , . . . , q1n ) be the induced coordinates on Q˙ × Q. − + expression of θ and θ is: θ− =

X

(dq0i + f0i dt0 ) ⊗

∂ , ∂q0i

θ+ =

X

(dq1i + f1i dt1 ) ⊗

∂ ∂q1i

i

i

and the condition T ∗ θ+ = θ− implies that f1i = f0i . Hence, the local expression of the contact 1-form is θ− =

X

(dq0i − ui dt0 ) ⊗

X

(dq1i

i

+

θ =

i

∂ , ∂q0i

∂ − u dt1 ) ⊗ i . ∂q1

(6)

i

Definition 6. The contact distribution D will be the kernel of the contact 1-form (θ− , θ+ ) D = {(D0 , D1 ) : (θ− (D0 ), θ+ (D1 )) = (0, 0)}.

90


From the local expresion of the contact 1-form (6), it follows that the contact distribution is locally spanned by the vector fields D− and D+ given by X ∂ ∂ D− = + ui i , ∂t0 ∂q0 i X ∂ ∂ D+ = + ui i . ∂t1 ∂q1 i

Theorem 3. There exists a unique contact 1-form (θ− , θ+ ) which contact distribution D is integrable and its first integrals are ϕ-invariant. Proof. From the local expression of the local vector fields D− and D+ , it follows that the integrability condition is X − + − j ∂ + j ∂ [D , D ] = D (u ) j − D (u ) j = 0 ∂q1 ∂q0 j and, hence, D− (uj ) = D+ (uj ) = 0,

j = 1, . . . , n

ui

and the functions are first integrals of the distribution. From this set of first integrals we can construct a set of n additional first integrals v0j = q0j − uj t0 ,

j = 1, . . . , n.

Now, by imposing the condition of ϕ-invariance of the first integrals, we have that, being

it should be ϕv0j = v0j , that is

ϕv0j = q1j − uj t1

q0j − uj t0 = q1j − uj t1 ,

⇒ uj =

q1j − q0j , t1 − t0

j = 1, . . . n.

From now onwards we are going to consider only this canonically determined contact 1-form.

4.

Discrete Lepage Congruences

Let L : Q˙ × Q˙ → R a discrete Lagrangian, and let (Θ− , Θ+ ) be its Poincaré-Cartan form as defined in Section 2.. In these conditions, we have: ˙ its PoincaréTheorem 4. If L is a first integral of the contact distribution D on Q˙ × Q, − + Cartan form (Θ , Θ ) is univocally determined by the following conditions: Θ− = p− · θ− + Ldt0 ;

Θ+ = p+ · θ+ + Ldt1 , Θ+ − Θ− = d L(t1 − t0 ) ,

(7)

where p− and p+ are functions on Q˙ × Q˙ with values on T ∗ Q0 and T ∗ Q1 respectively. And, conversely, if the Poincaré-Cartan form of a discrete Lagrangian L satisfies the conditions (7), then L is a first integral of the contact distribution D


91

Proof. Let (q 1 , . . . , q n ) be local coordinates on Q, and (t0 , q01 , . . . , q0n , t1 , q11 , . . . , q1n ) the ˙ In these coordinates, the local expression of the Poincaréinduced coordinates on Q˙ × Q. Cartan form is X ∂L ∂L + i Θ = (t1 − t0 )dq1 + (t1 − t0 ) + L dt1 = ∂t1 ∂q1i i X ∂L (t1 − t0 ) dq1i − ui dt1 + Ldt1 = i ∂q1 i X ∂L i ∂L + (t1 − t0 ) u + i ∂t1 ∂q 1 i X ∂L i ∂L + + , =p · θ + Ldt1 + (t1 − t0 ) u + ∂t1 ∂q1i i

where p+ =

X ∂L (t − t0 )dq1i i 1 ∂q 1 i

and the local expression of Θ− is X ∂L i ∂L Θ = p · θ + Ldt0 + (t1 − t0 ) , u + ∂t0 ∂q0i i −

−

−

where p− = −

X ∂L (t − t0 )dq0i . i 1 ∂q 0 i

From these two local expressions, it follows that the discrete Lepage’s congruences (7) hold if and only if the Lagrangian function L is a first integral of D. On the other hand, if a 1-form (ω − , ω + ) satisfies the Lepage’s congruences (7) we have that X X i + i p− = p− dq , p = p+ 0 i i dq1 i

i

with p− i

=ω

−

∂ , ∂q0i

p+ i

=ω

+

∂ ∂q1i

and, hence, p+ i

=ω

+

∂ ∂q1i

+ p− i =−ω

−

∂ ∂L = d L(t1 − t0 ) = i (t1 − t0 ), i ∂q1 ∂q 1 ∂ ∂ ∂L = −d L(t1 − t0 ) = − i (t1 − t0 ) ∂q0i ∂q0i ∂q1

∂ ∂q1i

−ω ∂ + ω− ∂q0i

and, then, it is ω − = Θ− ,

ω + = Θ+ .

92


Remark 1. Despite its apparent restriction, there is a rather natural way of obtaining local discrete Lagrangians that fullfil the hypothesis of Theorem (4): Starting from a continuous Lagrangian L(t, q, q), ˙ we can construct a family of local discrete Lagrangians, simply by evaluating the Lagrangian in an intermediate point of the segment joining the points (tk , qk ) and (tk+1 , qk+1 ) i Ld (tk , qki , tk+1 , qk+1 , α)

i qk+1 − qki i i . = L (1 − α)tk + αtk+1 , (1 − α)qk + αqk+1 , tk+1 − tk

Then, the condition of being Ld a first integral of the contact distribution can be expressed as: ! ∂L X i ∂L − D Ld =(1 − α) + uk i = 0, ∂t ∂q i ! ∂L X i ∂L + uk i = 0. D+ Ld =α ∂t ∂q i

On the other hand, as X ∂L dLd ∂L = (tk+1 − tk ) + (q i − qki ) dα ∂t ∂q i k+1 i

it follows that Ld is a first integral of the contact distribution if and only if dLd /dα = 0, that is, if and only if Ld is independent of the parameter α.

5.

Examples

In this section we are going to deal with discrete local Lagrangians obtained from continuous Lagrangians of the form n

L(ui , v i ) =

1X i 2 (u ) − Ψ(v 1 , . . . , v n ), 2 i=1

where ui = q˙i and v i = q i − tq˙i . The Euler-Lagrange equations of the variational problem defined by this Lagrangians are: d ∂L ∂L − i = 0, (8) i dt ∂ q˙ ∂q where

∂L ∂Ψ = ui + i t, i ∂ q˙ ∂v

∂L ∂Ψ =− i i ∂q ∂v

and, hence, equations (8) are: dui ∂Ψ d ∂Ψ +2 i + = 0. dt ∂v dt ∂v i

Lepage Congruences in Discrete Mechanics But, given that i v only:

dv i dt

93

i

= −t du dt we can express the Euler-Lagrange equations in terms of

∂Ψ dv i d 2 ∂Ψ 1 dv i d ∂Ψ +2 i + ⇔ = t 0=− t dt ∂v dt ∂v i dt dt ∂v i

(9)

∂Ψ that is, the variables v i are related by the equations v i = t2 ∂v i + λ. Let us assume that this i equations allow us to determine v as functions of t. Given that v i = q − tq, ˙ we have that Z i vi v (t) q i − tq˙i q ′ , q(t) = −t = = − dt. 2 2 t t t t2

By Theorem (4), the Poincaré-Cartan form (Θ− , Θ+ ) in each stage (t0 , q0 , t1 , q1 ), is given by the canonical momenta X ∂L ∂uj ∂L ∂L ∂v j ∂L ∂L + pi = i (t1 − t0 ) = (t1 − t0 ) + j i = − t0 i , i j i ∂u ∂v ∂u ∂v ∂q1 ∂q1 ∂q1 j X ∂L ∂uj ∂L ∂v j ∂L ∂L ∂L − + j i = − t1 i pi = − i (t1 − t0 ) = (t1 − t0 ) i j i ∂u ∂v ∂u ∂v ∂q1 ∂q0 ∂q0 j in our particular case, ∂Ψ , ∂v i ∂Ψ i p− . i = u + t1 ∂v i On the other hand, as we have seen, the Cartan equation (5) is equivalent to i p+ i = u + t0

(10)

φ∗k Θ− = Θ+ and, hence, by (7) to

φ∗k

X i

+ φ∗k p− i,k =pi,k , X i i p+ p− i,k uk − Lk = i,k−1 uk−1 − Lk−1 .

(11)

i

Example 1 (The P free particle). This mechanical system is given by the Lagrangian function L(ui ) = 21 i (ui )2 ; and its Euler-Lagrange equations are u ï = 0 ⇔ ui (t) = λi t + µi ,

i = 1, . . . , n.

(12)

On the other hand, the local discrete Lagrangians defined by this Lagrangian function are Lk (uik ) = and, hence, the momenta (10) are

1X i 2 (uk ) , 2

uik =

i

− i p+ i,k = uk = pi,k

i qk+1 − qki tk+1 − tk

94


and the discrete Cartan equations (11) are φ∗k uik = uik−1 ⇔

i i − qki qk+1 q i − qk−1 = k tk+1 − tk tk − tk−1

that is, the discrete Cartan equations do not impose conditions on all the discrete variables; in particular, every time discretization t0 , . . . , tN is compatible with them. If we choose a regular discretization, whith tk+1 − tk = tNN−t0 = h, we have the following discrete equations: i i qk+1 − 2qki + qk−1 =0 h that is the standard central-point discretization (multiplied by h) of the Euler-Lagrange equations (12). Example 2.

Let us now add a linear potential to the free particle Lagrangian: L(ui , v i ) =

1 1X i 2 X (u ) − αi v i = u · ut − A · v t , 2 2 i

i

where we have wrote u = (u1 , . . . , un ), v = (v 1 , . . . , v n ), A = (a1 , . . . , an ). The Euler-Lagrange equations are, then du + 2A = 0 ⇔ q¨ = −2A. dt The solutions of the Euler-Lagrange equations can be obtained by the general procedure previously introduced: Z i v (t) βi i 2 i v (t) = t αi + βi , q (t) = −t dt = −t tαi − + γi = −αi t2 − γi t + βi . t2 t The local discrete Lagrangians defined by this Lagrangian function can be written in vectorial form as 1 Lk (uik , vki ) = uk · utk − A · vkt 2 and, then, the canonical momenta are: p+ k = uk + Atk ,

p− k = uk + Atk+1 .

The first equation of (11) is, then φ∗k (uk + Atk+1 ) = uk−1 + Atk−1 . Let us now drop the φ+ k in order to simplify the explanation; and, then, we have that uk = uk−1 − A(hk + hk−1 ),

hk = tk+1 − tk , hk−1 = tk − tk−1 .

(13)

The second equation is 1 1 + t t t t t t p− k −Lk = pk−1 −Lk−1 ⇔ 2 uk ·uk +A·(uk tk+1 +qk+1 ) = 2 uk−1 ·uk−1 +A·(uk−1 tk−1 +qk−1 )


95

and, given that vk = qk − tk uk = qk+1 − tk+1 uk+1 , this equation is 1 1 uk · utk + A · utk hk = uk−1 · utk−1 − A · utk−1 hk−1 . 2 2 If we substitute uk using (13), we get, after a straightforward computation, that 1 A · At (h2k − h2k−1 ) = 0 2 that is, the time discretization should be uniform: hk = hk−1 = h =

tN − t0 N

and, returning to the first equation, we have that qk+1 − qk qk − qk−1 qk+1 − 2qk + qk−1 − = −2Ah ⇔ = −2Ah h h h that is, once again, the central-point discretization of the Euler Lagrange equations. Example 3.

Finally, we are going to consider a quadratic potential: L(ui , v i ) =

1X i 2 1X i 1 1 (u ) − v αij v j = u · ut − v · A · v t , 2 2 2 2 i

i,j

where A is a symmetric matrix. Once again, the solutions of the Euler-Lagrange equations can be easily obtained: v(t) =t2 v(t) · A + λ ⇒ v(t) = λ(I − t2 A)−1 , Z λ(I − t2 A)−1 q(t) = − t dt, t2 where λ = (λ1 , . . . , λn ) is a vector of constants and we have assumed that the matrix I − t2 A is invertible. The local discrete Lagrangians defined by this Lagrangian function can be written in vectorial form as 1 1 Lk (uk , vk ) = uk · utk − v · A · vkt 2 2 and, then, the canonical momenta are: p+ k = uk + tk vk · A,

p− k = uk + tk+1 vk · A.

Using the same conventions that in the previous example, the first equation of (11) is, then uk + tk+1 vk · A = uk−1 + tk−1 vk−1 · A and, hence uk = uk−1 Bk−1 · Bk−1 + (tk−1 − tk+1 )qk · A · Bk−1 ,

(14)

96


where we have denoted by Bk = I − tk+1 tk A. From this expression, it is easy to obtain that vk = vk−1 Bk−1 · Bk−1 . (15) The second equation is 1 1 + t t t t t p− k −Lk = pk−1 −Lk−1 ⇔ 2 uk ·uk +vk ·A·(uk tk+1 +vk ) = 2 uk−1 ·uk−1 +vk−1 ·A·qk−1 and, by substituting uk and vk using (14) and (15) it is possible to obtain tk+1 in terms of (tk−1 , qk−1 , tk , qk ); however, in this case the time distribution it is not the regular one tk = t0 + kh in general, as it is easy to see in the simplest case of being A a diagonal matrix.

Acknowledgements This work has been partially supported by the Spanish Ministerio de Ciencia y Tecnolog´ıa, project number MTM2004-01683.

References [1] E. Bibbona, L. Fatibene and M. Francaviglia, Gauge-natural parameterized variational problems, vakonomic field theories and relativistic hydrodynamics of a charged fluid, Int. J. Geom. Methods Mod. Phys. 3 (8) (2006) 1573–1608. [2] J.-B. Chen, H.-Y. Guo and K. Wu, Total variation and variational symplectic-energymomentum integrators, (2001); arXiv:hep-th/0109178. [3] J.-B. Chen, H.-Y. Guo and K. Wu, Discrete total variation calculus and Lee’s discrete mechanics, Appl. Math. Comput. 177 (1) (2006) 226–234. [4] Th. De Donder, Théorie invariantive du calcul des variations (Nuov. e´ d. GauthierVillars, Paris, 1935). [5] A. Fernández, P. L. Garc´ıa and C. Rodrigo, Lagrangian reduction and constrained variational calculus, In: Proceedings of the IX Fall Workshop on Geometry and Physics (Vilanova i la Geltrú, 2000, Publ. R. Soc. Mat. Esp., vol. 3, R. Soc. Mat. Esp., Madrid, 2001) 53–64. [6] M. Ferraris and M. Francaviglia, On the global structure of Lagrangian and Hamiltonian formalisms in higher order calculus of variations, In: Proceedings of the International Meeting on Geometry and Physics (Florence, 1982, Bologna, Pitagora, 1983) 43–70. [7] P. L. Garc´ıa, The Poincaré-Cartan invariant in the calculus of variations, In: Symposia Mathematica (Vol. XIV, Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973, Academic Press, London, 1974) 219–246.


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[8] P. L. Garc´ıa and J. Muñoz, On the geometrical structure of higher order variational calculus, In: Proceedings of the IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics (Vol. I, Torino, 1982, vol. 117, 1983) 127–147. [9] P. L. Garc´ıa and C. Rodrigo, The momentum map in vakonomic mechanics, In: Proceedings of the XII Fall Workshop on Geometry and Physics (Publ. R. Soc. Mat. Esp., vol. 7, R. Soc. Mat. Esp., Madrid, 2004) 111–123. [10] H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier (Grenoble) 23 (1) (1973) 203–267. [11] M. Horák and I. Kolárˇ, On the higher order Poincaré-Cartan forms, Czechoslovak Math. J. 33 (108) (3) (1983) 467–475. [12] D. Krupka, A geometric theory of ordinary first order variational problems in fibered manifolds. I. Critical sections, J. Math. Anal. Appl. 49 (1975) 180–206. [13] D. Krupka, A geometric theory of ordinary first order variational problems in fibered manifolds. II. Invariance, J. Math. Anal. Appl. 49 (1975) 469–476. ˇ epánková, On the Hamilton form in second order calculus of [14] D. Krupka and O. Stˇ variations, In: Proceedings of the International Meeting on Geometry and Physics (Florence, 1982, Bologna, Pitagora, 1983) 85–101. [15] Th. Lepage, Sur les champs géodésiques des intégrales multiples, Acad. Roy. Belgique. Bull. Cl. Sci. (5) 27 (1941) 27–46. [16] Th. Lepage, Champs stationnaires, champs géodésiques et formes intégrables. II, Acad. Roy. Belgique. Bull. Cl. Sci. (5) 28 (1942) 247–265. [17] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer. 10 (2001) 357–514. [18] H. Weyl, Geodesic fields in the calculus of variations for multiple integrals, Ann. of Math. (2) 36 (3) (1935) 607–629.


Chapter 6

F INITE O RDER VARIATIONAL S EQUENCES : A S HORT R EVIEW Raffaele Vitolo∗ Department of Mathematics “E. De Giorgi”, University of Lecce, Via per Arnesano, 73100 Lecce, Italy

Abstract Variational sequences are complexes of modules or sheaf sequences in which one of the maps is the Euler–Lagrange operator, i.e., the differential operator taking a Lagrangian into its Euler–Lagrange form. In this review paper we discuss variational sequences on finite order jets, with special emphasis on Krupka’s approach. We also discuss recent results on this topic as well as possible research directions.

2000 Mathematics Subject Classification. Primary 58J10, secondary 58A12, 58A20. Key words and phrases. Jet spaces, variational sequence, variational bicomplex.

Introduction In the Seventies, during a process of geometrization of the calculus of variations, it was realized that operations like passing from a Lagrangian to its Euler–Lagrange form were part of a complex, namely, the variational sequence. Foundational contributions to variational sequences are in the papers [3, 7, 12, 39, 40, 41, 42, 43, 44, 45]. Among the problems which were solved by the variational sequence was the so-called global inverse problem of the calculus of variations: given a set of Euler–Lagrange equations, the vanishing of Helmholtz conditions is a necessary and sufficient condition for the existence of a local Lagrangian for the given equations; does there exist a global Lagrangian? It was proved that the answer is in the cohomology of the variational sequence. More precisely, the cohomological obstruction for always having a global Lagrangian is the n + 1-st de Rham cohomology of the space of independent and dependent variables. ∗


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The geometric framework for variational sequences is that of jet spaces. Infinite order jet spaces were used as a rule, with the exception of [3]. There are some technical reasons for that choice: the first and most important is that on infinite order jet spaces the contact distribution is integrable and admits an intrinsic direct summand. This fact leads to much simpler computations. On the other hand, using infinite order jets one simply drops any information on the order of the objects involved in the computations. In this sense, the use of finite order jets can lead to finer results. A first approach in this sense was in [3]. In that paper the finite order variational sequence was truncated after the space of Euler–Lagrange forms. Moreover, in order to obtain the solution of the global inverse problem the authors resorted to infinite order jets. Another approach was through C-spectral sequences in [8, 9]. But it used one conjecture about the structure of contact forms (see Theorem 1.3). In [23] Krupka proved the above conjecture and was able to give the first formulation of the (long) variational sequence on finite order jets. The formulation was different from both the so-called variational bicomplex [2, 37] and the C-spectral sequence [7, 44]. The idea is rather simple: consider the de Rham complex on jets of order r. Then a subsequence of forms which yield trivial contribution to action-like functionals is defined. The quotient of the former sequence with the latter one yields the finite order variational sequence. In this paper, after a preliminary section on jet spaces and contact forms, we describe Krupka’s finite order variational sequence. In the final section we discuss the state of the research on this topic.

1.

Jet Spaces

Manifolds and maps between manifolds are C ∞ . All morphisms of fibred manifolds (and hence bundles) will be morphisms over the identity of the base manifold, unless otherwise specified. In particular, when speaking of ‘forms’ we will always mean ‘C ∞ differential forms’. We recall some basic facts on jet spaces. Our framework is a fibred manifold π : Y → X, with dim X = n, dim Y = n + m, and n, m ≥ 1. We have the vector subbundle def VY = ker T π of T Y , which is made by vectors which are tangent to the fibres of Y . For 1 ≤ r, we are concerned with the r-th jet space J r π; we also set J 0 π ≡ Y . For 0 ≤ s < r we recall the natural fibrings πr,s : J r π → J s π,

πr : J r π → X,

and the affine bundle πr,r−1 : J r π → J r−1 π associated with the vector bundle ⊙r T ∗ X ⊗J r−1 π V Y → J r−1 π. Charts on Y adapted to the fibring are denoted by (xi , y σ ). Latin indices i, j, . . . run from 1 to n and label base coordinates, Greek indices σ, τ , . . . run from 1 to m and label fibre coordinates, unless otherwise specified. We denote by (∂/∂xi , ∂/∂y σ ) and (dxi , dy σ ), respectively, the local bases of vector fields and 1-forms on Y induced by an adapted chart.

Finite Order Variational Sequences: A Short Review

101

We denoteP(symmetrized) multi-indices by capital letters: I = (i1 , . . . , in ) ∈ Nn . We def def also set |I| = k ik and I! = σ1 ! · · · σn !. The sum of a multiindex with a Latin index I + i will denote the sum of I and the multiindex (0, . . . , i, 0, . . . , 0), where 1 is at the i-th entry. def σ The charts induced on J r π are denoted by (xi , yIσ ), where 0 ≤ |I| ≤ r and y0σ = y . The local vector fields and forms of J r π induced by the fibre coordinates are denoted by (∂/∂yIσ ) and (dyIσ ), 0 ≤ |I| ≤ r, 1 ≤ i ≤ m, respectively. An r-th order (ordinary or partial) differential equation is, by definition, a submanifold S ⊂ J r π. We denote by jr s :X → J r π the jet prolongation of a section s : X → Y and by r J f : J r π → J r π the jet prolongation of a fibred morphism f : Y → Y over a diffeomorphism f¯: X → X. Any vector field ξ : Y → T Y which projects onto a vector field ξ : X → T X can be prolonged to a vector field ξ r : J r π → T J r π by prolonging its flow; its coordinate expression is well-known (see, e.g., [5, 37]). The fundamental geometric structure on jets is the contact distribution (or Cartan distribution) C r ⊂ T J r π. It is the distribution on J r π generated by all vectors which are tangent to the image jr s(X) ⊂ J r π of a prolonged section jr s. It is locally generated by the vector fields ∂ ∂ ∂ σ Di = + yI+i , , (1) σ i ∂x ∂yI ∂yJσ with 0 ≤ |I| ≤ r − 1, |J| = r. It is easy to show that this distribution is not involutive and does not admit any natural direct summand that complement it to T J r π. While the contact distribution has an essential importance in the symmetry analysis of PDE [5], in this context the dual concept of contact differential forms plays a central role. Let us denote by Fr the sheaf of smooth functions on J r π. We denote by Ωkr the sheaf of k-forms on J r π. We denote by Ω∗r the sheaf of forms of any degree on J r π. Definition 1.1. We say that a form α ∈ Ωkr is a contact k-form if (jr s)∗ α = 0 for all sections s of π. We denote by C 1 Ωkr the sheaf of contact k-forms on J r π. We denote by C 1 Ω∗r the sheaf of contact forms of any degree on J r π. Note that if k > n then every form is contact, i.e., C 1 Ωkr = Ωkr . It is obvious from the commutation of d and pull-back that dC 1 Ωkr ⊂ C 1 Ωk+1 . Morer 1 ∗ over, it is obvious that C Ωr is a sheaf of ideals (with respect to the exterior product) in Ω∗r . Unfortunately, C 1 Ω∗r does not coincide with the ideal generated by 1-forms which annihilate the contact distribution (for this would contradict the non-integrability). More precisely, the following lemma can be easily proved (see, e.g., [23]). Lemma 1.2. The sheaf C 1 Ω1r is locally generated (on Fr ) by the 1-forms σ ωIσ = dyIσ − yI+i dxi , def

0 ≤ |I| ≤ r − 1.

The above differential forms generate an ideal of Ω∗r . However, such an ideal is not differential, hence it does not coincide with C 1 Ω∗r . To realize it, the following formula can be easily proved σ dωIσ = −ωI+i ∧ dxi , (2)

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from which it follows that, when |I| = r − 1, then dωIσ , which is a contact 2-form, cannot σ contains derivatives of order be expressed through the 1-forms of lemma 1.2 because ωI+i r + 1. The following theorem is an important achievement by Krupka. It has been first conjectured in [9] (C 1 Ω-hypothesis), then proved in [23, 24]. Theorem 1.3. Let k ≥ 2. The sheaf C 1 Ωkr is locally generated (on Fr ) by the forms ωIσ ,

dωJσ ,

0 ≤ |I| ≤ r − 1,

|J| = r − 1.

We can consider forms which are generated by p-th exterior powers of contact forms. More precisely, we have the following definition. Definition 1.4. Let p ≥ 1. We say that a form α ∈ Ωkr is a p-contact k-form if it is generated by p-th exterior powers of contact forms. We denote by C p Ωkr the sheaf of p-contact k-forms on J r π. We denote by C p Ω∗r the sheaf of p-contact forms of any degree on J r π. def Finally, we set C 0 Ω∗r = Ω∗r . In other words, C p Ω∗r is the p-th power of the ideal C 1 Ω∗r in Ω∗r . Of course, a 1-contact form is just a contact form. We have the obvious inclusion C p+1 Ω∗r ⊂ C p Ω∗r . It follows that C p+1 Ω∗r is a sheaf of ideals of C p Ω∗r , hence of Ω∗r . Moreover, dC p+1 Ω∗r ⊂ C p+1 Ω∗r . Now, we would like to introduce a tool to extract from a form α ∈ Ωkr the non-trivial part (to the purposes of calculus of variations). In other words, we would like to introduce a map whose kernel is precisely the set of contact forms. Such forms yield no contribution to action-like functionals (see Remark 1.10). First of all, we observe that eq. (2) and Theorem 1.3 suggest that such a map can be constructed if we allow it to increase the jet order by 1. More precisely, it can be easily proved that the contact 1-forms ωIσ , with 0 ≤ |I| ≤ r − 1 generate a natural subbundle Cr∗ ⊂ J r π ×J r−1 π T ∗ J r−1 π ⊂ T ∗ J r π [46]. We have the following lemma (see [32, 37]). Lemma 1.5. We have the splitting J

r+1

r+1 ∗ ∗ π × T J π= J π×T X ⊕ Cr+1 , ∗ r

Jrπ

X

J r+1 π

with projections Dr+1 : J r+1 π → T ∗ X ⊗ T J r π, X

ω r+1 : J r+1 π → T ∗ J r π ⊗ V J r π, Jrπ

and coordinate expression ∂ ∂ σ + y , I+i ∂xi ∂yIσ ∂ ∂ σ = ωIσ ⊗ σ = (dyIσ − yI+i dxi ) ⊗ σ . ∂yI ∂yI

Dr+1 = dxi ⊗ Di = dxi ⊗ ω r+1

(3)


103

Note that the above construction makes sense through the natural inclusions V J r π ⊂ T J r π and J r+1 π ×X T ∗ X ⊂ J r+1 π ×J r π T ∗ J r π, the latter being provided by T ∗ πr . From elementary multilinear algebra it turns out that we have the splitting M r+1 k ∗ r r+1 q ∗ ∗ J π ×J r π ∧ T J π = J π×∧ T X ⊕ ∧p Cr+1 . p+q=k

J r+1 π

X

Now, we observe that a form α ∈ Ωkr fulfills ∗ πr+1,r (α) : J r+1 π → ∧k T ∗ J r π ⊂ ∧k T ∗ J r+1 π, ∗ (α) can be split where the inclusion is realized through the map T ∗ πr+1,r . Hence, πr+1,r into k + 1 factors which, respectively, have 0 contact factors, 1 contact factor, . . . , k contact factors. More precisely, let us denote by Hrq the set of q-forms of the type

α : J r π → ∧q T ∗ X. We have the following proposition (for a proof, see [23, 46, 48]). Proposition 1.6. We have the natural decomposition M q ∗ πr+1,r (Ωkr ) ⊂ C p Ωpr+1 ∧ Hr+1 , p+q=k

q with splitting projections prp,q : Ωkr → C p Ωpr+1 ∧ Hr+1 defined by p+q p,q p q ∗ pr (α) = ⊙ iDr+1 ⊙ ⊙ iωr+1 ◦ πr+1,r , q

where iDr+1 , iωr+1 stand for contractions followed by a wedge product. Note that the above maps prp,q are not surjective. See [46] for more details. Definition 1.7. We say the horizontalization to be the map q hp,q : C p Ωp+q → C p Ωpr+1 ∧ Hr+1 , r

We denote by

α 7→ prp,q (α).

p,q

def p,q Ωr = h (C p Ωp+q r )

(4) 0,q

the image of the horizontalization; we say an element α ¯ ∈ Ωr to be a horizontal form. Probably the first occurrence of horizontalization is in [22]. Of course, horizontalization is just the projection on forms which have no contact factors. Note that, if q > n, then horizontalization is the zero map. In coordinates, if 0 < q ≤ n, then σh σ1 ih+1 h α = ασI11···I ∧ · · · ∧ dxiq ...σh ih+1 ···iq dyI1 ∧ · · · ∧ dyIh ∧ dx

and the coordinate expression of the horizontalization is i1 iq h h0,q (α) = yIσ11+i1 · · · yIσhh+ih ασI11···I ...σh ih+1 ···iq dx ∧ · · · ∧ dx ,

(5)

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where 0 ≤ h ≤ q. The coordinate expressions of hp,q can be obtained in a similar way (see [3, 23, 24, 46]). Note that if n > 1 then the above form is not the most general polynomial in (r + 1)-st derivatives, even if q = 1. For q > 1 the skew-symmetrization in the indexes i1 ,. . . , ih yields a peculiar structure in the polynomial, in which the sums of all terms of the same degree are said to be hyperjacobians. Finally, we observe that if n = 1 then the horizontalization is surjective on the space of forms with affine coefficients with respect to r + 1-st derivatives [25]. The technical importance of horizontalization is in the next two results. Lemma 1.8. Let α ∈ Ωqr , with 0 ≤ q ≤ n, and s : X → Y be a section. Then (jr s)∗ (α) = (jr+1 s)∗ (h0,q (α))

Proposition 1.9. Let p ≥ 0. The kernel of hp,q coincides with p + 1-contact q-forms, i.e., C p+1 Ωq = ker hp,q .

For a proof of both results, see, for example, [48]. The above decomposition also affects the exterior differential. Namely, the pull-back of the differential can be split in two operators, one of which raises the contact degree by one, and the other raises the horizontal degree by one. More precisely, in view of proposition 1.6 and following [37], we introduce the maps iH : Ωkr → Ωkr+1 , iV :

Ωkr

→

Ωkr+1 ,

∗ iH = iDr+1 ◦ πr+1,r ,

iV = iωr+1 ◦

∗ πr+1,r .

(6a) (6b)

The maps iH and iV are two derivations along πr+1,r of degree 0. Together with the exterior differential d they yield two derivations along πr+1,r of degree 1, the horizontal and vertical differential def dH = iH ◦ d − d ◦ iH : Ωkr → Ωkr+1 , def dV = iV ◦ d − d ◦ iV : Ωkr → Ωkr+1 ,

It can be proved (see [37]) that dH and dV fulfill the properties d2H = d2V = 0, dH + ∗

(jr+1 s) ◦ dV = 0,

dH ◦ dV + dV ◦ dH = 0,

dV = (πrr+1 )∗ d ◦ (jr s)∗

◦ d,

∗

= (jr+1 s) ◦ dH .

(7a) (7b) (7c)

The action of dH and dV on functions f : J r Y → R and one–forms on J r Y uniquely characterizes dH and dV . We have the coordinate expressions ∂f ∂f σ i + yI+i σ dxi , (8a) dH f = Di f dx = ∂xi ∂yI dH dxi = 0,

σ dH dyIσ = −dyI+i ∧ dxi , ∂f dV f = σ ωIσ , ∂yI

dV dxi = 0 , We note that dH dyIσ = dH ωIσ .

σ dV dyIσ = dyI+i ∧ dxi ,

σ dH ωIσ = −ωI+i ∧ dxi ,

(8b) (8c)

dV ωIσ = 0.

(8d)

Finite Order Variational Sequences: A Short Review Remark 1.10. A form α ∈ Ωnr defines an action functional Z def A(s, U ) = (jr s)∗ α,

105

(9)

U

where U ⊂ X is any oriented n-dimensional submanifold of X with regular boundary. This is slightly more general than the usual notion, where a horizontal form of the type λ : J r π → ∧n T ∗ X is used (see, e.g., [37]). It follows that contact forms yield no contribution to action-like functionals. The definition (9) is a first motivation for the computations of the above section.

2.

Finite Order Variational Sequence

The first statement of a partial version of finite order variational sequence was in [3]. This finite order variational sequence stopped with a trivial projection to 0 just after the space of finite order source forms (see below). The local exactness of this sequence was proved, together with an original solution of the global inverse problem (despite the fact that in order to do that the authors used infinite order jets). For more detailed comments about that variational sequence see remark 2.8. The first formulation of a (long) variational sequence on finite order jet spaces is due to Krupka [23] (see [25] for the case n = 1). Below we will describe the main points of the approach of [23], and compare it with other approaches. In [23] a natural exact subsequence of the de Rham sequence on J r π is defined. This subsequence is made by contact forms and their differentials. Then we define the r–th order variational sequence to be the quotient of the de Rham sequence on J r π by means of the above exact subsequence. Local and global results about the variational sequence are proved using the fact that the above subsequence is globally exact and using the abstract de Rham theorem. Let us consider the sheaf of 1-contact forms C 1 Ω∗r , and denote by (dC p Ωkr )˜the sheaf generated by the presheaf dC p Ωkr . We set def 1 q Θqr = C Ωr + (dC 1 Ωq−1 r )˜ 0 ≤ q ≤ n,

def p p+n p p+n−1 Θp+n )˜ 1 ≤ p ≤ dim J r π. = C Ωr + (dC Ωr r

(10)

We observe that dC 1 Ωrq−1 ⊂ C 1 Ωqr , so that the second summand of the above first equation yields no contribution to C 1 Ωqr . The sheaves Θp+n become trivial when p + n > P , where r the value of P is computed in [23] using Theorem 1.3. Moreover, we have the following property (proved in [23]). Lemma 2.1. Let 0 ≤ k ≤ dim J r π. Then the sheaves Θkr are soft sheaves. We have the following natural soft subsequence of the de Rham sequence on J r π 0

Θ1r

d

Θ2r

d

...

d

ΘPr

d

0

Definition 2.2. The sheaf sequence (11) is said to be the contact sequence.

(11)

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Raffaele Vitolo

Theorem 2.3. The contact sequence is an exact soft resolution of C 1 Ω1r , hence the cohomology of the associated cochain complex of sections on any open subset of J r π vanishes. The above theorem is proved in [23] by first proving the local exactness of the contact sequence and then using standard results from sheaf theory (for which an adequate source is [50]). Standard arguments of homological algebra prove that the diagram in Figure 1 (p.106) is commutative, and its rows and columns are exact.

0

0

0

R

Ω0r

d

0

0

Θ1r

d

Θ2r

d

...

d

ΘPr

d

Ω1r

d

Ω2r

d

...

d

ΩPr

d EI

Ω1r /Θ1r 0

E1

Ω2r /Θ2r 0

E2

...

0

ΩPr +1

d

···0

EP −1

ΩPr /ΘPr 0

Figure 1. The r-th order variational bicomplex. Definition 2.4. The diagram in Figure 1 is said to be the r-th order variational bicomplex associated with the fibred manifold π : Y → X. We say the bottom row of the above diagram to be the r-th order variational sequence associated with the fibred manifold π : Y → X. Due to theorem 2.3 the finite order variational sequence is an exact sheaf sequence (this means that the sequence is locally exact, [50]). Hence both the de Rham sequence and the variational sequence are acyclic resolutions of the constant sheaf R (‘acyclic’ means that the sequences are locally exact with the exception of the first sheaf R). Next corollary follows by the abstract de Rham theorem. Corollary 2.5. The cohomology of the variational sequence is naturally isomorphic to the de Rham cohomology of J r π. The above finite order diagram yields a variational sequence which can be proved to be equal to the finite order variational sequence obtained from a finite order analogue of the C-spectral sequence [49]. Moreover, as one could expect, for 0 ≤ s < r pull-back via πr,s yields a natural inclusion of the s-th order variational bicomplex into the r-th order variational bicomplex. More precisely, we have the following lemma (see [23]).


107

Lemma 2.6. Let 0 ≤ s < r. Then we have the injective sheaf morphism ∗ χrs : Ωks /Θks → Ωkr /Θkr , [α] 7→ [πr,s α].

Hence, there is an inclusion of the s–th order variational bicomplex into the r–th order variational bicomplex. The inclusion commutes with the operators of the variational bicomplexes of orders s and r.

Having already dealt with local and global properties of the r-th order variational sequence, we are left with the problem of representing the quotient sheaves. This problem has been independently solved by many authors in the infinite order case. We recognize two different approaches to the problem: with differential forms (see for example [41, 42]) and with differential operators [43, 44]. The restriction to finite order jets of the former approach has been developed in [46] for p = 1, p = 2, and in [20, 21] for all p. See [49] for a finite order differential operator approach. We will describe the differential forms approach. First of all, it is obvious that, for 0 ≤ q ≤ n, horizontalization provides such a representation (see [23, 46]). Proposition 2.7. Let 0 ≤ q ≤ n. Then we have the isomorphism 0,q

Hq : Ωqr /Θqr → Ωr ,

[α] 7→ h0,q (α).

The quotient differential Eq reads through the above isomorphism as Hq+1 (Eq ([α])) = Hq+1 ([dα]) = h0,q+1 (dα) = dH h0,q (α). The last equality of the above equation is the least obvious, and was first proved in [3]. σ σ The proof depends on the fact that Di yI+j = yI+j+i , and that the indexes i, j are skew0,q symmetrized in the coefficients of dH h (α) (see the coordinate expression of h0,q ). Remark 2.8. In [3] the finite order variational sequence is developed starting from the idea of finding a subsequence of forms whose order do not change under dH . The authors prove that the above property characterizes the forms which are in the image of h0,q (see also [2]). Conversely, in [23] the idea is to start with forms on finite order jets, but the result is the same up to the degree q = n. When the degree of forms is greater than n we are able to provide isomorphisms of the quotient sheaves with other quotient sheaves made with proper subsheaves. This helps both to the purpose of representing quotient sheaves and to the purpose of comparing the current approach with others, as we will see. Proposition 2.9. Let p ≥ 1. The horizontalization hp,n induces the natural sheaf isomorphism p,n

Hp+n : Ωp+n /Θp+n → Ωr /hp,n ((dC p Ωp+n−1 )˜), r r r

[α] 7→ [hp,n (α)].

The quotient differential Ep+n reads through the above isomorphism as Hp+1+n (Ep+n ([α])) = Hp+1+n ([dα]) = [hp+1,n (dα)].

108

Raffaele Vitolo For a proof, see [46, 48]. Following [2, 41, 42], let us introduce the map 1 Ip (α) = ω σ ∧ (−1)|I| DI i∂/∂uσI α p

n Ip : C p Ωpr ∧ Hrn → C p Ωp2r ∧ H2r ,

(12)

where DI stands for the iterated Lie derivative (LD1 )i1 · · · (LDn )in . We say the map Ip to be the interior Euler operator. It can be proved [2, 20, 42] that the following properties of Ip hold - Ip is a natural map, i.e., LX 2r (Ip (α)) = Ip (LX r (α)), hence Ip is a global map; n , which is of the - if α ∈ C p Ωpr ∧ Hrn then there exists a unique form β ∈ C p Ωp2r ∧ H2r p n−1 type β = dH γ with γ ∈ C p Ω2r−1 ∧ H2r−1 , such that

α = I(α) + β.

(13)

Remark 2.10. The above form γ is not uniquely defined, in general. For p = 1, if the order of α is 1 it is easily proved that γ is uniquely defined; if the order of α is 2 then there exists a unique γ fulfilling a certain intrinsic property; if the order is 3 it is proved in [16, 17] that no natural γ of the above type exists. However, suitable linear connections on M and on the fibres of π : E → M can be used to determine a unique γ. See [1, 2] for the case of p > 1. n−1 It follows from the above theorem that if γ ∈ C p Ωp2r−1 ∧ H2r−1 then Ip (dH γ) = 0, so 2 that Ip = Ip .

Theorem 2.11. We have the isomorphism Ωp+n /Θp+n → Vrp , r r

[α] 7→ Ip (Hp+n ([α])),

n is a suitable subspace (see [46] for a characterization for where Vrp ⊂ C p Ωp2r+1 ∧ H2r+1 p = 1, p = 2).

For a proof, see [46] (p = 1, p = 2) and [20, 21] for any p. The above theorem also mean that, despite the fact that the denominator in proposition 2.9 is made by forms which are locally total divergences, only global divergences really matter. We say the elements of V p to be the p-th degree variational forms; for p = 1 they are also known as source forms. The map Ip+1 allows us to represent the differentials Ep+n through forms: Ip+1 (Hp+1+n (Ep+n ([α]))) = Ip+1 (Hp+1+n ([dα])).

(14)

From the coordinate expression of Ip it follows that En is just the Euler–Lagrange opdef erator and E1+n is just the Helmholtz operator. In fact, let ν = dx1 ∧ · · · ∧ dxn . Then, if 0,n λ ∈ Ωr , then λ = h0,n (α) = Lν, where L is a function with polynomial structure in r + 1-st order derivatives as in (5). Now we can use (14) on α, but if α is not known the computational problem of finding it can be technically difficult in principle. On the other hand, we can use the commutativity of the inclusion of Lemma 2.6 with the operators Ep+n and consider λ ∈ Ωnr+1 . Then h0,n (λ) = λ and En (λ) is the standard Euler–Lagrange


109

operator on the r + 1-st order Lagrangian λ. A similar reasoning proves that E1+n coincide with the Helmholtz operator. A different, computational approach to the problem of the representation of quotients is presented in [13, 14]. A further approach to the problem of representation appeared in [30] for the case n = 1. Here the concept of Lepagean equivalent is introduced in full generality (older version of this concept can be found e.g., in [22], with references to older foundational works). Namely, let α ∈ Ωp+n . Then a Lepage equivalent of [α] ∈ Ωp+n /Θp+n is a differential r r r p+n form β ∈ Ωr such that hp,n (β) = hp,n (α),

hp+1,n (dβ) = Ip+1 (hp+1,n (dα)).

The most important example of a Lepagean equivalent is the Poincaré–Cartan form of a Lagrangian (see, e.g., [22]).

3.

Some Related Problems

In this section we will briefly describe what are the most recent results which involve finite order variational sequences.

3.1.

Variationally Trivial Lagrangians

A variationally trivial Lagrangian is an element [α] ∈ Ωnr /Θnr such that En ([α]) = 0. If [α] is a variationally trivial Lagrangian, then by the local exactness of the variational sequence we have h0,n (α) = dH (h0,n−1 (β)) with [β] ∈ Ωn−1 /Θn−1 a local form. A r r n−1 n−1 global horizontal n − 1-form [β] ∈ Ωr /Θr such that [α] = dH [β] exists if and only if [α] induces the zero cohomology class in the variational sequence. A refinement of this result is the following theorem. Theorem 3.1. Let λ : J r π → ∧n T ∗ X induce a variationally trivial Lagrangian [λ]. Then, locally, λ = dH µ, where µ = h0,n−1 (α) and α ∈ Ωn−1 r−1 . In other words, λ = h0,n (dα), hence λ is the representative of a class En−1 ([α]) = r−1 [dα] ∈ Ωn−1 r−1 /Θn−1 . This means that λ depends on r-th order derivatives through hyperjacobians. This result has been proved in [3], [4] (here the proof is for the special case when the Lagrangian does not depend on (xi )), [14, 29] (here the proof uses the finite order variational sequence). See also [27] for another approach to the problem. Of course, the result is sharp: the order cannot be further lowered.

3.2.

Locally Variational Source Forms

such that E1+n ([α]) = 0. /Θn+1 A locally variational source form is an element [α] ∈ Ωn+1 r r If [α] is a locally variational source form, then by the local exactness of the variational sequence [α] is the Euler–Lagrange expression of a local Lagrangian, i.e., [α] = En ([β]) with [β] ∈ Ωnr /Θnr . A global Lagrangian [β] ∈ Ωnr /Θnr such that [α] = En ([β]) exists if and only if [α] = 0 ∈ H n+1 (Y ).

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The previous result is sharp with respect to the order [23, 46]. However, it can be very difficult to check that a source form is in the space Ωn+1 /Θn+1 . A result proved in [2] is r r (r) helpful in this sense. Let y denote all derivative coordinates of order r on a jet space. Let f ∈ C ∞ (J 2r π), and suppose that f (xi , y (0) , . . . , y (r) , ty (r+1) , t2 y (r+2) , . . . , tr y (2r) ) is a polynomial of degree less than or equal to r in y (s) , with r + 1 ≤ s ≤ 2r. Then f is said to be a weighted polynomial of degree r in the derivative coordinates of order r + 1 ≤ s ≤ 2r. Theorem 3.2. Let [∆] be a locally variational source form, with ∆ : J 2r π → C0∗ ∧∧n T ∗ X. Suppose that the coefficients of ∆ are weighted polynomials of degree less than or equal to r. Then ∆ = E(λ), where λ : J r π → ∧n T ∗ X. Again, the result is sharp with respect to the order of the jet space where the Lagrangian is defined. The above theorem is complemented in [2] by a rather complex algorithm for building the lowest order Lagrangian. This algorithm is an improvement of the well-known Volterra Lagrangian Z 1

L=

0

y σ ∆σ (xi , tyIτ )dt

for a locally variational source form ∆. In fact, the above Lagrangian is defined on the same jet space as ∆. The finite order variational sequence yields another method for computing lower order Lagrangians, provided we know that ∆ = [α] ∈ Ωn+1 /Θn+1 . Namely, we r r n+2 apply the contact homotopy operator to the closed form dα ∈ Θr , finding β ∈ Θn+1 r such that dβ = dα. Using the (standard) homotopy operator we find γ ∈ Ωnr such that def 0,n dγ = β − α, and λ = h (γ) is the required Lagrangian. Of course, the most difficult point is to invert the representation of quotients in the variational sequence, i.e., to find a least order α such that ∆ = [α]. The above theorem does not exhaust the finite order inverse problem. A locally variational source form ∆ on J 2r π seems to have a definite form of the coefficients with respect to its derivatives of order s, with r + 1 ≤ s ≤ 2r. It is an open problem to determine such a structure, e.g. prove that such forms always lie in Ωn+1 /Θn+1 for a minimal value of s; a s s least order Lagrangian would follow from the local exactness of the variational sequence. Finally, we recall that recently some geometric results on variational first-order partial differential equations have been obtained in [15]. Such equations arise in multisymplectic field theories.

3.3.

Contact Elements

Let Y be an n + m-dimensional manifold, and x ∈ Y . We say that two n-dimensional submanifolds L1 , L2 such that x ∈ L1 ∩ L2 are r-equivalent if they have a contact of order r at x. It is possible to choose a chart of Y at x of the form (xi , y σ ), 1 ≤ i ≤ n, 1 ≤ i ≤ m, where both L1 and L2 can be expressed as graphs y σ = f1σ (xi ), y σ = f2σ (xi ). Then the contact condition is the equality of the derivatives of the above functions f1 , f2 at x up to the order r. This is an equivalence relation whose quotient set is J r (Y, n), the manifold of r-th order n-dimensional contact elements. This construction was first formalized in [10], and is also known as r-th order jet space of n-dimensional submanifolds of Y [45] or extended jet bundle [34]). If Y is endowed with a fibring π, then J r π is the open and dense subspace of J r (Y, n) which is made by submanifolds which are transverse to the fibring at


111

a point (which, of course, can be locally identified with the images of sections, hence with local sections themselves). Of course, manifolds of contact elements have a contact distribution, hence a variational sequence can be formulated through the C-spectral sequence [7, 43, 44]. Manifolds of contact elements can also be seen as jets of parametrizations of submanifolds (i.e., jets of local n-dimensional immersions) up to the action of the reparametrization group [18]. In this setting another approach to the variational sequence is [38]. In [33] the finite-order C-spectral sequence on the manifold of contact elements is computed. Research based on Krupka’s approach on a variational sequence on finite order contact elements is in progress [31]. Another interesting research topic is the development of finite order variational structures on differential equations, i.e. submanifolds of jet spaces. This would possibly lead to a classification of their conservation laws of a certain order [35].

3.4.

Variational Sequence and Symmetries

The Lie derivative of variational forms is interesting for the determination of symmetries of Lagrangians and source forms. However, the result of a Lie derivative with respect to a prolonged vector field is a form which, in general, contains dH -exact terms. For this reason it is natural to use a new operator, the variational Lie derivative, which is defined up to dH -exact terms. Such a formula first appeared in [45] (‘infinitesimal Stokes’formula’) in the infinite order formalism. The finite order case has been dealt with in [11, 28]. See also [6] for symmetries of source forms which are locally but not globally variational. This topic has clear connections with Noether’s theorem, for which we invite the reader to consult the above literature.

3.5.

Further Topics

We already mentioned that other approaches to variational sequences exist in literature, mostly on infinite order jets. It can be proved that there exists an infinite order analogue of Krupka’s r-th order variational bicomplex [47]. This is defined in view of Lemma 2.6 via a direct limit of the injective family of r-th order variational bicomplexes. Nonetheless the direct limit infinite order bicomplex will be a bicomplex of presheaves, because gluing forms defined on jets of increasing order provides ‘forms’ which are only locally of finite order. The C-spectral sequence on jets of fibrings yields an infinite order variational sequence [7, 43, 44]. See [26, 48] for a comparison with Krupka’s approach and [49] for some finite order C-spectral sequence computations. In [36] the relationship between a part of the finite order variational sequence and the Spencer sequence are stressed. This relationship was already explored in [43, 44] in the case of infinite order jet spaces.

Acknowledgements It is a pleasure for me to acknowledge Professor Demeter Krupka. His hospitality and his inspiring seminars during my Ph.D. studies have been an invaluable contribution to my

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mathematical education. This research has been supported by PRIN 2005/2007 “Simmetrie e supersimmetrie classiche e quantistiche”, by the section GNSAGA of the Istituto Nazionale di Alta Matematica http://www.altamatematica.it, and by the Dipartimento di Matematica “E. De Giorgi” of the University of Salento.

References [1] R. Alonso Blanco, On the Green-Vinogradov formula, Acta Appl. Math. 72 (1-2) (2002) 19–32. [2] I. M. Anderson, The Variational Bicomplex (book preprint, technical report of the Utah State University, 1989); available at http://www.math.usu.edu/˜fg mp/). [3] I. M. Anderson and T. Duchamp, On the existence of global variational principles, American J. of Math. 102 (1980) 781–868; http://www.jstor.org/. [4] J. M. Ball, J. C. Currie and P. J. Olver, Null Lagrangians, weak continuity and variational problems of arbitrary order, J. Funct. Anal. 41 (1981) 135–174. [5] A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor′ kova, I. S. Krasil′ shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky and A. M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics ((I. S. Krasil′ shchik and A. M. Vinogradov, Eds.) Amer. Math. Soc., 1999). [6] J. Brajerˇc´ık and D. Krupka, Variational principles for locally variational forms, J. Math. Phys. 46 (2005) 052903. [7] P. Dedecker, On applications of homological algebra to calculus of variations and mathematical physics, In: Proceedings of the IV international colloquium on differential geometry (Santiago de Compostela, Universidad de Santiago de Compostela, 1978, Cursos y Congresos de la Universidad de Santiago de Compostela 15) 285– 294. [8] S. V. Duzhin, C-spectral sequence on the manifold J 1 M , Uspekhi Math. Nauk 38 (1983) 165–166 (in Russian); English translation: Russian Math. Surveys 38 (1) (1983) 179–181. [9] S. V. Duzhin, C-spectral sequence on manifolds of jets of finite order, preprint VINITI UDK 514.763.8 (1982) (in Russian). [10] C. Ehresmann, Les prolongements d’une variete differentiable. IV. Elements de contact et elements d’enveloppe, C. R. Acad. Sci. Paris 234 (1952) 1028–1030. [11] M. Francaviglia, M. Palese and R. Vitolo, Symmetries in finite order variational sequences, Czech. Math. J. 52 (127) (2002), 197–213; http://poincare.unile.it/vitolo.


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[12] I. M. Gel′ fand and L. A. Dikii, Asymptotic behaviour of the resolvent of Sturm– Liouville equations and the algebra of the Kortexeg–De Vries equations, Uspekhi Mat. Nauk 30 (5) (1975) 67–100 (in Russian); Russian Math. Surveys 30 (5) (1975) 77–113 (English translation). [13] D. R. Grigore, The Variational Sequence on Finite Jet Bundle Extensions and the Lagrangian Formalism, Diff. Geom. Appl. 10 (1999) 43–77. [14] D. R. Grigore, Variationally trivial Lagrangians and locally variational differential equations of arbitrary order, Diff. Geom. Appl. 10 (1999) 79–105. [15] A. Haková and O. Krupková, Variational first-order partial differential equations, J. Diff. Equat. 191 (2003) 67–89. [16] I. Koláˇr, A geometrical version of the higher order Hamilton formalism in fibred manifolds, J. Geom. Phys. 1 (2) (1984) 127–137. [17] I. Koláˇr, Natural operators related with the variational calculus, In: Proc. Conf. Diff. Geom. Appl. 1992 (Silesian Univ. Opava, 1993) 461–472. [18] I. Koláˇr, P. Michor and J. Slovák, Natural Operations in Differential Geometry (Springer-Verlag, 1993); http://www.emis.de/monographs/index.html. [19] I. S. Krasil′ shchik and A. M. Verbovetsky, Homological methods in equations of mathematical physics, Open Education and Sciences, Opava (Czech Rep.) 1998; arXiv:math.DG/9808130. [20] M. Krbek and J. Musilová, Representation of the variational sequence by differential forms, Rep. Math. Phys. 51 (2/3) (2003) 251–258. [21] M. Krbek and J. Musilová, Representation of the variational sequence by differential forms, Acta Appl. Math. 88 (2) (2005) 177–199. [22] D. Krupka, Some geometric aspects of the calculus of variations in fibred manifolds, Folia Fac. Sci. Nat. UJEP Brunensis, Brno University, 14 (1973); arXiv:mathph/0110005. [23] D. Krupka, Variational sequences on finite order jet spaces, In: Proc. Conf. Diff. Geom. Appl. 1989 (World Scientific, New York, 1990) 236–254. [24] D. Krupka, The contact ideal, Diff. Geom. Appl. 5 (1995) 257–276. [25] D. Krupka, Variational sequences in mechanics, Calc. Var. 5 (1997) 557–583. [26] D. Krupka, Variational sequences and variational bicomplexes, In: Differential Geometry and Applications (Proc. Conf., (I. Koláˇr, O. Kowalski, D. Krupka and J. Slovák, Eds.) Brno, August 1998, Masaryk Univ., Brno, Czech Republic, 1999) 525– 531; http://www.emis.de/proceedings. [27] D. Krupka, The total divergence equation, Lobachevskii Journal of Mathematics 23 (2006) 71–93.

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[28] D. Krupka, O. Krupková, G. Prince and W. Sarlet, Contact symmetries and variational sequences, In: Differential Geometry and Its Applications (Proc. Conf., (J. Bureˇs, O. Kowalski, D. Krupka and J. Slovák, Eds.) Prague, August 2004, Charles University in Prague, Czech Republic, 2005) 605–615. [29] D. Krupka and J. Musilová, Trivial Lagrangians in Field Theory, Diff. Geom. and Appl. 9 (1998) 293–505. ˇ enková, Variational sequences and Lepage forms, In: Differen[30] D. Krupka and J. Sedˇ tial Geometry and Its Applications (Proc. Conf., (J. Bureˇs, O. Kowalski, D. Krupka and J. Slovák, Eds.) Prague, August 2004, Charles University in Prague, Czech Republic, 2005) 617–627. [31] D. Krupka and Z. Urban, Differential invariants and higher order Grassmann bundles, In: Differential Geometry and its Applications (Proc. 10th Int. Conf. on Diff. Geom. and Appl., Olomouc 2007, World Scientific, Singapore, 2008) 463–473. [32] L. Mangiarotti and M. Modugno, Fibered Spaces, Jet Spaces and Connections for Field Theories, In: Proc. of the Int. Meeting on Geometry and Physics (Pitagora Editrice, Bologna, 1983) 135–165. [33] G. Manno and R. Vitolo, Variational sequences on finite order jets of submanifolds, In: Proc. of the VIII Conf. on Diff. Geom. and Appl. (Opava 2001, Czech Republic, http://www.emis.de/proceedings); see also the longer preprint arXiv:math.DG/0602127. [34] P. J. Olver, Applications of Lie Groups to Differential Equations (GTM 107, 2nd edition, Springer 1993). [35] P.J. Olver, private communication (2007). [36] J. F. Pommaret, Spencer sequence and variational sequence, Acta Appl. Math. 41 (1995) 285–296. [37] D. J. Saunders, The Geometry of Jet Bundles (Cambridge Univ. Press, 1989). [38] D. J. Saunders, Homogeneous variational complexes and bicomplexes; arXiv:math.DG/0512383. [39] F. Takens, A global version of the inverse problem of the calculus of variations, J. Diff. Geom. 14 (1979) 543–562. [40] W. M. Tulczyjew, Sur la différentielle de Lagrange, C. R. Acad. Sc. Paris, série A 280 (1975) 1295–1298. [41] W. M. Tulczyjew, The Lagrange Complex, Bull. Soc. Math. France 105 (1977) 419– 431. [42] W. M. Tulczyjew, The Euler–Lagrange Resolution, In: Internat. Coll. on Diff. Geom. Methods in Math. Phys. (Aix–en–Provence, 1979; Lecture Notes in Mathematics, n. 836, Springer–Verlag, Berlin, 1980) 22–48.


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[43] A. M. Vinogradov, On the algebro–geometric foundations of Lagrangian field theory, Soviet Math. Dokl. 18 (1977) 1200–1204; http://diffiety.ac.ru/. [44] A. M. Vinogradov, A spectral sequence associated with a non–linear differential equation, and algebro–geometric foundations of Lagrangian field theory with constraints, Soviet Math. Dokl. 19 (1978) 144–148; http://diffiety.ac.ru/. [45] A. M. Vinogradov, The C–Spectral Sequence, Lagrangian Formalism and Conservation Laws I and II, J. Math. Anal. Appl. 100 (1) (1984); http://diffiety.ac.ru/. [46] R. Vitolo, Finite order variational bicomplexes, Math. Proc. of the Camb. Phil. Soc. 125 (1998) 321–333; http://poincare.unile.it/vitolo; a more detailed version is available at arXiv:math-ph/0001009. [47] R. Vitolo, A new infinite order formulation of variational sequences, Arch. Math. Un. Brunensis 4 (34) (1998) 483–504; http://poincare.unile.it/vitolo. [48] R. Vitolo, On different geometric formulations of Lagrangian formalism, Diff. Geom. and its Appl. 10 (3) (1999) 225–255; updated version at http://poincare.unile.it/vitolo. [49] R. Vitolo, Finite order formulation of Vinogradov C–spectral sequence, Acta Appl. Math. 70 (1-2) (2002) 133–154; updated version at http://poincare.unile.it/vitolo. [50] R. O. Wells, Differential Analysis on Complex Manifolds (GTM, n. 65, Springer– Verlag, Berlin, 1980).


Chapter 7

C ONCATENATING VARIATIONAL P RINCIPLES AND THE K INETIC S TRESS -E NERGY-M OMENTUM T ENSOR Marco Castrillón López1∗, Mark J. Gotay2† and Jerrold E. Marsden3‡ Departamento de Geometr´ıa y Topolog´ıa, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain, 2 Department of Mathematics, University of Hawai‘i, Honolulu, Hawai‘i 96822, USA, 3 Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, California 91125, USA

1

Abstract We show how to “concatenate” variational principles over different bases into one over a single base, thereby providing a unified Lagrangian treatment of interacting systems. As an example we study a Klein–Gordon field interacting with a mesically charged particle. We employ our method to give a novel group-theoretic derivation of the kinetic stress-energy-momentum tensor density corresponding to the particle.

1.

Introduction and Setup

Let us recall the geometric setting of a classical variational principle [3]: We are given a fibration Y → X, with dim X = n + 1, and we wish to extremize an action of the form Z S(ψ) = L(j 1 ψ) X

where ψ : X → Y is a section and L : J 1 Y → Λn+1 X is a specified Lagrangian density.1 ∗

E-mail address: [email protected] E-mail address: [email protected] ‡ E-mail address: [email protected] 1 For simplicity we consider only first order theories. We also ignore technical issues and proceed formally. †

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Marko Castrillón López, Mark J. Gotay and Jerrold E. Marsden

One commonly encounters several (say K) such variational principles simultaneously, for instance when one studies the Newtonian dynamics of a swarm of charged particles (in a background electromagnetic field), or the interaction between Dirac and Yang–Mills fields. In the cases cited, the relevant fibrations have the form Yi → X for the ith variational principle; the key point being that each fibration has the same base X. To combine these variational principles into a single principle is a straightforward matter: one builds the fiber product Y1 ×X · · · ×X YK → X, and then on the first jet of this bundle one takes as the Lagrangian density an expression of the form L1 + · · · + LK + Lint for some interaction terms Lint . It is less clear how to deal with variational principles with disparate bases, that is, fibrations Yi → Xi in which the Xi are all different. A simple example is a nucleon moving in a dynamic Klein–Gordon field. (Here the configuration bundle for the nucleon is X ×R → R, where X is 4-dimensional spacetime and R is the material world line of the nucleon. The fibration for the Klein–Gordon field is R × X → X, sections of which are scalar fields on spacetime.) Even if the bases are identical, it may be desirable to distinguish them. This is the case, for instance, in relativistic multiparticle systems (cf. [1]), when one wants to parametrize each particle’s trajectory by its own proper time, as opposed to a single “universal” time. In this context of disparate bases, one standard way to proceed is as follows. Construct an action functional using sections ψi : Xi → Yi for the ith bundle by setting S(ψi , . . . , ψK ) =

K Z X i=1

Xi

1

Li (j ψi ) +

Z

X1 ×···×XK

Lint (j 1 ψ1 , . . . , j 1 ψK ).

(1.1)

Then varying these fields ψi , one obtains the Euler–Lagrange equations for the problem. See equation (2.1) for a specific example. However, while producing the Euler–Lagrange equations, this approach has the unsatisfactory feature of not yielding a field theory in the usual sense, in which the fields are sections of a single bundle and which has a well-defined Lagrangian density. This or some other formalism is needed if one wishes to tap into the machinery of multisymplectic geometry, multimomentum maps, stress-energy-momentum (“SEM”) tensors, and constraint theory, etc. To concatenate variational principles with disparate bases in such a way as to recapture a genuine field theory, we proceed as follows. To begin, construct the product bundle Y1 × · · · × YK → X1 × · · · × XK , which we denote Y → X for short. In agreement with experience we restrict attention to product sections of this bundle of the form ψ = (ψ1 , . . . , ψK ), where each ψi is a section of Yi → Xi . With ψ = (ψ1 , . . . , ψK ) such a section, j 1 ψ(x) = j 1 ψ1 (x1 ), . . . , j 1 ψK (xK ) where x = (x1 , . . . , xK ). Denote by J¯1 Y the subbundle of J 1 Y consisting of all such jets; equivalently, J¯1 Y = J 1 Y1 × · · · × J 1 YK . Given Lagrangian densities Li on the jet bundles J 1 Yi , it is simple enough to lift them to maps, still denoted by Li , on the concatenated jet bundle J¯1 Y by composing with projections: j 1 ψ(x) 7→ Li j 1 ψi (xi ) .

Concatenating Variational Principles

119

But how do we concatenate these Li into a single Lagrangian density? Even ignoring interaction terms, we cannot just add the Li as they take values in different spaces, viz. Λni +1 Xi and so need not be forms of equal rank. The trick is to “suspend” the Li : J¯1 Y → Λni +1 Xi to maps J¯1 Y → ΛN +K X, where N = n1 +· · ·+nK , by inserting suitable tensor densities in the Li to “even out” their ranks in the target. First, we pull Li back via the projection X → Xi to an (ni + 1)-form on X. Secci · · · × XK . Now in ond, for each i choose scalar densities Di of weight 1 on X1 × · · · X n +1 Li = Li d i xi the coefficient Li transforms as a scalar density of weight 1 on Xi , so the coefficient in ni +1x ∧ · · · ∧ d nK +1x = ±L D d N +Kx L¯i := Li d ni +1xi ∧ Di d n1 +1x1 ∧ · · · ∧ d\ i i i K

will also transform as a scalar density of weight 1 on X under the subgroup Diff(X1 ) × · · · × Diff(XK ) ⊂ Diff(X) (which is sufficient for our purposes). The densities Di are to be chosen by hand, depending on the precise structure of the system; see the examples in §§2. and 3.. Thus modified, we may assemble L¯1 + · · · + L¯K into a map L¯ : J¯1 Y → ΛN +K X. Interaction terms, which are typically defined over several of the bases Xi (again, see the following examples) are treated similarly. Finally, it is straightforward to deal with composite situations in which some of the bases are identical and others are not. Ultimately, the specific choice of the Di will not matter as long as Z Z N +K Li d ni +1x Li Di d x= Xi

X

for each i, that is, the concatenated action reduces to the original action. Specifically, this means that Z ¯ 1 ψ) = S(ψ) L(j X

where the right hand side is given by (1.1). In particular, the Euler–Lagrange equations remain unaltered when the Lagrangian L¯ is used in place of the action functional (1.1). Once we have a total Lagrangian density in hand (albeit possibly a distributional one), we may proceed in the usual fashion. Thus we may compute the equations of motion and various geometric objects, such as SEM tensors. To extract physical information from these objects, however, it will normally be necessary to “project” them from X to some Xi or products thereof; this projection is accomplished by integration over the remaining Xj . Rather than continuing to try to describe the procedure in generality, it is more instructive to illustrate it via a simple example. (It really is easier done than said!) In §2. we apply this method to a system consisting of a Klein–Gordon field interacting with a mesically charged particle. (Think of a pion field interacting with a nucleon.) Beyond illustrating concatenation, this example has interesting features which are worth elucidating. In particular, we study the SEM tensor density of this system. Its computation, following

120


[5], is interesting in that it naturally produces the Minkowski, or kinetic, SEM tensor for a moving particle as a matter of course. To our knowledge, this SEM tensor has never been derived via a Lagrangian from first principles; it has always been inserted into the formalism in an ad hoc manner. An important point therefore is that our method is not merely a ‘tidy’ means of packaging variational principles; it is capable of providing, in an entirely straightforward fashion, quantities which otherwise cannot be obtained except in makeshift ways. Finally in §3. we briefly indicate some other contexts in which our results should be useful.

2.

Motion of a Mesically Charged Particle in a Klein–Gordon Field

Let X be an oriented spacetime with metric G. We consider a real Klein–Gordon field φ : X → R of mass M interacting with a particle of mass m and mesic charge ε. The particle’s trajectory in spacetime (or “placement field”) is z : R → X. The base for the system is thus X × R, the second factor being thought of as a time axis,2 and the configuration bundle Y is then (R × X) × (X × R) → X × R with coordinates (φ, X a ) on the fiber and (xµ , λ) on the base. We set z a = X a ◦ z. Our presentation is based upon the excellent exposition in Chapter 8 of [1], to which we refer the reader for further information. The action (1.1) for the system in this case is usually written Z p 1 µν S(φ, z) = G (x)φ,µ (x)φ,ν (x) − M 2 φ(x)2 −G(x) d 4x 2 X Z Z 4 4 εφ(x)kz(λ)k − ˙ δ (x − z(λ)) d x dλ − mkz(λ)k ˙ dλ, (2.1) X×R

R

p where the dot denotes differentiation with respect to λ and kzk ˙ = −Gab z˙ a z˙ b . Observe that the bases for the free Klein–Gordon term and the free particle term are different, and that the interaction term in the middle lives on the product of these. Before proceeding, there are two technical issues that need to be resolved, stemming from the presence of the two factors of X in the configuration bundle. First, note that in the leading term of S, G is regarded as living on the X in the base, while in the last term it evidently resides on the X in the fiber. It is necessary to know precisely where G lives, as this has an effect on the subsequent analysis: if on the base, then G is treated as a field, while if on the fiber it is simply thought of as a geometric object. We reconcile these two interpretations by taking G to be anchored to the base, and then pulling it back to the fiber by means of the following construction.3,4 Introduce yet another factor of X in the 2

Not necessarily proper time. This is a variant of the Kuchaˇr method of parametrizing a classical field theory; see [6] and [2] for details. 4 At the end of this section we will briefly examine what happens if instead we anchor G to the fiber.

3


121

fiber along with diffeomorphisms η : X → X, viewed as sections of X × X → X, with corresponding configuration and multivelocity variables η a = X a ◦ η and η a µ = ∂(X a ◦ η)/∂xµ , respectively. (We can, and do, regard the two copies of X in the fiber as identical.) We use these auxiliary nondynamic fields, the diffeomorphism fields (or “diffields” for short, to (i) identify the copies of X in the fiber with that in the base, and (ii) endow the new copy of X in the fiber with the metric g = η∗ G with components gab = Gµν κµ a κν b , where (κµ a ) = (η a µ )−1 . All this is summarized in the following figure.

The general set up for the introduction of diffeomorphism fields.

Second, the delta function δ 4 (x − z(λ)) must be modified, as it compares elements x in the base with elements z(λ) in the fiber. As just indicated we can use the diffields fields to remedy this problem as well: we need only write δ 4 x − η −1 (z(λ)) instead. It is sometimes convenient to replace δ 4 x − η −1 (z(λ)) = δ 4 (η(x) − z(λ))(det η∗ )

(2.2)

using the properties of delta functions (cf. the Appendix), where η∗ is the Jacobian of η. From this we see that δ 4 x − η −1 (z(λ)) (i) is a scalar density on X (again, see the Appendix), and (ii) depends upon the spacetime derivatives of η, even though this is not obvious at first glance. The reason we do not insist on a fixed identification of the base X with the fiber X, and instead allow a variable identification by means of the diffields, is to allow some gauge freedom in the fields; see also footnote 8 below. Remark. Analogous fields η, called “covariance fields,”, are introduced in [6] and [2], but there they have a different purpose, namely, to make a field theory on a given background generally covariant and in doing so, they are introduced as dynamic fields.

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In addition to the diffields η, we introduce a (positive-definite) metric K on R as a nondynamic field. We suppose that K is chosen so that R has metric volume 1. With these fixes we may concatenate the three action terms over the composite R now 4 ¯ base X × R as S(φ, z) = L d x dλ, with the Lagrangian √ √ ¯ µ , λ, φ, φ,µ , z b , z˙ b ; η a , η a µ , Gρσ , K) = 1 Gµν φ,µ φ,ν − M 2 φ2 −G K L(x 2 4 − m + εφ kzk ˙ δ (η − z)(det η∗ ). (2.3)

Notice that the interaction term itself needs no essential modification, as the corresponding term in (2.1) is already an integral over X × R,but informs our choice of scalar density in the free particle term, viz. δ 4 x − η −1 (z(λ)) , when we suspend the latter to X × R. ¯ is defined We have also written this delta function in the form (2.2) to make it clear that L pointwise. √ Remark. The choice of R scalar density D = K in the Klein–Gordon term is hardly unique; all we require is that R D dλ = 1. For instance, we could instead take δ(λ) for D with no essential difference. As evident from (2.3), the modified configuration bundle is taken to be Y ′ = Y ×X X 2 ×X Lor(X) ×R Riem(R), where we abbreviate the bundle X × X → X by X 2 , Lor(X) is the bundle whose sections are Lorentz metrics on X and, similarly, Riem(R) is the bundle whose sections are Riemannian metrics on R. However, in our approach φ and z are variational, while η, G and K are nondynamic fields. As per the above, we now regard q kzk ˙ = −Gµν κµ a κν b z˙ a z˙ b . Remark. Occasionally, as in [9], one encounters what one might call “noncovariant concatenations.” In the current example, this amounts to writing the terms in the action as integrals over X alone and is effectively accomplished by imposing the coordinate condition x0 = λ. As this procedure is not covariant, it can lead to problems [10]. We compute the Euler–Lagrange equations. Varying with respect to φ and employing (2.2), we obtain p p −M 2 φ(x) −G(x) K(λ) − εkz(λ)k ˙ δ 4 x − η −1 (z(λ)) p √ − ∂µ Gµν φ,ν −G (x) K(λ) = 0.

Integrating with respect to λ, using the fact that volK (R) = 1, and rearranging, this reduces to the Klein–Gordon equation ∇µ ∇µ φ + M 2 φ = −ρ where ∇ denotes the G-covariant derivative and Z − 12 ρ(x) = ε(−G) kz(λ)k ˙ δ 4 x − η −1 (z(λ)) dλ R

(2.4)


123

is the source density. Similarly, varying with respect to z and employing (2.2) yield i ∂ h 4 −1 m + εφ(x) k z(λ)k ˙ δ x − η (z(λ)) ∂z a gab (z(λ))z˙ b (λ) ∂ −1 m + εφ(x) x − η (z(λ)) = 0. + ∂λ kz(λ)k ˙

Carrying out the differentiation and then integrating over X, some manipulations give " # b (λ) d g (z(λ)) z ˙ ab m + εφ η −1 (z(λ)) dλ kz(λ)k ˙ = −εκµ a φ,µ η −1 (z(λ)) kz(λ)k ˙ gbc,a (z(λ))z˙ b (λ)z˙ c (λ) −1 + m + εφ η (z(λ)) . (2.5) 2kz(λ)k ˙ To give insight into these equations, note that in the special case when (X, G) is Minkowski spacetime, η = IdX , and λ is taken to be proper time along the particle’s world line, these equations simplify in a global Lorentz frame to i d h m + εφ(z(λ)) z˙a (λ) = −εφ,a (z(λ)). dλ

This is the mesic analogue of the Lorentz force law in electrodynamics. Neither K, the Gµν , nor the η a have field equations, since they are not variational. Thus one is free to assign them whatever values one wishes in (2.4) and (2.5). Often, however, one has specific values of G and η in mind, e.g., the given spacetime metric for G and IdX for η. Turning now to the SEM tensor, let Diffc (X) × Diffc (R) (that is, the group of diffeomorphisms that are the identity outside a compact set) act on the modified configuration bundle Y ′ according to (σ × f ) · (x, λ, φ, z; η, G, K) = σ(x), f (λ), φ, z; η, σ∗ G, f∗ K . (We assume that all diffeomorphisms are positively oriented.) The Lagrangian density L¯ = ¯ d 4x dλ is then visibly equivariant with respect to the induced action on J¯1 Y ′ , that is,5 L L¯ (σ × f ) · j 1 (φ, z; η, G, K = (σ × f )∗ L¯ j 1 (φ, z; η, G, K) . We may thus use equation (3.12) in [5] to compute the 5-dimensional SEM tensor den-

sity T = 5

T µν T 4ν T µ4 T 44

Even though the pointwise action of Diff(X) on the fiber of the“diffeomorphism bundle” X × X → X is taken to be trivial, its action on sections thereof is not: σ · η = η ◦ σ −1 . Thus the identification of the factor of X in the base with that in the fiber can fluctuate, which is one of the reasons we allow η to be variable in the first place.

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of the interacting system (where x4 = λ).6 Integrating over λ and raising an index, we project out the spacetime SEM tensor density: Z µν T = T µν dλ = tµν + (m + ǫφ)Θµν , (2.6) R

where

√ 1 (2Gµα Gνβ − Gµν Gαβ )φ,α φ,β + Gµν M 2 φ2 −G 2 is the canonical SEM tensor density of the (free) Klein–Gordon field and tµν = −

µν

µ

ν

Θ (x) = κ a κ

b

z˙ a (λ)z˙ b (λ) 4 δ x − η −1 (z(λ)) dλ kz(λ)k ˙

Z

R

is the Minkowski tensor density. (mΘµν is then the kinetic SEM tensor density). As well, we compute T 4 ν = 0 = T µ 4 . Finally, we find that when integrated over X, T 4 4 is effectively the Klein–Gordon action: Z √ √ 1 4 4 µν 2 2 −G d x K. T 4= G φ,µ φ,ν − M φ 2 X Remark. The kinetic SEM tensor density is a familiar object in microscopic continuum mechanics, cf. Chapter 8 of [1] and §33 of [9]. Minkowski [11] originally introduced it in flat-spacetime electrodynamics in order to recover the continuity equation Tµν ,ν = 0 in view of the fact that Tµν EM ,ν 6= 0 when currents are present. In the continuum limit of a noninteracting clutch of particles, Θµν goes over to the SEM tensor density for a perfect fluid as in §§9.1-2 of [1]. It is interesting that in this limit, the infinite time integrals in the kinetic SEM tensor density disappear and one is left with a local tensor density. To our knowledge, ours is the first genuine derivation of the Minkowski tensor density from first principles in a variational context, once again illustrating the power of multisymplectic geometry in classical field theory and in particular, the usefulness of having a concatenated theory for which one can make use of concepts such as the SEM tensor. As we have defined it, the Minkowski tensor density depends upon the diffields as well as the particle placement field. However, note that when η = IdX , Θ reduces to the more familiar expression ab

Θ (x) =

Z

R

z˙ a (λ)z˙ b (λ) 4 δ (x−z(λ)) dλ. kz(λ)k ˙

Remark. Suppose we focus solely on the particle dynamics so that the (original) configuration bundle is X × R → R. The corresponding Lagrangian density −mkz(λ)kdλ ˙ is Diffc (R)-covariant, and so we may compute the corresponding SEM scalar density as in Example a, Interlude II of [4]. We obtain T = −E, the energy of the particle, which vanishes as the Lagrangian is time reparametrization-invariant. (This is reflected by the 6

Using the product metric G ⊕ K on X × R, one could also compute T via the Hilbert formula (4.2) in [5]. See also [10].


125

vanishing of T 4 4 in the 5-dimensional context when there is no Klein–Gordon field.) Thus only when the spacetime X is part of the base of the variational principle do we encounter the kinetic SEM tensor density; it does not appear in standard particle dynamics per se. To reiterate, even in the absence of other fields, our technique yields yet another (5dimensional!) treatment of the relativistic free particle that has the advantage of automatically incorporating the Minkowski tensor. Remark. Note that the term εφΘµν in (2.6) arises from the interaction of φ with the mesically charged particle. This term has no analogue in the electrodynamics of particles; there we get simply µν Tµν = Tµν EM + mΘ . Charged strings behave similarly, as we show in §3A (cf. equation (3.1)). This can be traced to the fact that the electromagnetic field is a covector, while the Klein–Gordon field is a scalar. The SEM tensor density T is symmetric. It is also divergence-free, as can be seen from general principles (cf. Proposition 5 in [5]). One may verify this directly, via a long calculation. We end with a discussion of an alternate treatment of this system. Suppose we consider the physical metric as a geometric object g on the fiber as opposed to a field on spacetime. Then we would define G = η ∗ g with components Gµν = η a µ η b ν gab . Proceeding as in the above, the Lagrangian density would be ¯ µ , λ, φ, φ,µ , z b , z˙ b ; η a , η a µ , K) L(x √ √ 1 µ ν ab κ a κ b g φ,µ φ,ν − M 2 φ2 −g (det η∗ ) K = 2 − m + εφ kzk ˙ δ 4 (η − z)(det η∗ )

p √ √ ˙ = −gab (z)z˙ a z˙ b . where −G = −g (det η∗ ) and kzk Computing the SEM tensor density in this formulation, we obtain T µ ν ≡ 0 and the other components as before. That the spacetime components vanish is actually a consequence of the generalized Hilbert formula (3.13) in [5], since the nondynamic fields η and K do not transform under Diff(X). (In the original formulation, the nondynamic metric G on X does transform under the spacetime diffeomorphism group with the result that (2.6) is nonzero.) The difference between this SEM tensor density and the previous one stems from: (i) the spacetime metric no longer being regarded as a field, so that it cannot contribute to the energy, momentum, and stress content of the system, and (ii) the subtly different manners in which the diffields appear in the two formulations. That one can encounter several SEM tensor densities for the ‘same’ system may seem surprising, but is unavoidable and can also be regarded as different “packaging” of the same information. What the SEM tensor density turns out to be depends upon what the fields are, whether they are dynamic, and precisely how they appear in the Lagrangian. And even the size of the SEM tensor density depends upon how the system is formulated! For instance, for something as simple as a relativistic free particle, we can have a 1 × 1 SEM tensor density (which vanishes identically)—as noted in a previous remark, or a 5 × 5 SEM tensor

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density (which does or doesn’t, depending on where the spacetime metric is anchored). And in the latter case, the 5 × 5 object reduces to the 4 × 4 Minkowski tensor density! Thus how the system is formulated plays a substantial role insofar as how various quantities, and in particular the SEM tensor density, are to be understood.

3.

Further Examples and Outlook

To conclude we briefly mention some other systems for which our techniques should prove helpful. We begin by upping the dimension of the matter from 1 to 2, that is, we replace the particle by a string. For variety, we also replace the mesic interaction by an electromagnetic one.

Charged Strings. We closely follow the exposition in §2. Let (X, G) and (W = R × B, H = −HR ⊕ HB ) be 4- and 2-dimensional Lorentzian spacetimes, respectively. We consider a string worldsheet in X, this being a map z : W → X. We use coordinates xµ , λA = (τ, σ) as coordinates on X × W . The configuration bundle Y is correspondingly Λ1 X ×W (X × W ) ×X Lor(X) ×W Lor(W ) → X × W. Assume that the string carries a charge density ρ : B → R and interacts with a dynamic electromagnetic field described by a potential 1-form A on X. We also take the metric H on W to be dynamic; thus we adopt the Polyakov approach as in [7]. The action for this system is 1 S(A, z, H) = − 4

Z

ZX

p F µν (x)Fµν (x) −G(x) d 4x

p ∂z µ (τ, σ) δ 4 (x − z(τ, σ)) HB (σ) d 4x dτ dσ ∂τ X×W Z p T − H AB (λ) Gµν (z(λ))z µ (λ),A z ν (λ),B −H(λ) d 2λ 2 W +

Aµ (x)ρ(σ)

where T is the tension. As in the case of the meson, we see that z takes values in X, which is the base for the electromagnetic field. So we need to introduce diffields as before. As well, we take the spacetime metric to reside on the factor of X in the base. Finally, let K be a nondynamic Riemannian metric on W with total volume 1. The modified configuration bundle for the concatenated variational principle is then Y ×X X 2 ×W Riem(W ) → X × W


127

and the Lagrangian reads ¯ µ , τ, σ, Aµ , Aµ,ν , z a , z a ,A , HAB ; η a , η a ν , Gµν , KAB ) L(x √ √ 1 = − Gµα Gνβ Fµν Fαβ −G K 4 p ∂z a 4 + Aµ κµ a ρ δ η − z)(det η∗ ) HB ∂τ √ T AB − H Gµν κµ a κν b z a ,A z b ,B δ 4 (η − z)(det η∗ ) −H. 2 Now build the product metric G ⊕ K on X × W . Using the Hilbert formula and then integrating as before, we compute the 6-dimensional SEM tensor density as follows: the spacetime components are µν Tµν = Tµν (3.1) EM + T Θ where Tµν EM

√ 1 µν αβ νβ αµ = − G Fαβ F + G F Fβα −G 4

is the free electromagnetic SEM tensor density, and Z √ µν µ ν Θ = κ aκ b H AB z a ,A z b ,B δ 4 x − η −1 (z(σ)) −H d 2σ W

is the analogue of the Minkowski tensor density for strings. The ‘extra’ T µA and T Aµ components are zero and, after integrating over X, the T AB subblock reduces to Z √ √ 1 µα νβ AB 4 G G Fµν Fαβ −G d x K AB K. T =− 4 X Continua. Another intriguing example that we intend to pursue in future works is a charged elastic body, fluid, or plasma, in which one concatenates a continuum with electromagnetism on a given background metric spacetime. Such theories will likely have significant differences with the particle and string examples presented above. Amongst these differences, we expect that, unlike mesically or electrically charged particles, continua should have well-defined initial value problems (see also the discussion of this point in [1]). Evidence for this can be found in works such as [8] and [12]. One other interesting aspect of a charged elastic body is the following. If B is the body manifold, then its motion in spacetime is determined by a map z : R × B → X. The main difference from our previous two examples is that rather than the delta functions 4 −1 δ (η(x) − z(λ)) we must now use characteristic functions χ η (z(R × B)) . We expect that examples such as this will be key players in the future development of the point of view given in this paper.

Appendix Let M be a manifold with coordinates x = (x1 , . . . , xm ). Here we prove that the delta function δ m (x − x0 ) transforms as a scalar density of weight 1.

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Marko Castrillón López, Mark J. Gotay and Jerrold E. Marsden Let η : M → M be a diffeomorphism and f ∈ C ∞ (X). On the one hand, Z (f ◦ η)(x0 ) = (f ◦ η)(x) δ m (x − x0 ) d m x. X

On the other hand, by the change of variables theorem with y = η(x), Z f (η(x0 )) = f (y) δ m (y − η(x0 )) d m y X Z = f (η(x)) δ m (η(x) − η(x0 )) |J(x)| d m x. X

where J is the Jacobian determinant of η. Since f is arbitrary the desired result follows upon comparing these two formulæ.

References [1] J. L. Anderson, Principles of Relativity Physics (Academic Press, New York, 1967). [2] M. Castrillón López, M. J. Gotay and J. E. Marsden, Parametrization and stressenergy-momentum tensors in metric theories, J. Phys. A:Math. Theor. (2008); to appear. [3] M. J. Gotay, J. A. Isenberg, J. E. Marsden and R. Montgomery, Momentum maps and classical fields, I: Covariant field theory, (1998); arXiv:physics/9801019. [4] M. J. Gotay, J. A. Isenberg and J. E. Marsden, Momentum maps and classical fields, II: Canonical analysis of field theories, (2004); arXiv:math-ph/0411032. [5] M. J. Gotay and J. E. Marsden, Stress-energy-momentum tensors and the Belinfante– Rosenfeld formula, Contemp. Math. 132 (1992) 367–391. [6] M. J. Gotay and J. E. Marsden, Parametrization theory, (2008); in preparation. [7] M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Volume I: Introduction (Cambridge Univ. Press, Cambridge, 1987). [8] M. Kunzinger, G. Rein, R. Steinbauer and G. Teschl, On classical solutions of the relativistic Vlasov–Klein–Gordon system, Electronic J. Differential Equations (1) (2005) pp. 17. [9] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Fourth revised English ed., Permagon Press, Aberdeen, 1979). [10] M. Leclerc, Canonical and gravitational stress-energy tensors, (2006); arXiv:grqc/0510044. [11] H. Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, Nach. Ges. Wiss. Göttingen (1908) 53–111. [12] G. Rein, Generic global solutions of the relativistic Vlasov–Maxwell system of plasma physics, Comm. Math. Phys. 135 (1990) 41–78.


Chapter 8

A G EOMETRIC H AMILTON -JACOBI T HEORY FOR C LASSICAL F IELD T HEORIES∗ Manuel de León1†, Juan Carlos Marrero2‡ and David Mart´ın de Diego3§ 1 Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Serrano 123, 28006 Madrid, Spain 2 Departamento de Matemática Fundamental, Universidad de La Laguna, La Laguna, Canary Islands, Spain, 3 Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Serrano 123, 28006 Madrid, Spain

Abstract In this paper we extend the geometric formalism of the Hamilton-Jacobi theory for hamiltonian mechanics to the case of classical field theories in the framework of multisymplectic geometry and Ehresmann connections.

2000 Mathematics Subject Classification. 70S05, 49L99. Key words and phrases. Multisymplectic field theory, Hamilton-Jacobi equations.

1.

Introduction

The standard formulation of the Hamilton-Jacobi problem is to find a function S(t, q A ) (called the principal function) such that ∂S A ∂S + H q , A = 0. (1.1) ∂t ∂q ∗

To Prof. Demeter Krupka in his 65th birthday E-mail address: [email protected] ‡ E-mail address: [email protected] § E-mail address: [email protected] †

130

M. de León, J.C. Marrero and D. Mart´ın de Diego

If we put S(t, q A ) = W (q A ) − tE, where E is a constant, then W satisfies A ∂W H q , A = E; ∂q

(1.2)

W is called the characteristic function. Equations (1.1) and (1.2) are indistinctly referred as the Hamilton-Jacobi equation. There are some recent attempts to extend this theory for classical field theories in the framework of the so-called multisymplectic formalism [15, 16]. For a classical field theory the hamiltonian is a function H = H(xµ , y i , pµi ), where (xµ ) are coordinates in the spacetime, (y i ) represent the field coordinates, and (pµi ) are the conjugate momenta. In this context, the Hamilton-Jacobi equation is [17] µ ∂S µ ν i ∂S =0 + H x ,y , ∂xµ ∂y i

(1.3)

where S µ = S µ (xν , y j ). In this paper we introduce a geometric version for the Hamilton-Jacobi theory based in two facts: (1) the recent geometric description for Hamiltonian mechanics developed in [6] (see [8] for the case of nonholonomic mechanics); (2) the multisymplectic formalism for classical field theories [3, 4, 5, 7] in terms of Ehresmann connections [9, 10, 11, 12]. We shall also adopt the convention that a repeated index implies summation over the range of the index.

2.

A Geometric Hamilton-Jacobi Theory for Hamiltonian Mechanics

First of all, we give a geometric version of the standard Hamilton-Jacobi theory which will be useful in the sequel. Let Q be the configuration manifold, and T ∗ Q its cotangent bundle equipped with the canonical symplectic form ωQ = dq A ∧ dpA

where (q A ) are coordinates in Q and (q A , pA ) are the induced ones in T ∗ Q.

Let H : T ∗ Q −→ R a hamiltonian function and XH the corresponding hamiltonian vector field: iXH ωQ = dH The integral curves of XH , (q A (t), pA (t)), satisfy the Hamilton equations: ∂H dpA ∂H dq A = , =− A. dt ∂pA dt ∂q Theorem 2.1 (Hamilton-Jacobi Theorem). Let λ be a closed 1-form on Q (that is, dλ = 0 and, locally λ = dW ). Then, the following conditions are equivalent:

Hamilton-Jacobi Theory for Classical Field Theories

131

(i) If σ : I → Q satisfies the equation ∂H dq A = dt ∂pA then λ ◦ σ is a solution of the Hamilton equations; (ii) d(H ◦ λ) = 0. To go further in this analysis, define a vector field on Q: λ XH = T π Q ◦ XH ◦ λ

as we can see in the following diagram: XH

T ∗Q

T (T ∗ Q)

πQ

λ

T πQ λ XH

Q

TQ

Notice that the following conditions are equivalent: (i) If σ : I → Q satisfies the equation ∂H dq A = dt ∂pA then λ ◦ σ is a solution of the Hamilton equations; λ , then λ ◦ σ is an integral curve of X ; (i)’ If σ : I → Q is an integral curve of XH H λ are λ-related, i.e. (i)” XH and XH λ T λ(XH ) = XH ◦ λ

so that the above theorem can be stated as follows: Theorem 2.2 (Hamilton-Jacobi Theorem). Let λ be a closed 1-form on Q. Then, the following conditions are equivalent: λ and X are λ-related; (i) XH H

(ii) d(H ◦ λ) = 0.

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3.

The Multisymplectic Formalism

3.1.

Multisymplectic Bundles

The configuration manifold in Mechanics is substituted by a fibred manifold π : E −→ M such that (i) dim M = n, dim E = n + m (ii) M is endowed with a volume form η. We can choose fibred coordinates (xµ , y i ) such that η = dx1 ∧ · · · ∧ dxn . We will use the following useful notations: dn x = dx1 ∧ · · · ∧ dxn , dn−1 xµ = i

∂ ∂xµ

dn x .

Denote by V π = ker T π the vertical bundle of π, that is, their elements are the tangent vectors to E which are π-vertical. Denote by Π : Λn E −→ E the vector bundle of n-forms on E. The total space Λn E is equipped with a canonical n-form Θ: Θ(α)(X1 , . . . , Xn ) = α(e)(T Π(X1 ), . . . , T Π(Xn )), where X1 , . . . , Xn ∈ Tα (Λn E) and α is an n-form at e ∈ E. The (n + 1)-form

Ω = −dΘ ,

is called the canonical multisymplectic form on Λn E. Denote by Λnr E the bundle of r-semibasic n-forms on E, say Λnr E = {α ∈ Λn E | iv1 ∧···∧vr α = 0, whenever v1 , . . . , vr are π-vertical}. Since Λnr E is a submanifold of Λn E it is equipped with a multisymplectic form Ωr , which is just the restriction of Ω. Two bundles of semibasic forms play an special role: Λn1 E and Λn2 E. The elements of these spaces have the following local expressions: Λn1 E

:

p0 dn x,

Λn2 E

: p0 dn x + pµi dy i ∧ dn−1 xµ


133

which permits to introduce local coordinates (xµ , y i , p0 ) and (xµ , y i , p0 , pµi ) in Λn1 E and Λn2 E, respectively. Since Λn1 E is a vector subbundle of Λn2 E over E, we can obtain the quotient vector space denoted by J 1 π ∗ which completes the following exact sequence of vector bundles: 0 −→ Λn1 E −→ Λn2 E −→ J 1 π ∗ −→ 0 . We denote by π1,0 : J 1 π ∗ −→ E and π1 : J 1 π ∗ −→ M the induced fibrations.

3.2.

Ehresmann Connections in the Fibration π1 : J 1 π ∗ −→ M

A connection (in the sense of Ehresmann) in π1 is a horizontal subbundle H which is complementary to V π1 ; namely, T (J 1 π ∗ ) = H ⊕ V π1 where V π1 = ker T π1 is the vertical bundle of π1 . Thus, we have: (i) there exists a (unique) horizontal lift of every tangent vector to M ; (ii) in fibred coordinates (xµ , y i , pµi ) on J 1 π ∗ , then ∂ ∂ V π1 = span , , H = span {Hµ } , ∂y i ∂pµi where Hµ is the horizontal lift of

∂ ∂xµ .

(iii) there is a horizontal projector h : T J 1 π ∗ −→ H.

3.3.

Hamiltonian Sections

Consider a hamiltonian section h : J 1 π ∗ −→ Λn2 E of the canonical projection µ : Λn2 E −→ J 1 π ∗ which in local coordinates read as h(xµ , y i , pµi ) = (xµ , y i , −H(x, y, p), pµi ) . Denote by Ωh = h∗ Ω2 , where Ω2 is the multisymplectic form on Λn2 E. The field equations can be written as follows: ih Ωh = (n − 1) Ωh ,

(3.1)

where h denotes the horizontal projection of an Ehresmann connection in the fibred manifold π1 : J 1 π ∗ −→ M . The local expressions of Ω2 and Ωh are:

Ω2 = −d(p0 dn x + pµi dy i ∧ dn−1 xµ ),

Ωh = −d(−H dn x + pµi dy i ∧ dn−1 xµ ) .

134


3.4.

The Field Equations

Next, we go back to the Equation (3.1). The horizontal subspaces are locally spanned by the local vector fields ∂ ∂ ∂ ∂ Hµ = h = + Γiµ i + (Γµ )νj , ∂xµ ∂xµ ∂y ∂pνj where Γiµ and (Γµ )νj are the Christoffel components of the connection. Assume that τ is an integral section of h; this means that τ : M −→ J 1 π ∗ is a local section of the canonical projection π1 : J 1 π ∗ −→ M such that T τ (x)(Tx M ) = Hτ (x) , for all x ∈ M . If τ (xµ ) = (xµ , τ i (x), τiµ (x)) then the above conditions becomes ∂τ i ∂H ∂τiµ ∂H = µ , =− i µ µ ∂x ∂y ∂pi ∂x which are the Hamilton equations.

4.

The Hamilton-Jacobi Theory

Let λ be a 2-semibasic n-form on E; in local coordinates we have λ = λ0 (x, y) dn x + λµi (x, y) dy i ∧ dn−1 xµ . Alternatively, we can see it as a section λ : E −→ Λn2 E, and then we have λ(xµ , y i ) = (xµ , y i , λ0 (x, y), λµi (x, y)) . A direct computation shows that ∂λ0 ∂λµi − µ dy i ∧ dn x dλ = ∂y i ∂x µ ∂λi + dy j ∧ dy i ∧ dn−1 xµ . ∂y j Therefore, dλ = 0 if and only if ∂λ0 ∂y i

=

∂λµi ∂y j

=

∂λµi , ∂xµ ∂λµj . ∂y i

(4.1) (4.2)

Using λ and h we construct an induced connection in the fibred manifold π : E −→ M by defining its horizontal projector as follows: ˜e h

: Te E −→ Te E, ˜ he (X) = T π1,0 ◦ h(µ◦λ)(e) ◦ ǫ(X)


135

where ǫ(X) ∈ T(µ◦λ)(e) (J 1 π ∗ ) is an arbitrary tangent vector which projects onto X. From the above definition we immediately proves that

˜ is a well-defined connection in the fibration π : E −→ M . (i) h (ii) The corresponding horizontal subspaces are locally spanned by ∂ ∂ ˜ ∂ ˜µ = h H = + Γiµ ((µ ◦ λ)(x, y)) i . ∂xµ ∂xµ ∂y The following theorem is the main result of this paper. ˜ is a flat conTheorem 4.1. Assume that λ is a closed 2-semibasic form on E and that h nection on π : E −→ M . Then the following conditions are equivalent: ˜ then µ ◦ λ ◦ σ is a solution of the Hamilton equations. (i) If σ is an integral section of h (ii) The n-form h ◦ µ ◦ λ is closed. Before to begin with the proof, let us consider some preliminary results. We have (h ◦ µ ◦ λ)(xµ , y i ) = (xµ , y i , −H(xµ , y i , λµi (x, y)), λµi (x, y)) , that is h ◦ µ ◦ λ = −H(xµ , y i , λµi (x, y)) dn x + λµi dy i ∧ dn−1 xµ . Notice that h ◦ µ ◦ λ is again a 2-semibasic n-form on E.

A direct computation shows that d(h ◦ µ ◦ λ) = −

∂λµi ∂H ∂λνj ∂H + + ∂y i ∂pνj ∂y i ∂xµ

!

dy i ∧ dn x +

Therefore, we have the following result. Lemma 4.2. Assume dλ = 0; then d(h ◦ µ ◦ λ) = 0 if and only if

∂λµi ∂H ∂H ∂λνj + + =0. ∂y i ∂pνj ∂y i ∂xµ

Proof of the Theorem. (i) ⇒ (ii)

It should be remarked the meaning of (i). Assume that σ(xµ ) = (xµ , σ i (x))

∂λµi dy j ∧ dy i ∧ dn−1 xµ . ∂y j

136


˜ then is an integral section of h;

∂σ i ∂H = µ . ∂xµ ∂pi

(i) states that in the above conditions, (µ ◦ λ ◦ σ)(xµ ) = (xµ , σ i (x), σ ¯jν = λνj (σ(x))) is a solution of the Hamilton equations, that is, ∂σ ¯iµ ∂λµi ∂λµi ∂σ j ∂H = + = − i. ∂xµ ∂xµ ∂y j ∂xµ ∂y Assume (i). Then ∂λµi ∂H ∂H ∂λνj + + ∂y i ∂pνj ∂y i ∂xµ =

∂λµi ∂H ∂H ∂λνi + + , ∂y i ∂pνj ∂y j ∂xµ

∂λµi ∂H ∂σ j ∂λνi + + , ∂y i ∂xν ∂y j ∂xµ =0 (since (i))

=

(since dλ = 0) (since the first Hamilton equation)

which implies (ii) by Lemma 4.2. (ii) ⇒ (i)

Assume that d(h ◦ µ ◦ λ) = 0. ˜ is a flat connection, we may consider an integral section σ of h. ˜ Suppose that Since h σ(xµ ) = (xµ , σ i (x)). Then, we have that

∂σ i ∂H = µ. ∂xµ ∂pi

Thus, ∂σ ¯jµ ∂xµ

∂λµj

∂λµj ∂σ i = + , ∂xµ ∂y i ∂xµ ∂λµj ∂λµi ∂σ i = + , (since dλ = 0) ∂xµ ∂y j ∂xµ ∂λµj ∂λµi ∂H = + , (since the first Hamilton equation) ∂xµ ∂y j ∂pµi ∂H =− j , (since (ii)). 2 ∂y

Assume that λ = dS, where S is a 1-semibasic (n − 1)-form, say S = S µ dn−1 xµ

Hamilton-Jacobi Theory for Classical Field Theories Therefore, we have λ0 =

137

∂S µ ∂S µ µ , λ = i ∂xµ ∂y i

and the Hamilton-Jacobi equation has the form µ µ ∂S ∂ ν i ∂S + H(x , y , ) =0. ∂y i ∂xµ ∂y i The above equations mean that µ ∂S µ ν i ∂S = f (xµ ) + H x , y , ∂xµ ∂y i ˜ = H − f we deduce the standard form of the Hamilton-Jacobi equation so that if we put H ˜ (since H and H give the same Hamilton equations): µ ∂S µ ν i ∂S ˜ + H x ,y , =0. ∂xµ ∂y i An alternative geometric approach of the Hamilton-Jacobi theory for Classical Field Theories in a multisymplectic setting was discussed in [15, 16].

5.

Time-dependent Mechanics

A hamiltonian time-dependent mechanical system corresponds to a classical field theory when the base is M = R. We have the following identification Λ12 E = T ∗ E and we have local coordinates (t, y i , p0 , pi ) and (t, y i , pi ) on T ∗ E and J 1 π ∗ , respectively. The hamiltonian section is given by h(t, y i , pi ) = (t, y i , −H(t, y, p), pi ) , and therefore we obtain Ωh = dH ∧ dt − dpi ∧ dy i . If we denote by η = dt the different pull-backs of dt to the fibred manifolds over M , we have the following result. The pair (Ωh , dt) is a cosymplectic structure on E, that is, Ωh and dt are closed forms and dt ∧ Ωnh = dt ∧ Ωh ∧ · · · ∧ Ωh is a volume form, where dimE = 2n + 1. The Reeb vector field Rh of the structure (Ωh , dt) satisfies iRh Ωh = 0 , iRh dt = 1. The integral curves of Rh are just the solutions of the Hamilton equations for H. The relation with the multisymplectic approach is the following: h = Rh ⊗ dt ,

138


or, equivalently, ∂ = Rh . h ∂t A closed 1-form λ on E is locally represented by λ = λ0 dt + λi dy i . Using λ we obtain a vector field on E: (Rh )λ = T π1,0 ◦ Rh ◦ µ ◦ λ such that the induced connection is ˜ = (Rh )λ ⊗ dt h Therefore, we have the following result. Theorem 5.1. The following conditions are equivalent: (i) (Rh )λ and Rh are (µ ◦ λ)-related. (ii) The 1-form h ◦ µ ◦ λ is closed. Remark 5.2. An equivalent result to Theorem 5.1 was proved in [14] (see Corollary 5 in [14]). ⋄ Now, if λ = dS =

∂S ∂S dt + i dy i , ∂t ∂y

then we obtain the Hamilton-Jacobi equation ∂ ∂S i ∂S + H t, y , =0. ∂y i ∂t ∂y i

Acknowledgement This work has been partially supported by MEC (Spain) Grants MTM 2006-03322, MTM 2007-62478, project “Ingenio Mathematica” (i-MATH) No. CSD 2006-00032 (ConsoliderIngenio 2010) and S-0505/ESP/0158 of the CAM.

References [1] R. Abraham and J. E. Marsden, Foundations of Mechanics (2nd edition, BenjaminCumming, Reading, 1978). [2] V. I. Arnold, Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics 60, Springer-Verlag, Berlin, 1978).


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[3] E. Binz, J. Sniatycki and H. Fischer, Geometry of Classical Fields (North-Holland Mathematics Studies, 154, North-Holland Publishing Co., Amsterdam, 1988). [4] F. Cantrijn, A. Ibort and M. de León, On the geometry of multisymplectic manifolds, J. Austral. Math. Soc. (Series A) 66 (1999) 303–330. [5] F. Cantrijn, A. Ibort and M. de León, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino 54 (3) (1996) 225–236. [6] J. F. Cariñena, X. Gracia, G. Marmo, E. Mart´ınez, M. Muñoz-Lecanda and N. RománRoy, Geometric Hamilton-Jacobi theory, Int. J. Geom. Meth. Mod. Phys. 3 (7) (2006) 1417–1458. [7] M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery, Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory; arXiv:physics/9801019v2. [8] M. de León, D. Iglesias-Ponte and D. Mart´ın de Diego, Hamilton-Jacobi Theory for Nonholonomic Mechanical Systems, J. Phys. A: Math. Theor. 41 (2008) 015205, pp. 14. [9] M. de León, J. C. Marrero and D. Mar´ın, A geometrical approach to Classical Field Theories: a constraint algorithm for singular theories, In: New Developments in Differential Geometry (Debrecen, 1994, Mat. Appl. 350, Kluwer, Dordrecht, 1996) 291– 312. [10] M. de León, J. C. Marrero and D. Mar´ın, Ehresmann connections in Classical Field Theories, In: Differential Geometry and its Applications (Granada, 1994, Anales de F´ısica, Monograf´ıas 2, 1995) 73–89. [11] M. de León, D. Mart´ın de Diego and A. Santamar´ıa-Merino, Tulczyjew’ s triples and lagrangian submanifolds in classical field theory, In: Applied Differential Geometry and Mechanics ((W. Sarlet and F. Cantrijn, Eds.) Universiteit Gent, 2003) 21–47. [12] M. de León, D. Mart´ın de Diego and A. Santamar´ıa-Merino, Symmetries in Field Theory, Int. J. Geom. Meth. Mod. Phys. 1 (5) (2004) 651–710. [13] M. de León, P. R. Rodrigues, Generalized Classical Mechanics and Field Theory (North-Holland, Amsterdam, 1985). [14] J. C. Marrero and D. Sosa, The Hamilton-Jacobi equation on Lie Affgebroids, Int. J. Geom. Meth. Mod. Phys. 3 (3) (2006) 605–622. [15] C. Paufler and H. Romer, De Donder-Weyl equations and multisymplectic geometry, In: XXXIII Symposium on Mathematical Physics (Torún, 2001, Rep. Math. Phys. 49, 2002) 325–334. [16] C. Paufler and H. Romer, Geometry of Hamiltonean n-vector fields in multisymplectic field theory, J. Geom. Phys. 44 (2002) 52–69.

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[17] H. Rund, The Hamilton-Jacobi theory in the Calculus of Variations (Robert E. Krieger Publ. Co., Nuntington, N.Y. 1973). [18] D. J. Saunders, The Geometry of Jet Bundles (London Mathematical Society Lecture Notes Ser. 142, Cambridge Univ. Press, Cambridge, 1989).

Part II

N ATURAL B UNDLES AND D IFFERENTIAL I NVARIANTS

141


Chapter 9

N ATURAL L AGRANGIAN S TRUCTURES Josef Janyˇska∗ Department of Mathematics and Statistics Masaryk University, Janácˇ kovo nám 2a, 602 00 Brno Czech Republic

Abstract We use the theory of natural and gauge-natural bundles and natural differential operators to give a general description of invariant and gauge invariant Lagrangian structures on natural and gauge natural bundles.

2000 Mathematics Subject Classification. 58A32, 53C05, 53C80, 70G45, 70G50, 70S05, 70S15 Key words and phrases. Natural bundle, gauge-natural bundle, natural differential operator, invariant Lagrangian, infinitesimal symmetr

1.

Introduction

Let π : Y → M be a fibered manifold and π r : J r Y → M its r-jet prolongation, Saunders [43]. An r-order Lagrangian is a fibered morphism, over M , L : J r Y → ∧m T ∗ M .

(1.1)

A vector field η on Y projectable on a vector field ξ on M is said to be an infinitesimal symmetry of L if the Lie derivative of L with respect to the r-jet lift J r η of η vanishes, Krupka and Trautman [35], i.e. LJ r η L(y r ) = 0 , where y r ∈ J r Y . But from the general theory of Lie derivatives LJ r η L(y r ) := L(J r η,∧m T ∗ ξ) L(y r ) = (T L ◦ J r η)(y r ) − ∧m T ∗ ξ(L(y r )) , ∗


(1.2)

144

Josef Janyˇska

where ∧m T ∗ ξ, m = dim M , is the vector field on ∧m T ∗ M obtained by the flow prolongation of ξ, see Subsection 2.13.. So the condition for infinitesimal symmetries LJ r η L = 0 reads as T L(J r η)(y r ) = ∧m T ∗ ξ(L(y r )) .

(1.3)

Usually, in physical theories, Y is a bundle (tensor bundle, bundle of connections, gauge bundle, ...) with a geometrical structure given by an action of a Lie group. Then, if η is a vector field preserving the given structure, a Lagrangian satisfying (1.3) is said to be invariant with respect to the given structure. There are two main geometrical structures of Y : 1. Y is a natural bundle (in the sense of Nijenhuis [40]) with a geometrical structure given by the differential group Grm = invJ0r (Rm , Rm )0 . Examples of such bundles are tensor bundles (r = 1) and the bundle of classical (linear) connections (r = 2). Invariant Lagrangians on tensor bundles were studied by many authors, see for instance Krupka [26, 27, 28, 30, 31] and Novotný [41]. 2. Y is a gauge-natural bundle (in the sense of Eck [6]) with a geometrical structure induced by a gauge group G. In this case a special attention is devoted to invariant Lagrangians on the bundle of principal connections, see for instance Betounes [1], Eck [6], Horndeski [10], Utiyama [51] and others. In this paper we will study natural (invariant and gauge invariant) Lagrangian structures on natural and gauge-natural bundles. To describe invariant Lagrangians on natural or gauge-natural bundles we can use general properties of natural differential operators on natural bundles or gauge-natural bundles. The paper is organized as follows. In Section 2. we recall basic properties of natural bundles and natural differential operators on natural bundles. In Section 3. we study invariant (natural) Lagrangians on natural bundles, especially on the natural bundle of classical connections. In Section 4. we recall basic properties of gauge-natural bundles and natural differential operators on gauge-natural bundles and finally, in Section 5., we study gauge invariant (natural) Lagrangians on gauge-natural bundles, especially on the gaugenatural bundle of general linear and principal connections by using higher order versions of Utiyama’s reduction method, Janyˇska [15, 17, 18]. In what follows we will use the following notations. M is the category of all smooth manifolds and smooth mappings, Mm is the category of all m-dimensional smooth manifolds and local diffeomorphisms, F M m is the category of all fibred manifolds with mdimensional bases and smooth fibred morphisms covering local diffeomorphisms of bases, V B m (A B m ) is the category of all vector (affine) bundles with m-dimensional bases and smooth linear (affine) fibred morphisms covering local diffeomorphisms of bases and, finally, PB m (G) is the category of all principal G-bundles with m-dimensional bases and smooth principal fibred morphisms covering local diffeomorphisms of bases. In what follows all manifolds and maps are supposed to be smooth.

Natural Lagrangian Structures

2.

145

Natural Bundles and Operators

We recall here definitions and basic properties of the theory of natural bundles and natural differential operations, for details see [9, 11, 19, 24, 33, 40, 48, 53]. As examples we mention functors and operators which will be used later. Natural differential operators on natural bundles are closely related with the term geometric invariant (or concomitant) which has been used in differential geometry since the end of the 19th century. In the 1930’s Schouten and his collaborators, [45], used the notion of geometric object and invariant operations with geometric objects. A modern functorial approach to the theory of geometrical objects and invariant operations with geometric objects was introduced by Nijenhuis [39] in the 1950’s. Starting from the famous paper by Nijenhuis [40] geometrical objects and invariant operations with geometrical objects have been very intensively studied by using the concepts of natural bundles and natural differential operators. Nijenhuis defined natural bundles as lifting functors on the category Mm . Lifting functors are supposed to satisfy three conditions: prolongation, localization and regularity (continuity). Koláˇr [22] generalized lift functors to the category M of all differentiable manifolds and their smooth mappings. Such functors are called prolongation functors. Later various prolongation functors on subcategories of M were studied. Main problem of the theory of natural differential operators is to give a complete classification of them for concrete underlying geometric structures. Such classification is based on the one-to-one correspondence of natural differential operators and equivariant maps between standard fibres. To classify equivariant maps we can use several methods. Formerly the method of Lie equations was used, Krupka and Janyˇska [33], recently we use the algebraic method by Koláˇr, Michor and Slovák [24]. In literature it is possible to find many examples and applications of natural operations used in geometry and physics. For wide list of references we recommend to see Fatibene and Francaviglia [9], Koláˇr, Michor and Slovák [24] and Krupka and Janyˇska [33].

2.1.

Natural Bundles

Natural bundles were introduced by Nijenhuis [40] over the category Mm . We use more general definition by Koláˇr, Michor and Slovák [24]. Definition 2.1. A natural bundle functor on a subcategory C of M is a covariant functor F from C to the category F M satisfying (i) (prolongation) for each manifold M ∈ Ob C , pM : F M → M is a fibred manifold over M , (ii) (localization) for each f ∈ Mor C , F f is a fibred manifold map covering f such that F ι(U ) = ι(F U ) for any open subset ι : U ֒→ M . A natural bundle functor on the subcategory Mm of M , for a certain m, is a natural lift functor, Nijenhuis [40]. In literature natural bundle functors on M are also called prolongation functors, Koláˇr [22]. In original definitions of natural lift and prolongation functors there is the regularity condition saying that a smoothly parameterized family of diffeomorphisms is prolonged

146

Josef Janyˇska

into a smoothly parameterized family of diffeomorphisms. But this condition turns out to be a consequence of remaining prolongation and localization properties. This was proved by Epstein and Thurston [7] for natural lift functors and by Koláˇr and Slovák [25] for natural prolongation functors. A natural bundle is then a triplet (F M, pM , M ). Later (Theorem 2.2) we will see that pM : F M → M is indeed a bundle.

2.2.

Geometrical Object

A geometrical object on a manifold M is now an element from F M , where F is a natural bundle functor. A section σ : M → F M is a field of geometrical objects on M .

2.3.

Order of Natural Bundle Functors

We say that a natural lift functor F is of finite order r if r is the smallest number such that jxr f = jxr g ⇒ F f |Fx M = F g|Fx M ¯ ) ∈ Mor Mm and any x ∈ M . for any (f, g : M → M

Palais and Terng [42] proved that the order r of a natural lift functor is finite r < 2n+1 where n is the dimension of the fiber F0 Rm . Later Epstein and Thurston [7] gave much better bound. They proved r ≤ 2n + 1 and that this bound is sharp for m = 1. Finally Zajtz n n [54] proved r ≤ max{ n−1 ;m + 1}. Mikulski [38] has shown that a natural prolongation functor with infinite order exists.

2.4.

Differential Group

Let us denote by Grm the Lie group Grm = invJ0r (Rm , Rm )0 of invertible r-jets (with source and target 0) of diffeomorphisms of Rm which preserve 0. The group multiplication is given by the jet composition. The canonical coordinates on Grm will be denoted by (aλµ , . . . , aλµ1 ...µr ) and tilde will refer to the inverse element.

2.5.

Standard Fiber

Let F be an r-order natural lift functor and let F0 = F0 Rm . Because of (ii) of Definition 2.1 F0 is diffeomorphic with Fx M for any x ∈ M , M ∈ ObMm . F0 will be called the standard fiber of F . Applying F on origin-preserving diffeomorphisms of Rm we get a left action of Grm on F0 which defines on F0 a structure of a left smooth Grm -manifold, Krupka [29] and Terng [48].

2.6.

Natural Fibred Coordinate Chart

Local coordinate charts (xλ ) on M and (y p ) on F0 induce a fibred coordinate chart (xλ , y p ) on F M , which is said to be the natural fibred coordinate chart.


2.7.

147

Examples

1. The tangent functor T is a natural bundle functor of order one on the category M with values in the category V B. In dimension m the corresponding standard fiber is Rm on which G1m = Gl(m, R) acts in the standard way by the matrix multiplication. The natural fibred coordinate chart on T M will be denoted by (xλ , x˙ λ ). 2. The cotangent functor T ∗ is a natural lift functor of order one with values in the category V B m . The standard fiber is Rm∗ with the standard action of G1m . 3. The functor ∧p T ∗ of p–forms is a natural lift functor of order one with values in the category V B m . The standard fiber is ∧p Rm∗ on which G1m acts in the standard tensor way. The natural fibred coordinate chart on ∧p T ∗ M will be denoted by (xλ , ωλ1 ...λp ), 1 ≤ λ1 < · · · < λp ≤ m. Especially, for p = m, we obtain the natural lift functor of volume forms. 4. The functor of pseudo–Riemannian metrics pRm is a natural lift functor of order one such that pRm(M ) are subbundles of bundles from the category V B m . The standard fiber (pRm)0 is the subspace in ⊙2 Rm∗ of non-degenerate symmetric matrices with the tensor action of G1m . The natural fibred coordinate chart on pRm(M ) will be denoted by (xλ , gλµ ), gλµ = gµλ , det(gλµ ) 6= 0. 5. The functor of k r -velocities Tkr is a natural bundle functor of order r on the category M . For any M ∈ Ob M , we define Tkr M = J0r (Rk , M ) and, for any f ∈ Mor M , ¯ , we define T r f (j r α) = j r (f ◦ α), where j r α ∈ T r M . The standard fiber of f :M →M 0 0 0 k k r Tk , in dimension m, is J0r (Rk , Rm )0 and the action of Grm on the standard fiber is given by the jet composition. 6. The functor of r-order frames F r is a natural lift functor of order r. For any M ∈ Ob Mm , we define F r M = invJ0r (Rm , M ) and, for any f ∈ Mor Mm , F r f is defined as in Example 2.7..5. The values of the functor F r are in the category PB m (Grm ). 7. The functor Cla of classical (linear) connections on a given manifold is a natural lift functor of order two with values in the category A B m . Its standard fiber is R∗m ⊗ Rm ⊗ Rm∗ on which G2m acts via the well known transformation relations of the Christoffel symbols, Christoffel [3], ¯ µ λ ν = aλρ (Λσ ρ τ a Λ ˜σµ a ˜τν − a ˜ρµν ) . The natural fibred coordinate chart on Cla M will be denoted by (xλ , Λµ λ ν ). By Claτ will be denoted the functor of torsion free linear connections. In natural fibred coordinates Claτ is characterized by Λµ λ ν = Λν λ µ . 8. Let F be a natural lift functor of order r and J s be the functor of s-jet prolongation, Saunders [43]. Then J s F ≡ J s ◦ F is a natural lift functor of order (r + s). If F0 is the standard fiber of F , then the standard fiber of J s F is (J s F )0 = Tns F0 and the action of s r Gr+s m on (J F )0 is obtained by the jet prolongation of the action of Gm on F0 .

2.8.

The Bundle Structure

In the theory of natural lift functors the functor of r-order frames, defined in Example 2.7..6, plays a fundamental role. Namely, we have the following theorem, Krupka [29] and Terng [48].

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Josef Janyˇska

Theorem 2.2. Any natural lift functor F of order r, with the standard fiber F0 , is canonically represented by F M = [F r M, F0 ],

F f = [F r f, idF0 ],

where M ∈ Ob Mm , f ∈ Mor Mm , and [F r M, F0 ] = (F r M, F0 )/Grm is the bundle associated with F r M . This theorem implies that there is the one-to-one correspondence between r-order natural lift functors and left Grm -manifolds.

2.9.

Natural Differential Operators

¯ be a mapping in MorMm and σ : M → F M Let F be a natural lift functor, f : M → M ∗ ¯ → FM ¯ by f ∗ σ = F f ◦ σ ◦ f −1 . be a section. Then we define the section f σ : M Definition 2.3. A natural differential operator D from a natural lift functor F1 to a natural lift functor F2 is a family of differential operators ¯ )}M ∈ObM {D(M ) : C ∞ (F1 M ) → C ∞ (F2 M m such that ¯ )(f ∗ σ) = f ∗ D(M )(σ) for every section σ ∈ C ∞ (F1 M ) and every f : M → (i) D(M ¯ M in MorMm , (ii) DU (σ|U ) = (DM σ)|U for every section σ ∈ C ∞ (F1 M ) and every open submanifold U ⊂ M , (iii) every smoothly parameterized family of sections of F1 M is transformed into a smoothly parametrized family of sections of F2 M .

2.10.

Order of Natural Differential Operator

A natural differential operator is of order k, 0 ≤ k ≤ ∞, if all D(M ), M ∈ ObMm , are of order k. Thus, a k-order natural differential operator D from F1 to F2 is characterized by the associated fibred manifold morphisms D(M ) : J k F1 M → F2 M , over M , according to the formula D(M )(jxk σ) = D(M )(σ)(x). The family D = {D(M )}M ∈ObMm defines a natural transformation of the functors J k F1 and F2 . In what follows we will identify korder natural differential operators with the corresponding natural transformations and use the same symbol.

2.11.

Equivariant Mappings Given by Natural Operators

Coordinate independent geometrical constructions are in fact natural differential operators between natural lift functors. The study of natural differential operators is based on relations between natural differential operators and equivariant mappings. The basic tool is the following theorem, Terng [48]. Theorem 2.4. There is a bijective correspondence between the set of k-order natural differential operators from a natural lift functor F1 to a natural lift functor F2 and equivariant mappings from the standard fiber of J k F1 to the standard fiber of F2 .


2.12.

149

Examples

1. The exterior differential d is a first order natural differential operator from ∧p T ∗ , p ≥ 0, to ∧p+1 T ∗ . The corresponding G2n -equivariant mapping from J 1 (∧p T ∗ )0 = Tn1 (∧p Rm∗ ) to (∧p+1 T ∗ )0 = ∧p+1 Rm∗ is given, in the canonical coordinate chart (ωλ1 ...λp ), 1 ≤ λ1 < · · · < λp ≤ m, on (∧p Rm∗ ), by ωλ1 ...λp+1 ◦ d = ω[λ1 ...λp ,λp+1 ] , where [...] denotes the anti-symmetrization. For p ≥ 1 the naturality determines d up to a constant multiple while in classical proofs the linearity was supposed, see for instance Koláˇr [23], Krupka and Janyˇska [33] and Krupka and Mikolásˇová [34]. 2. The Levi-Civita connection is a first order natural differential operator from pRm to Claτ . The corresponding G2m -equivariant mapping from J 1 (pRm)0 to Cla0 is given by the formal Christoffel symbols, Christoffel [3], Λµ λ ν = − 12 g λρ (gρµ,ν + gρν,µ − gµν,ρ ), where (g λµ ) is the inverse matrix of (gλµ ). The uniqueness of the Levi–Civita connection is the classical geometrical problem. The proof of the uniqueness by using natural techniques can be found in Krupka and Janyˇska [33], Krupka and Mikolásˇová [34] and Slovák [47]. 3. The curvature tensor of a classical connection is a first order natural differential operator from Cla to T ∗ ⊗ T ⊗ (∧2 T ∗ ). The corresponding G3m -equivariant mapping from J 1 Cla0 to (T ∗ ⊗ T ⊗ (∧2 T ∗ ))0 = R∗m ⊗ Rm ⊗ (∧2 Rm∗ ) is given by wν κ λµ = Λν κ λ,µ − Λν κ µ,λ + Λρ κµ Λν ρλ − Λρ κ λ Λν ρ µ . The curvature tensor is not unique operator of this type and plays an important role in classification of natural operators defined on classical connections, see for instance Koláˇr [23], Koláˇr, Michor and Slovák [24] and Schouten [44].

2.13.

Infinitesimal Properties of Natural Lift Functors

The regularity property of lift functors allows us to lift vector fields on a manifold M to projectable vector fields on a natural bundle F M by using flows. Namely if exp(tξ) is the flow of a vector field ξ on M then F (exp(tξ)) = exp(tFξ) is the flow of the vector field Fξ on F M which is said the flow lift of ξ. Moreover, if F is of order r, then Fξ depends on r-jets of ξ. For instance the flow lift of a vector field ξ = ξ λ ∂x∂ λ with respect to the tangent functor is the vector field ∂ ∂ξ λ ∂ T ξ = ξ λ λ + ρ x˙ ρ λ ∂x ∂x ∂ x˙

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on T M and the flow lift of ξ with respect to the natural lift functor of classical connections, see Example 2.4..7, is the vector field ∂ξ λ ∂ Cla ξ = ξ λ λ + Λµ ρ ν (2.1) ∂x ∂xρ ρ ∂ξ ρ ∂2ξλ ∂ λ ∂ξ − Λρ λ ν − Λ + µ ρ ∂xµ ∂xν ∂xµ ∂xν ∂Λµ λ ν on Cla M .

2.14.

Infinitesimal Properties of Natural Operators

If σ : M → F M is a section of an r-th order natural bundle (a field of geometrical objects) then we can define the Lie derivative of σ with respect to a vector field ξ by the formula d |0 {exp(−tξ)∗ σ} . (2.2) dt Lξ σ is a section of V F M . Natural differential operators D from a natural lift functor F1 to a natural lift functor F2 are infinitesimally characterized by the commutativity with the Lie derivatives, Kirillov [21] and Janyˇska and Modugno [19], in the following sense. Lξ σ =

Theorem 2.5. A k-order differential operator D(M ) from a natural bundle F1 M to a natural bundle F2 M is natural if and only if Lξ D(M )(σ) = T D(M )(Lξ σ)

(2.3)

for all σ ∈ C ∞ (F1 M ) and all vector fields ξ on M . If we identify D with the corresponding natural transformation D, then the above condition (2.3) is equivalent with T D(M )(J k F1 ξ)(j k σ) = F2 ξ(D(M )(j k σ)) .

3.

(2.4)

Natural Lagrangian Structures on Natural Bundles

If we assume in (2.4) the natural bundle of volume forms as the target bundle, i.e. F2 = ∧m T ∗ , the condition (2.4) means that natural Lagrangians on the natural bundle F1 M are just invariant Lagrangians on F1 M (in the sense of (1.2)) and infinitesimal symmetries of such Lagrangians are vector fields F1 ξ, for all vector fields ξ on M . So a k-order Lagrangian L : J k F1 M → ∧m T ∗ M is invariant (natural) if and only if LJ k F1 ξ L = 0

(3.1)

for any vector field ξ on M . The above infinitesimal characterization of invariant Lagrangians on natural bundles leads in natural fibered local coordinates to a system of partial differential equations which is generally difficult to solve. Much more simple is to use algebraical method of classification of equivariant mappings corresponding to natural operators. Special situation we obtain in the case of invariant Lagrangians on the natural bundle of classical symmetric connections. In this case we can use the reduction theorems by Schouten [44].


3.1.

151

The First and the Second Reduction Theorems

Let us assume the natural bundle of classical symmetric connections on manifolds given by the lift functor Claτ , see Example 2.7..7. Schouten [44] proved that all finite order polynomial concomitants on Claτ M and a bundle of geometrical object of order one with values in an other bundle of geometrical objects of order one factorize through the curvature tensor, a given field of geometrical objects and their covariant differentials. By using the theory of natural differential operators on natural bundles the results of Schouten were generalized by Koláˇr, Michor and Slovák [24] and we obtain the first reduction theorem in the form. Theorem 3.1. All natural differential operators of order r with values in a natural bundle of order one of a classical symmetric connection are natural differential operators of order zero of the curvature tensor and its covariant differentials up to the order (r − 1), i.e. e (r−1) R[Λ]) , D(j r Λ) = D(∇

where ∇(r−1) = (id, ∇, . . . , ∇r−1 ).

The second reduction theorem can be formulated as follows. Theorem 3.2. Let Φ be a field of geometrical objects of order one and s ≥ r − 1. All natural differential operators with values in a natural bundle of order one of a classical symmetric connection (in order s) and Φ (in order r) are natural differential operators of the curvature tensor, Φ and their covariant differentials up to the orders (s − 1) and r, respectively, i.e. e (s−1) R[Λ], ∇(r) Φ) . D(j s Λ, j r Φ) = D(∇

Remark 3.3. If Λ is a classical non-symmetric connection on M , then there exists its unique e + T , where Λ e is the classical symmetric connection obtained by the symsplitting Λ = Λ metrization of Λ and T is the torsion tensor of Λ. Then all natural operators of Φ and Λ are of the form e j s T, j r Φ) = D( e ∇ e (s−1) R[Λ], e ∇ e (s) T, ∇ e (r) Φ) , D(j s Λ, j r Φ) = D(j s Λ,

e refers to the connection Λ. e where ∇

3.2.

Invariant Lagrangians of Classical Symmetric Connections

Now, as a simple consequence of Theorem 3.1, we have Theorem 3.4. All invariant (natural) Lagrangians of order r on the natural bundle of classical symmetric connections are of the type e (r−1) R[Λ]) . L(j r Λ) = L(∇

Remark 3.5. r-order invariant Lagrangians on classical symmetric connections are infinitesimally given as solutions of the system of partial differential equations given by LJ r Cla ξ L = 0

(3.2)

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for all vector fields ξ on M and Cla ξ given by (2.1). From Theorem 3.4 it follows that all solutions of (3.2) factorize through the curvature tensor and its covariant differentials up to the order (r − 1). As a consequence of Theorem 3.2, we have Theorem 3.6. All invariant Lagrangians on the natural bundle of classical symmetric connections (in order s) and a natural bundle of order one (in order r) are of the type

3.3.

e (s−1) R[Λ], ∇(r) Φ) . L(j s Λ, j r Φ) = L(∇

Invariant Lagrangians of Metric Fields

In physical theories M is usually an oriented spacetime with a metric g. In this case we have the Levi Civita connection Λ(g) and the canonical volume form p ω(g) = |g| dx1 ∧ . . . ∧ dxm .

Any invariant Lagrangian is of the form L(j r g, j r Φ) = l(j r g, j r Φ) ω(g) where l(j r g, j r Φ) is an invariant Lagrangian function. Then, as a consequence of Theorem 3.6, invariant Lagrangian functions are of the type l(j r g, j r Φ) = e l(g, ∇(r−2) R[Λ(g)], ∇(r) Φ) .

Invariant Lagrangians on the natural bundle of metrics were studied for instance by Krupka [27, 28] and Novotný [41].

4.

Gauge-Natural Bundles

Natural differential operators on natural bundles describe the invariance of geometrical or physical theories with respect to changes of local coordinates. But in physical theories another sort of invariance plays an important role, the so called gauge invariance. Invariant gauge theory has been introduced in the book by H. Weyl [52] in 1918 as a generalization of the Einstein’s general relativity. Weyl considered operators on a spacetime invariant not only with respect to isomorphisms of spacetime but also with respect to gauge transformations g 7→ eα g (the term ”gauge” was used for the first time by H. Weyl). The original invariant physical gauge theories was related with the gauge group U (1) acting on wave functions and electromagnetic potentials. In early 1950’s the concept of gauge invariance was generalized first for the spin group, see for instance Yang and Mills [55], and then for any Lie group G playing the role of the gauge group. The first geometrical interpretation of gauge invariance with respect to a general gauge group can be found in the famous paper by Utiyama [51]. Gauge invariant theories can be described geometrically by using the concepts of gauge-natural bundle functors and natural differential operators between gaugenatural bundles. So in Section 4. we recall basic definitions and properties of gauge-natural bundle functors, see Eck [6], Fatibene and Francaviglia [9], Koláˇr [23] and Koláˇr, Michor and Slovák [24].


4.1.

153

Gauge-natural Bundle Functors

Gauge-natural bundle functors was introduced by Eck [6]. We recall here the definition of Koláˇr, Michor and Slovák [24]. Let us recall that B is the base functor from the category F M to the category M . Definition 4.1. A gauge-natural bundle functor over m-dimensional manifolds is a functor F : PB m (G) → F M m such that (i) every PB m (G)-object π : P → BP is transformed into a fibered manifold qP : F P → BP over BP , (ii) every PB m (G)-morphism f : P → P¯ is transformed into a fibered morphism F f : F P → F P¯ over Bf , (iii) for every open subset U ⊂ BP , the inclusion ι : π −1 (U ) ֒→ P is transformed into the inclusion F ι : qP−1 (U ) ֒→ F P . A gauge-natural bundle is then a quadruple (F P, πP , M, π : P → BP ). Later (Theorem 3.3.) we will see that F P is actually a bundle. In the original definition by Eck [6] there is one more regularity (continuity) condition which says that a smoothly parametrized family of diffeomorphisms of P is ”transformed” into a smoothly parameterized family of isomorphisms of F P . But this condition is a consequence of (i), (ii) and (iii), Koláˇr, Michor and Slovák [24].

4.2.

Functor W r

r Let (π : P → M ) ∈ Ob PB m (G), let W r P be the space of all r-jets j(0,e) ϕ, where m m ϕ : R × G → P is in Mor PB m (G), 0∈ R and e is the unity in G. The space W r P is r G = Jr m m a principal fibre bundle over M with the structure group Wm (0,e) (R × G, R × G) of all r-jets of principal fibre bundle isomorphisms Ψ : Rm × G → Rm × G covering the diffeomorphisms ψ : Rm → Rm such that ψ(0) = 0. The group Wnr G is the semidirect r G of Gr and T r G with respect to the action of Gr on T r G given by product Grm ⋊ Tm m m m m the jet composition. Let (ϕ : P → P¯ ) ∈ Mor PB m (G), then we can define the principal bundle morphism W r ϕ : W r P → W r P¯ by the jet composition. The rule transforming r G) and any ϕ ∈ Mor PB (G) into any P ∈ Ob PB m (G) into W r P ∈ Ob PB m (Wm m r r W ϕ ∈ Mor PB m (Wm G) is a gauge-natural bundle functor, Koláˇr [23].

4.3.

Bundle Structure

The gauge-natural bundle functor W r described in Subsection 4.2. plays a fundamental role in the theory of gauge-natural bundle functors. We have, Eck [6], Theorem 4.2. Every gauge-natural bundle F P is a fibred bundle associated with the gauge-natural bundle W r P for a certain order r.

4.4.

Order of Gauge-natural Bundle Functors

The number r from Theorem 4.2 is called order of the gauge-natural bundle functor F . So if F is an r-order gauge-natural bundle functor then F P = [W r P, F0 ],

F ϕ = [W r ϕ, idF0 ],

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r G-manifold called the standard fibre of F . where F0 is a left Wm

4.5.

Gauge and Total Order of Gauge-natural Functors

Let F be an s-order gauge-natural bundle functor and let r ≤ s be the minimal number s G = Gs ⋊ T s G on F can be factorized through the canonical such that the action of Wm 0 m m s G → T r G, s ≥ r. Then s is said to be the total order of F , r is projection πrs : Tm m the gauge order and we say that F is of order (s, r). In what follows we shall denote by (s,r) r G and by W (s,r) P the corresponding principal bundle. Wm G = Gsm ⋊ Tm

4.6.

Gauge-natural Fibred Coordinate Chart

A local fibred coordinate chart (xλ , pσ ) on P and a coordinate chart (y p ) on F0 induce a fibred coordinate chart (xλ , y p ) on F P which is said to be the gauge-natural fibred coordinate chart.

4.7.

Examples

1. Any r-order natural lift functor in the sense of Definition 2.1 is the (r,0)-order gaugenatural bundle functor with the trivial gauge action, i.e. the action (Grm × G) × F0 −→ F0 does not depend on G. 2. Let (π : P → M ) ∈ Ob PB m (G) and let us denote by Pri P → M the bundle of principal connections on P . Then Pri is a (1,1)-order gauge-natural bundle functor with 1 G given by, Kol´ the standard fibre G ⊗ Rm∗ and with the action of Wm aˇr [23], (X, g, Z)(Y ) = ad(g)(Y + Z)X −1 .

In particular, let G = Grn , then Pri P can be viewed as the bundle Lin E of linear connections on an associated vector bundle E → M with n-dimensional fibres. The standard fibre of Lin is Lin0 = R∗n ⊗ Rn ⊗ Rm∗ with coordinates (Kj i λ ), i, j = 1, ..., n, (1,1) 1 G1 on Lin is given, in the canonical λ = 1, ..., m, and the action of Wm G1n = G1m ⋊ Tm 0 n λ i i 1 1 1 coordinates (aµ , aj , ajλ ) on Gm ⋊ Tm Gn , by ¯ j i λ = aip Kq p ρ a K ˜qj a ˜ρλ + aipρ a ˜pj a ˜ρλ , where tilde refers to the inverse element. 3. Let F0 be a left G–manifold. The associated gauge-natural bundle functor is defined by assF0 (P ) = [P, F0 ], assF0 (ϕ) = [ϕ, idF0 ] , where P ∈ ObPB m (G), ϕ ∈ MorPB m (G). assF0 (P ) is a 0-order gauge-natural bundle. Especially the adjoint bundle ad P is the 0-order gauge-natural bundle given by the adjoint action of G on its Lie algebra G . 4. If F is a gauge-natural bundle functor of order (s, r) then J k F is a gauge-natural bundle functor of order at most (s + k, r + k). The number (s + k) is exact, but (r + k) may be too big. For instance if F is an s-order natural lift functor, i.e. an (s,0)-order gaugenatural bundle functor, then J k F is an (s+k)-order natural lift functor, i.e. an (s+k,0)-order gauge-natural bundle functor. 5. ad P ⊗ (∧p T ∗ M ) is a (1,0)-order gauge-natural vector bundle.


4.8.

155

Natural Operators

¯ , F be a gauge-natural bundle Let (ϕ, f ) ∈ Mor PB m (G), ϕ : P → P¯ over f : M → M ¯ → F P¯ by functor and σ : M → F P be a section. Then we define the section ϕ∗ σ : M ∗ −1 ϕ σ = Fϕ ◦ σ ◦ f . Definition 4.3. A natural differential operator D from a gauge-natural bundle functor F1 to a gauge-natural bundle functor F2 is a family of differential operators {D(P ) : C ∞ (F1 P ) → C ∞ (F2 P )}P ∈ObPBm (G) such that (i) D(P¯ )(ϕ∗ σ) = ϕ∗ D(P )(σ) for every sec∞ tion σ ∈ C (F1 P ) and every (ϕ, f ) ∈ MorP ¯, Bm (G), ϕ : P → P¯ over f : M → M −1 (ii) D(π (U ))(σ|U ) = (D(P )(σ))|U for every section σ ∈ C ∞ (F1 P ) and every open subset U ⊂ M , (iii) every smoothly parameterized family of sections of F1 P is transformed into a smoothly parameterized family of sections of F2 P .

4.9.

Order of Natural Differential Operators

A natural differential operator D from F1 to F2 is of a finite order k if all D(P ), (π : P → M ) ∈ObPB m (G), depend on k-order jets of sections of F1 P . Thus, a k-order natural differential operator from F1 to F2 is characterized by the associated fibred manifold morphism D(P ) : J k F1 P → F2 P , over M , such that the family D = {D(P )}P ∈ObPBm (G) is a natural transformation of J k F1 to F2 . In what follows we will identify k-order natural differential operators with the corresponding natural transformations and use the same symbol. Theorem 4.4. Let F1 and F2 be gauge-natural bundle functors of order ≤ r. Then we have a one-to-one correspondence between natural differential operators of order k from F1 to r+k G-equivariant mappings from (J k F ) to (F ) . F2 and Wm 1 0 2 0 This theorem is due to Eck [6], see also Koláˇr, Michor and Slovák [24].

4.10.

Curvature Operator

The curvature operator of principal connections is a 1-order natural differential operator (2,2) from Pri to ad ⊗ (∧2 T ∗ ) with the associated Wm G-equivariant morphism (ua λµ ) ◦ R = Γa λ,µ − Γa µ,λ + cabd Γb λ Γd µ , where c = (cabd ) are the structure constants of G.

156

4.11.

Josef Janyˇska

Infinitesimal Properties of Gauge-natural Bundle Functors

The regularity (continuity) property of gauge-natural bundle functors allows to transform right G-invariant vector fields on a principal G-bundle P to projectable vector fields on a gauge-natural bundle F P by using flows. Namely if exp(tΞ) is the flow of a right Ginvariant vector field Ξ on P , projectable on the vector field ξ on M , then F (exp(tΞ)) = exp(tFΞ) is the flow of the vector field FΞ on F P which is said the flow transformation of Ξ. Moreover, if F is of order r, then FΞ depends on r-jets of Ξ. ea ) is the base of vertical right G-invariant vector fields on P given by For instance if (B a base (Ba ) of the Lie algebra G and Ξ is a right G-invariant vector field Ξ = ξ λ (x) ∂x∂ λ + ea on P . Then the flow transformation of Ξ with respect to the gauge-natural bundle Ξa (x)B functor of principal connections, see Example 4.7..2, is the vector field ∂ ∂ξ ρ ∂Ξa ∂ P ri Ξ = ξ λ λ + cabd Ξb Γd λ − Γa ρ λ + (4.1) ∂x ∂x ∂xλ ∂Γa λ on Pri P .

4.12.

Infinitesimal Properties of Natural Differential Operators

If σ : M → F P is a section of an r-th order gauge-natural bundle then we can define the Lie derivative of σ with respect to a right G-invariant vector field Ξ on P , over the vector field ξ on M , by the formula d |0 {exp(−tΞ)∗ σ} . dt LΞ σ is a section of V F P . Natural differential operators D from a gauge-natural bundle functor F1 to a gauge-natural functor functor F2 can be infinitesimally characterized by the commutativity with the Lie derivatives, Janyˇska and Modugno [19], in the following sense. LΞ σ =

Theorem 4.5. A k-order differential operator D(P ) from a gauge-natural bundle F1 P to a gauge-natural bundle F2 P is natural if and only if LΞ D(P )(σ) = T D(P )(LΞ σ) ,

(4.2)

for all right G-invariant vector fields Ξ on P and all sections σ ∈ C ∞ (F1 P ). If we identify D with the corresponding natural transformation D, then the above condition (4.2) is equivalent with T D(P )(J k F1 Ξ)(j k σ) = F2 Ξ(D(P )(j k σ)) .

5.

(4.3)

Natural Lagrangians on Gauge-Natural Bundles

Gauge invariant theories can be very efficiently formulated by using the theory of gaugenatural bundles and natural differential operators and have wide applications in gauge field theories, see Fatibene and Francaviglia [9]. As applications of natural differential operators on gauge-natural bundles we will generalize the Utiyama’s reduction method for the linear group Gl(n, R) as a gauge group, Janyˇska [15], and for a general Lie group G as a gauge group, Janyˇska [17, 18].


5.1.

157

Gauge Invariant Lagrangians on Electromagnetic Potentials

Originally gauge invariant Lagrangians were studied on a complex wave function (a particle field) φ and a gauge field A = (Aλ (x)). The gauge invariance was considered with respect to the gauge transformation of φ given in the form φ(x) 7→ eiα(x) φ(x). In order to obtain a gauge invariant Lagrangian of φ of order one it is necessary to consider also a µ gauge field A with the gauge transformation Aλ 7→ (Aµ + ∂µ α) ∂x . Then a gauge invariant ∂x ¯λ Lagrangian of order one is of the form e ∇φ) , L(j 1 φ, A) = L(φ,

(5.1)

e ). L(j 1 A) = L(F

(5.2)

where ∇λ φ = ∂λ φ − i Aλ φ. In physical theories A is an electromagnetic potential and the above result is known as the minimal coupling principle. Further, a gauge invariant Lagrangian of order one of the gauge field A only factorizes through the 2-form (electromagnetic 2-form) Fλµ = ∂λ Aµ − ∂µ Aλ , i.e.

The above classical gauge invariance corresponds to the invariance with respect to the gauge group U (1). The above results can be interpreted geometrically as follows: let Q → M be a complex line bundle (a quantum bundle) with a Hermitian product h and local fibered coordinates (xλ , z). Then φ : M → Q is a section and A = dxλ ⊗ (

∂ ∂ + iAλ (x) ) λ ∂x ∂z

is a linear connection such that ∇h = 0 (a Hermitian connection). Then ∇φ is the standard covariant differential and the 2-form F is given by the curvature tensor of A.

5.2.

Utiyama’s Reduction Theorems for Principal Connections

In [51] Utiyama generalized the results of Subsection 5.1. for a general Lie group G as a gauge group. A particle field Φ is now a section of a vector bundle associated with a principle G-bundle P and a gauge field is a principal connection Γ on P . Then any gauge invariant first order Lagrangian on Γ is given by a gauge invariant Lagrangian of the curvature tensor R[Γ], i.e. e L(j 1 Γ) = L(R[Γ]) .

(5.3)

e ∇Φ) . L(Γ, j 1 Φ) = L(Φ,

(5.4)

This result is in literature cited as the Utiyama’s theorem. Further, Utiyama generalized the minimal coupling principle as the invariant interaction of the particle field Φ and the gauge field Γ in the form

In his original paper [51] Utiyama considered his theorems only locally with gauge transformations described in coordinates. Later the Utiyama’s theorem was reproved by many authors also globally, see for instance Castrillón, Muñoz and Ratiu [2], Eck [6] and Mangiarotti and Modugno [36]. The Utiyama’s results can be very simply generalized

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for operators with values in a gauge-natural bundle of order (1, 0). In this case we shall use the term Utiyama-like theorem instead of the Utiyama’s theorem. The Utiyama-like theorem was proved (in order 1) in Koláˇr, Michor and Slovák [24]. A generalization of the invariant interaction (5.4) was proved globally by Betounes [1] who proved L(j 1 Γ, j 1 Φ) = e L(R[Γ], Φ, ∇Φ).

5.3.

First Reduction Theorem for General Linear Connections

In Janyˇska and Modugno [20], see also Janyˇska [12, 13, 14], we have studied second order natural quantum Lagrangians and second order natural Schrödinger operators on the quantum bundle, i.e. we studied second order operators on a gravitational field (a classical symmetric connection on spacetime), an electromagnetic potential (a quantum connection, i.e. a general linear connection on the quantum bundle preserving the Hermitian product) and a wave function (a section of the quantum bundle). In both situations such operators are factorized through the covariant differentials of sections of the quantum bundle where the first order covariant differentials are given by the quantum connection only but the second order covariant differentials are given by both quantum and spacetime connections. This fact was a motivation how to generalize the reduction theorems, see Subsection 3.1., for general linear connections on vector bundles. Let E → M be a vector bundle with a m-dimensional base and n-dimensional fibres. Local linear fiber coordinate charts on E will be denoted by (xλ , y i ). We define a linear connection on E to be a linear splitting K : E → J 1 E . Considering the contact morphism J 1 E → T ∗ M ⊗ T E over the identity of T M , a linear connection can be regarded as a T E-valued 1-form K : E → T ∗ M ⊗ T E projecting onto the identity of T M . The coordinate expression of a linear connection K is of the type K = dλ ⊗ ∂λ + Kj i λ y j ∂i , with Kj i λ ∈ C ∞ (M, R) .

Linear connections can be regarded as sections of a (1,1)-order G = Gl(n, R)-gaugenatural bundle Lin E → M described in Example 4.7..2. The curvature of a linear connection K on E turns out to be the vertical valued 2–form R[K] = −[K, K] : E → V E ⊗ ∧2 T ∗ M , where [, ] is the Froelicher-Nijenhuis bracket. If we consider the identification V E = E × E and linearity of R[K], the curvature R[K] can M

be considered as the curvature tensor field R[K] : M → E ∗ ⊗ E ⊗ ∧2 T ∗ M and R[K] : C ∞ (LinE) → C ∞ (E ∗ ⊗ E ⊗ ∧2 T ∗ M )

is a natural differential operator which is of order one. p,r Let us set Eq,s = ⊗p E ⊗ ⊗q E ∗ ⊗ ⊗r T M ⊗ ⊗s T ∗ M . Then a classical connection Λ on M and a linear connection K on E induce the linear tensor product connection Kqp ⊗ Λrs = p,r ⊗p K ⊗ ⊗q K ∗ ⊗ ⊗r Λ ⊗ ⊗s Λ∗ on Eq,s p,r p,r Kqp ⊗ Γrs : Eq,s → T ∗ M ⊗ T Eq,s M p Kq ⊗

p,r p,r → J 1 Eq,s . Let Φ ∈ which can be considered as a linear splitting Λrs : Eq,s p,r ∞ C (Eq,s ). We define the covariant differential of Φ with respect to the pair of connecp,r tions (K, Λ) as a section of Eq,s ⊗ T ∗ M defined by, Janyˇska [16],

∇(K,Λ) Φ = j 1 Φ − (Kqp ⊗ Λrs ) ◦ Φ .


159

The iterated rth order covariant differential applied on the curvature tensor of the linear connection K is a natural differential operator which is of order (r − 1) with respect to the classical connection Λ and of order (r + 1) with respect to the linear connection K. Let (s) (r) us denote by CLr E the image of this operator and by CC M ×M CL E the (s, r)-order curvature bundle of classical and linear connections given as the image of the pair of the operators (∇(s+1) R[Λ], ∇(r+1) R[K]), s ≥ r − 2, ∇(s) = (id, ∇, . . . , ∇s ), defined on Claτ M × Lin E. Let us assume a (1, 0)-order Gl(n, R)-gauge natural bundle F E, then the first reduction theorem for linear and classical connections can be formulated as follows, Janyˇska [15]. Theorem 5.1. Let s ≥ r − 2, r ≥ 0. All natural differential operators D : C ∞ (Claτ M × Lin E) → C ∞ (F E) M

which are of order s with respect to classical symmetric connections and of order r with respect to general linear connections are of the form e (s−1) R[Λ], ∇(r−1) R[K]) D(j s Λ, j r K) = D(∇

e is a zero order natural operator where D

e : C ∞ (C (s−1) M × C (r−1) E) → C ∞ (F E) . D C L M

5.4.

Second Reduction Theorem for General Linear Connections

Let us assume the kth order covariant differential of sections of Eqp11,q,p22 (particle fields). It is a natural operator of order k with respect to sections of Eqp11,q,p22 and of order (k − 1) with respect to classical and linear connections. Let us define the k-th order Ricci bundle Z (k) E as the image of the triplet of the operators (∇(k−2) R[Λ], ∇(k−2) R[K], ∇(k) Φ) defined on Claτ M × Lin E × Eqp11,q,p22 . Then the second reduction theorem for linear and classical connections can be formulated as follows, Janyˇska [15]. Theorem 5.2. Let s, r ≥ k − 1, s ≥ r − 2. All natural differential operators D : C ∞ (Claτ M × Lin E × Eqp11,q,p22 ) → C ∞ (F E) M

M

which are of order s with respect to classical symmetric connections, of order r with respect to general linear connections and of order k with respect to sections of Eqp11,q,p22 are of the form e (s−1) R[Λ], ∇(r−1) R[K], ∇(k) Φ) D(j s Λ, j r K, j k Φ) = D(∇

e is a zero order natural operator where D e : C ∞ ((C (s−1) M × C (r−1) E) D C L M

×

(k−2) (k−2) CC M × CL E M

Z (k) E) → C ∞ (F E) .

160

5.5.

Josef Janyˇska

Higher Order Utiyama’s Theorem

Higher order local version of the Utiyama-like theorem was studied by Horndeski [10] who generalized the replacement theorem of Thomas and his collaborators, [49, 50], for gauge fields. The results obtained in [10] are local and not complete since only concomitants obtained from the covariant differentials of the curvature tensor of the gauge field are assumed, while concomitants obtained from classical connections on the base only are not considered. By using the methods of gauge-natural bundles we obtain complete and global coordinate free description of higher order Utiyama-like theorem. Let G be an n-dimensional Lie group, P ∈ Ob PBm (G), Γ be a principal connection on P and ad P is the adjoint vector bundle associated with the principal bundle P . Then we have the induced adjoint linear connection ad(Γ) on ad P . If Γ has the coordinate expression Γ = dλ ⊗ (

∂ ea ) , + Γa λ (x) B ∂xλ

(5.5)

then ad(Γ) has, in the induced fibered coordinates (xλ , ua ) on ad P , the coordinate expression ∂ λ a b d ∂ ad(Γ) = d ⊗ + cbd Γ λ (x) u . (5.6) ∂xλ ∂ua The curvature tensors of principal connections are given by a 1-order natural operator from Pri P into ad P ⊗ ∧2 T ∗ M . The covariant differential of the curvature tensor R[Γ] with respect to Γ and a classical connection Λ on the base M is then defined as the covariant differential with respect to ad(Γ) and Λ, see Subsection 5.2. and Janyˇska [17]. Then the iterated rth order covariant differential ∇r R[Γ] is a natural operator on Cla M × Pri P which is of order (r − 1) with respect to classical connections and of order (r + 1) with (s) (r) respect to principal connections. Let us denote by CC M × CP P , s ≥ r − 2, (s, r)order curvature bundle for classical and principal connections obtained as the image of the pair of the operators (∇(s) R[Λ], ∇(r) R[Γ]) defined on Claτ × Pri P . Then higher order Utiyama-like theorem for principal and classical connections can be formulated as follows, Janyˇska [17]. Theorem 5.3. Let s ≥ r − 2, r ≥ 0, and let F be a (1, 0)-order G-gauge-natural bundle functor. All natural differential operators D : C ∞ (Claτ M × Pri P ) → C ∞ (F P ) M

which are of order s with respect to classical symmetric connections and of order r with respect to principal connections are of the form e (s−1) R[Λ], ∇(r−1) R[Γ]) D(j s Λ, j r Γ) = D(∇

e is a zero order natural operator where D

e : C ∞ (C (s−1) M × C (r−1) P ) → C ∞ (F P ) . D C P M


161

Remark 5.4. The curvature bundle of classical symmetric and principal connections is given by identities depending on the structure constants of the group G. So all natural operators defined on the curvature bundle depend also on the structure constants, i.e. e ∇(s−1) R[Λ], ∇(r−1) R[Γ]) . D(j s Λ, j r Γ) = D(c,

For instance cbab ∇ρr . . . ∇ρ1 Ra λµ is an example of a natural tensor field of the type (0, r+2) on M given by Λ (in order (r − 1)) and Γ (in order (r + 1)). In the case of (general) linear connections the structure constants are given by the Kronecker deltas and they contract with the curvature tensor fields, i.e. they are not ”visible”.

5.6.

Higher Order Utiyama’s Invariant Interaction

Let E → M be a vector bundle obtained as the bundle associated with P with respect to a linear representation ℓ : G → Gl(n, R), ℓ = (ℓij (g)), and let us denote by ℓ′ the set ∂ℓi

of constants ℓija = ∂gja |e, g a beeing the local coordinates on G and e ∈ G is the unit element. Then any principal connection Γ on P induces the linear connection ℓ(Γ) on E by ℓ(Γ)j i λ = ℓijb Γb λ . Let us assume Eqp11,q,p22 defined as in Subsection 5.3., then the covariant differential of sections of Eqp11,q,p22 (particle fields) with respect to the pair of connection (Γ, Λ) is defined as the covariant differential with respect to the pair of connections (ℓ(Γ), Λ), see Subsection 5.3.. The k-order covariant differential is then a natural differential operator of order k with respect to sections of Eqp11,q,p22 and of order (k − 1) with respect to classical and principal connections. Let us define the k-th order Ricci bundle Z (k) P as the image of the triplet of the operators (∇(k−2) R[Λ], ∇(k−2) R[K], ∇(k) Φ) defined on Claτ M × Lin E × Eqp11,q,p22 . Then the higher order invariant interaction of particle fields, principal and classical symmetric connections can be formulated as follows, Janyˇska [18]. Theorem 5.5. Let s, r ≥ k − 1, s ≥ r − 2. All natural differential operators D : C ∞ (Claτ M × Pri P × Eqp11,q,p22 ) → C ∞ (F P ) M

M

which are of order s with respect to classical symmetric connections, of order r with respect to principal connections and of order k with respect to sections of Eqp11,q,p22 are of the form e (s−1) R[Λ], ∇(r−1) R[Γ], ∇(k) Φ) D(j s Λ, j r Γ, j k Φ) = D(∇

e is a zero order natural operator where D e : C ∞ ((C (s−1) M × C (r−1) P ) D C P M

×

(k−2) (k−2) CC M × CP P M

Z (k) P ) → C ∞ (F P ) .

Remark 5.6. The Ricci identities and the identities on the curvature bundle of classical symmetric and principal connections depend on the structure constants of the group G and the constants ℓ′ . So all natural differential operators defined on the Ricci bundle depend also on the structure constants and ℓ′ , i.e. e ℓ′ , ∇(s−1) R[Λ], ∇(r−1) R[Γ]) . D(j s Λ, j r Γ) = D(c,

162

5.7.

Josef Janyˇska

Gauge Invariant Lagrangians

As direct consequences of Theorems 5.3 and 5.5 we obtain higher order gauge invariant Lagrangians in the form. Theorem 5.7. All natural Lagrangians of orders s in Λ, r in Γ and k in Φ are of the form e ∇(s−1) R[Λ], ∇(r−1) R[Γ]) , L(j s Λ, j r Γ) = L(c, e ′ , c, ∇(s−1) R[Λ], ∇(r−1) R[Γ], ∇(k) Φ) , L(j s Λ, j r Γ, j k Φ) = L(ℓ

where Le is a unique zero order gauge invariant Lagrangian.

Remark 5.8. Gauge invariant Lagrangians on classical symmetric connections (in order s) and principal connections (in order r) are infinitesimally given as solutions of the system of partial differential equations given by LJ s Cla ξ+J r P ri Ξ L = 0

(5.7)

for all right G-invariant vector fields Ξ on P over ξ on M , where Cla ξ is given by (2.1) and P ri Ξ is given by (4.1). From Theorem 5.3 it follows that all solutions of (5.7) factorize through the curvature tensors of both connections and their covariant differentials up to the orders (s − 1) and (r − 1). Example 5.9. Let (M, g) be a (pseudo-)Riemannian oriented manifold. Any natural Lagrangian of g, Γ and Φ is given by an gauge invariant Lagrangian function e l(ℓ′ , c, ∇(s−2) R[Λ(g)], ∇(r−1) R[Γ], g, ∇(k) Φ) .

All natural Lagrangians used in fields theories are of this type (for r, s ≤ 1), Fatibene and Francaviglia [9]. It is easy to see that the invariant Lagrangian function e l = g λ1 µ1 g λ2 µ2 g ρ1 σ1 . . . g ρr σr cdae cebd Ra λ1 λ2 ;ρ1 ;...;ρr Rb µ1 µ2 ;σ1 ;...;σr

defines a higher order Yang-Mills Lagrangian which is of order (r+1) with respect to Γ and of order r with respect to g. For r = 0 we obtain just the classical Yang-Mills Lagrangian.

5.8.

Final Remarks

Let us note that in literature as gauge invariant operators are sometimes considered operators invariant with respect to principal bundle morphisms over the identity of base manifolds, see for instance Etayo, Garc´ıa, Muñoz and Pérez [8] and Manno, Pohjanpelto and Vitolo [37]. In this case the corresponding mapping between standard fibres is equivariant with r G ⊂ Gs ⋊ T r G. Then for r-order (r ≥ 2) operators respect to the subgroup {e} × Tm m m on the gauge-natural bundle of principal connections it is not necessary to use auxiliary classical connections on base manifolds and all r-order natural differential operators (with values in a gauge-natural bundle of order (r + 1, 0)) factorize through gauge covariant differentials of the curvature tensor up to the order r. The gauge covariant differentiation was defined intrinsically by Eck [6], for local formulas see also normal gauge concomitants


163

by Horndeski [10]. For instance the gauge covariant differential of the curvature tensor R[Γ] is given in coordinates by Γ

∇ν R[Γ]a λµ = R[Γ]a λµ;ν = ∂ν R[Γ]a λµ − cabd Γb ν R[Γ]a λµ

and it is a (2, 0)-order field. Then we have Theorem 5.10. All Lagrangians of orders r in Γ and k in Φ which are invariant with respect to principal bundle morphisms over the identity of the base manifold are of the form e Γ ∇(r−1) R[Γ]) , L(j r Γ) = L(c, e ′ , c, Γ ∇(r−1) R[Γ], Γ ∇(k) Φ) , L(j r Γ, j k Φ) = L(ℓ

where Le is a unique zero order Lagrangian.

For instance, if in Example 5.9 we replace covariant differentials with respect to (Γ, Λ) by gauge covariant differentials with respect to Γ only, then the corresponding higher-order Yang-Mills Lagrangian is invariant with respect to principal bundle morphisms over the identity of the base manifold.

Acknowledgement This research has been supported by the Ministry of Education of the Czech Republic under the project MSM0021622409, by the Grant agency of the Czech Republic under the project GA201/05/0523.

References [1] D. Betounes, The geometry of gauge-particle field interaction: a generalization of Utiyama’s theorem, J. Geom. Phys. 6 (1989) 107–125. [2] M. Castrillón López, J. Muñoz Masqué and T. Ratiu, Gauge invariance and variational trivial problems on the bundle of connection, Diff. Geom. Appl. 19 (2003) 127–145. [3] E. B. Christoffel, Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades, Journal für die reine und angewandte Mathematik, Crelles’s Journals 70 (1869) 46–70. [4] J. A. Dieudonné and J. B. Carrell, Invariant Theory, Old and New (Academic Press, New-York-London, 1971). [5] W. Drechsler and M. E. Mayer, Fibre Bundle Techniques in Gauge Theories (SpringerVerlag, New York, 1977). [6] D. E. Eck, Gauge-natural bundles and generalized gauge theories, Mem. Amer. Math. Soc. 33 (247) (1981).

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[7] D. B. A. Epstein and W. P. Thurston, Transformation groups and natural bundles, Proc. London Math. Soc. (3) 38 (1979) 219–236. [8] F. Etayo Gordejuela, P. L. Garc´ıa Pérez, J. Muñoz Masque and J. Pérez Alvarez, Higher-order Utiyama-Yang-Mills Lagrangians, J. Geom. Phys. 57 (2007) 1089–1097. [9] L. Fatibene and M. Francaviglia, Natural and Gauge Natural Formalism for Classical Field Theories, a Geometric Perspective including Spinors and Gauge Theories (Kluwer Academic Pub. 2003). [10] G. W. Horndeski, Replacement Theorems for Concomitants of Gauge Fields, Utilitas Math. 19 (1981) 215–146. ˇ [11] J. Janyˇska, Geometrical properties of prolongation functors, Cas. pˇest. mat. 110 (1985) 77–86. [12] J. Janyˇska, Natural quantum Lagrangians in Galilei quantum mechanics, Rendiconti di Matematica (Roma), Serie VII 15 (1995) 457–468. [13] J. Janyˇska, Natural Lagrangians for quantum structures over 4-dimensional spacetime, Rendiconti di Matematica (Roma), Serie VII 18 (1998) 623–648. [14] J. Janyˇska, A remark on natural quantum Lagrangians and natural generalized Schrödinger operators in Galilei quantum mechanics, In: The Proceedings of the Winter School Geometry and Topology (Srn´ı, 2000, Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II – No. 66, 2001) 117–128. [15] J. Janyˇska, Reduction theorems for general linear connections, Diff. Geom. Appl. 20 (2004) 177–196. [16] J. Janyˇska, On the curvature of tensor product connections and covariant differentials, In: The Proceedings of the 23rd Winter School Geometry and Physics (Srn´ı, 2003, Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II - Numero 72, 2004) 135–143. [17] J. Janyˇska, Higher order Utiyama-like theorem, Rep. Math. Phys. 58 (2006) 93–118. [18] J. Janyˇska, Higher order Utiyama invariant interaction, Rep. Math. Phys. 59 (2007) 63–81. [19] J. Janyˇska and M. Modugno, Infinitesimal natural and gauge-natural lifts, Diff. Geom. and its Appl. 2 (1992) 99–121. [20] J. Janyˇska and M. Modugno, Covariant Schroedinger operator, J. Phys. A: Math. Gen. 35 (2002) 8407–8434. [21] A. A. Kirillov, Invariant operators over geometric quantities, (Russian), Current problems in mathematics 16, VINITI, Moscow (1980) 3–29. [22] I. Koláˇr, Prolongations of generalized connections, In: Diff. Geom. (Math. Soc. János Bolyai, 31, Budapest Hungary, North Holland, 1979) 317–325.


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[23] I. Koláˇr, Some natural operators in differential geometry, In: Proc. Conf. Diff. Geom. and Its Appl. (Brno 1986, D. Reidel, 1987) 91–110. [24] I. Koláˇr, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry (Springer–Verlag, 1993). [25] I. Koláˇr and J. Slovák, On the geometric functors on manifolds, Suppl. Rend. Circ. Mat. Palermo 21 (1989) 223–233. [26] D. Krupka, A setting for generally invariant Lagrangian structures in tensor bundles, Bull. Acad. Polon. Sci., Sér. Math. Astronom. Phys. 22 (1974) 967–972. [27] D. Krupka, A theory of generally invariant Lagrangians for the metric fields I, Internat. J. Theoret. Phys. 17 (1978) 359–368. [28] D. Krupka, A theory of generally invariant Lagrangians for the metric fields II, Internat. J. Theoret. Phys. 15 (1976) 949–959. [29] D. Krupka, Elementary theory of differential invariants, Arch. Math. 4, Scripta Fac. Sci. Nat. UJEP Brunensis, XIV, (1978) 207–214. [30] D. Krupka, Pˇrirozené Lagrangeovy struktury, DrSc–thesis, Pˇr´ırodovˇedecká fakulta UJEP, Brno, 1978. [31] D. Krupka, Natural Lagrangian structures, Banach Center Publications 12 (1984) 185–210. [32] D. Krupka, Local invariants of a linear connection, In: Diff. Geom. (Coll. Math. Soc. Janos Bolyai 31, Budapest, Hungary, 1979, North Holland, 1982) 349–369. [33] D. Krupka and J. Janyˇska, Lectures on Differential Invariants (Folia Fac. Sci. Nat. Univ. Purkynianae Brunensis, Brno 1990). [34] D. Krupka and V. Mikolásˇová, On the uniqueness of some differential invariants: d, { , }, ∇, Czechoslovak Math. J. 34 (1984) 588–597. [35] D. Krupka and A. Trautman, General invariance of Lagrangian structures, Bull. Acad. Polon. Sci., Sér. Math. Astronom. Phys. 22 (1974) 207–211. [36] L. Mangiarotti and M. Modugno, On the geometric structure of gauge theories, J. Math. Phys. 26 (1985) 1373–1379. [37] G. Manno, J. Pohjanpelto and R. Vitolo, Gauge invariance, charge conservation, and variational principles, preprint 2007. ˇ pˇest. mat. 114 [38] W. Mikulski, There exists a prolongation functor of infinite order, Cas. (1989) 57–59. [39] A. Nijenhuis, Theory of the geometric object, Thesis, University of Amsterdam, 1952. [40] A. Nijenhuis, Natural bundles and their general properties, In: Diff. Geom. (in honour of K. Yano, Kinokuniya, Tokyo, 1972) 317–334.

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[41] J. Novotný, On the generally invariant Lagrangians for the metric field and other tensor fields, Internat. J. Theoret. Phys. 17 (1978) 677–684. [42] R. S. Palais and C. L. Terng, Natural bundles have finite order, Topology 16 (1977) 271–277. [43] D. J. Saunders, The Geometry of Jet Bundles (London Math. Soc., Lecture Note Series 142, Cambridge University Press, 1986). [44] J. A. Schouten, Ricci Calculus (Berlin-Göttingen, 1954). [45] J. A. Schounten and J. Haantjes, On the theory of the geometric object, Proc. London Math. Soc. 42 (1937) 356–376. [46] J. Slovák, On finite order of some operators, In: Proc. Conf. Diff. Geom. and Its Appl. (Communications, Brno 1986) J. E. Purkynˇe University, 1987 283–294. [47] J. Slovák, On natural connections on Riemannian manifolds, Comment. Math. Univ. Carolinae 30 (1989) 389–393. [48] C. L. Terng, Natural vector bundles and natural differential operators, Am. J. Math. 100 (1978) 775–828. [49] T. Y. Thomas, Differential Invariants of Generalized Spaces (Cambridge University Press, Cambridge, 1934). [50] T. Y. Thomas and A. D. Michal, Differential invariants of affinely connected manifolds, Ann. Math. 28 (1927) 196–236. [51] R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. 101 (1956) 1597–1607. [52] H. Weyl, Raum - Zeit - Materie (Space - Time - Matter) (1918, translated from the 4th German Edition, London: Methuen 1922). [53] A. Zajtz, Foundations of Differential Geometry on Natural Bundles (Caracas, 1983). [54] A. Zajtz, The sharp upper bound on the order of natural bundles of given dimensions, Bull. Soc. Math. Belgique 3 (1987) 347–357. [55] C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Physical Rev. (2) 96 (1954) 191–195.


Chapter 10

C ONNECTIONS ON H IGHER O RDER F RAME B UNDLES AND T HEIR G AUGE A NALOGIES∗ Ivan Kolárˇ† Institute of Mathematics and Statistics Faculty of Science, Masaryk University Janácˇ kovo nám. 2a, 602 00 Brno Czech Republic

Abstract In the first part of the paper, we present a survey of the basic properties of connections on the r-th order frame bundle of a manifold. Special attention is paid to the torsion and torsion-free connections. In the second part, connections on the r-th principal prolongation of a principal bundle are treated from similar points of view. The case of the first principal prolongation is discussed in detail.

2000 Mathematics Subject Classification. 58A20, 53C05, 58A32 Key words and phrases. r-th order frame bundle, connection, torsion, natural operator, semiholonomic 2-jet, r-th principal prolongation of principal bundle, gauge-natural operator In the present paper, connection means a principal connection on a principal bundle unless otherwise specified. Several properties of connections on the r-th order frame bundle P r M of a manifold M appear in the framework of the general theory of natural bundles and operators. This is described in the book by D. Krupka and J. Janyˇska, [21], and in the monograph [18]. So the first part of the present paper is devoted to a survey of some more specific properties of connections on P r M that are mostly related with the idea of torsion. In Section 1 we underline that the Lie algebroid version of a connection on the principal bundle P r M (M, Grm ) is a linear r-th order connection on T M . So we have two different approaches to the concept of torsion. Proposition 2 reads that both approaches are equivalent. ∗ †

To Demeter Krupka, on the occasion of his 65th birthday E-mail address: [email protected]

Ivan Koláˇr

168

In Section 3 we clarify that the torsion-free connections on P r M are in bijection with the reductions of P r+1 M to the canonical injection of G1m into Gr+1 m . This enables us to define the r-th exponential operator transforming every torsion-free connection Λ on P 1 M into a torsion-free connection on P r M . In particular, this implies that Λ determines a general connection on every natural bundle over m-manifolds. In Section 5, the Lie algebroid construction of the exponential operator is based on two interesting lemmas concerning the r-jet of the commutator of vector fields on M . Then we deduce that the torsion-free connections on P r M are in bijection with the splittings from the cotangent bundle T ∗ M into the bundle of (1, r + 1)-covelocities on M . In Section 7 we discuss a connection on P r M from the viewpoint of the theory of higher order G-structures and we characterize its integrability in this sense. In Section 8 we present a recent result by W. Mikulski, [27], who determined all natural operators transforming a torsion-free connection on P 1 M into a connection on P r M . Section 9 is devoted to the basic properties of semiholonomic 2-jets, that represent a useful tool for several problems of the present paper. The principal prolongation W r P of an arbitrary principal bundle P (M, G) is defined in Section 10 in a formally slightly different way to [18]. We hope this could be useful in applications. Then we summarize the basic properties of connections on W r P and their torsions. In Section 11 we clarify that every connection Φ on W 1 P is canonically identified with the triple (Γ, Λ, D) of a connection Γ on P , a connection Λ on P 1 M and a section D 2 N

of L0 P ⊗ T ∗ M , where L0 P is the adjoint bundle of P . In Section 12 we present the list of all gauge-natural operators transforming every pair (Γ, Λ) of a connection Γ on P and a torsion-free connection Λ on P 1 M into a connection on W 1 P . Finally we outline how the semiholonomic 2-jets can be used in the theory of connections on W 1 P . In particular, ˜ to every connection Φ on W 1 P by using the we introduce the conjugate connection Φ canonical involution of semiholonomic 2-jets. All manifolds and maps are assumed to be infinitely differentable. Unless otherwise specified, we use the terminology and notations from the book [18].

1.

The Algebroid form of Connections on P r M

First we recall that the Lie algebroid LP → M of an arbitrary principal bundle P (M, G) is defined by LP = T P/G. So the elements of LP are the right invariant families of tangent vectors along the individual fibers of P , every section σ : M → LP is identified with a right invariant vector field σ : P → T P and the bracket [σ 1 , σ 2 ] of right invariant vector fields on P induces the bracket [[σ1 , σ2 ]] of LP . The canonical projection q : LP → T M is called the anchor map. Clearly, a connection Γ on P can be interpreted as a linear morphism γ : T M → LP satisfying q ◦ γ = id T M , [26]. We write P r M for the r-th order frame bundle of an m-dimensional manifold M . This is a principal bundle over M with structure group Grm = inv J0r (Rm , Rm )0 . Every local diffeomorphism f : M1 → M2 induces a principal bundle morphism P r f : P r M1 → P r M2 , so that P r is a bundle functor on the category Mfm of m-dimensional manifolds and their local diffeomorphisms, [18]. For every vector field X : M → T M , its flow

Connections on Higher Order Frame Bundles and Their Gauge Analogies

169

prolongation ∂ r P (F ltX ) (1) ∂t 0 is a right invariant vector field on P r M . This follows directly from the fact that the values of P r are in the category PBm (Grm ) of principal Grm -bundles over m-manifolds and their local principal bundle isomorphisms. Since P r is an r-th order bundle functor, the restriction P r X | Pxr M depends on jxr X only, x ∈ M . P r X :=

Proposition 1. The rule identifies J r T M with LP r M .

r IM (jxr X) = P r X | Pxr M

(2)

r is a diffeomorphism. But P r M = reg T r M is an open Proof. We have to prove that IM m r M of all (m, r)-velocities on M and P r X is the restriction of subset of the bundle Tm r follows from the the flow prolongation Tmr X to this subset. Hence the bijectivity of IM r r existence of an exchange isomorphism κM : Tm T M → T Tm M such that Tmr X = κM ◦ r M , [16], [18]. Tm

A linear r-th order connection on T M is a linear morphism T M → J r T M that splits the target jet projection. According to Proposition 1, every connection Γ : P r M → J 1 P r M is identified with a linear splitting γ : T M → J r T M . We say that γ is the algebroid form of Γ.

2.

Two Approaches to the Torsion on P r M

r m The canonical (Rm ×gr−1 m )-valued 1-form ϕr on P M is defined as follows. We have R × r−1 r−1 r−1 m r m gm = Ter−1 P R , where er−1 = j0 id Rm . Every u = j0 f , f : R → M , induces P r−1 f : P r−1 Rm → P r−1 M . The tangent map u ˜ := Ter−1 P r−1 f : Ter−1 P r−1 Rm → r (u), depends on u only. Then one defines Tur−1 P r−1 M , ur−1 = πr−1 r (A) , ϕr (A) = u ˜−1 T πr−1

A ∈ Tu P r M .

P. C. Yuen introduced the torsion of a connection Γ on P r M as the exterior covariant differential DΓ ϕr of ϕr , [33]. Since DΓ ϕr is a horizontal 2-form on P r M , it can be 2 ∗ interpreted as a map P r M → (Rm ×gr−1 m )⊗Λ T M . Taking into account the identification m r u ˜1 : R → Tx M , u1 = π1 (u), we construct 2 m∗ DΓ ϕr : P r M → (Rm × gr−1 . m )⊗Λ R

(3)

On the other hand, the (r − 1)-jet at x ∈ M of the bracket [ξ, η] of two vector fields ξ, η on M depends on the r-jets jxr ξ and jxr η. This defines a map [ , ]r−1 : J r T M ×M J r T M → J r−1 T M . Let γ : T M → J r T M be the algebroid form of Γ. According to A. Zajtz, [28], the torsion of γ is a map τ γ : T M ×M T M → J r−1 T M defined by

τ γ(A, B) = γ(A), γ(B)

r−1

,

A, B ∈ Tx M .

(4)

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Clearly, τ γ can be interpreted as a section of J r−1 T M ⊗ Λ2 T ∗ M . This is a fiber bundle 2 m∗ . So the frame form of τ γ is associated to P r M with standard fiber (Rm × gr−1 m )⊗Λ R a map 2 m∗ τ γ : P r M → (Rm × gr−1 . m )⊗Λ R In [14], we deduced Proposition 2. We have DΓ ϕr = 12 τ γ. Let Γ and ∆ be two connections on P r M over the same connection on P r−1 M . Conr sider their algebroid forms γ, δ : T M → J r T M . Since the kernel of πr−1 : J rT M → J r−1 T M is T M ⊗ S r T ∗ M , [18], the difference of γ and δ is a section γ − δ : M → T M ⊗ Sr T ∗M ⊗ T ∗M .

(5)

Proposition 3. If both γ and δ are torsion-free, then the values of γ − δ lie in T M ⊗ S r+1 T ∗ M . Proof. If xi are some local coordinates on M , X i = dxi are the induced coordinates on T M and Xαi are the jet coordinates on J r T M , then the equations of γ or δ are Xαi = Γiαj (x)X j

or

Xαi = ∆iαj (x)X j ,

1 ≤ |α| ≤ r ,

where α is a multi-index of range m. The difference γ − δ can be interpreted as a map T M ×M T M → T M ⊗ S r−1 T ∗ M of the form Γiβjk − ∆iβjk ξ j η k ,

|β| = r − 1 .

The only r-th order terms in j r−1 [ξ, η] are ξj

r i ∂ r ηi j ∂ ξ − η , ∂xj ∂β x ∂xj ∂β x

|β| = r − 1 .

(6)

If γ is torsion-free, then (6) yields Γiβjk = Γiβkj . If δ is also torsion-free, (5) is symmetric in the last two subscripts. From the proof one sees directly that γ − δ is an arbitrary section of T M ⊗ S r+1 T ∗ M .

3.

Torsion-Free Connections on P r M as Reductions of P r+1 M

Every a ∈ G1m is a matrix, which defines a linear map l(a) : Rm → Rm . This induces a group homomorphism lr−1 : G1m → Grm ,

lr−1 (a) = j0r l(a) .

S. Kobayashi proved, [8], that the torsion-free connections on P 1 M are in bijection with the reductions of P 2 M to the subgroup l1 (G1m ) ⊂ G2m . We deduce an analogous result for arbitrary order r. This is based on the following injection irM : P r+1 M ֒→ J 1 P r M . Every u = j0r+1 f ∈ P r+1 M determines a local section ψ of P r M → M ψ(y) = j0r f ◦ tf −1 (y) ,

(7)


171

where y lies in a neighbourhood of f (0) ∈ M and tf −1 (y) : Rm → Rm is the translation x 7→ x + f −1 (y). Then we set irM (u) = jf1(0) ψ. If xi , xij , . . . , xiji ...jr are the standard coordinates on P r Rm , xij,k , . . . , xij1 ...jr ,k are the induced coordinates on J 1 P r Rm and xij1 ...jr+1 are the additional coordinates on P r+1 Rm , then (7) implies directly the following coordinate form of ir xij,k = xijl x ˜lk , . . . , xij1 ...jr ,k = xij1 ...jr l x ˜lk , (8) where x ˜ij is the inverse matrix to xij . Every X ∈ J 1 P r M over βX ∈ P r M is identified with an m-plane in the tangent space TβX P r M , which will be denoted by the same symbol X. Hence we can consider the restriction dϕr | X of the exterior differential of ϕr to X. Denote by X1 ∈ J 1 P r−1 M the underlying element of X. The following lemma from [14] is close to a result by Yuen, [33]. r r+1 M ) if and only Lemma 1. Let X ∈ J 1 P r M satisfy X1 = ir−1 M (βX). Then X ∈ iM (P if dϕr | X = 0.

For r = 1 we have no X1 and the claim dϕ1 | X = 0 if and only if X ∈ i1M (P 2 M ) was used in [8]. For every torsion-free connection Γ on P r M we define a map µ(Γ) : P 1 M → P r+1 M by the following induction. Consider a connection Γ : P r M → J 1 P r M such that the underlying connection Γ1 : P r−1 M → J 1 P r−1 M is torsion-free, so that Γ1 determines a map µ(Γ1 ) : P 1 M → P r M be the induction hypothesis. Proposition 4. Γ is torsion-free, if and only if the values of Γ ◦ µ(Γ1 ) lie in irM (P r+1 M ). Then we define µ(Γ) = (irM )−1 ◦ Γ ◦ µ(Γ1 ) : P 1 M → P r+1 M . Proof. By Lemma 1, we have dϕr | X = 0 for all X ∈ Γ µ(Γ1 )(P 1 M ) . But ϕr is a pseudotensorial form, [18], so that dϕr | A = 0 holds for every A ∈ Γ(P r M ). This is equivalent to DΓ ϕr = 0.

For every principal bundle P (M, G), we have an induced right action of G on J 1 P , jx1 s(y), g 7→ jx1 s(y)g , where s is a local section of P on a neighbourhood of x ∈ M and g ∈ G. This action will be denoted by (X, g) 7→ X̺(g). Lemma 2. For every v ∈ P r+1 M and a ∈ G1m , we have irM vlr (a) = irM (v)̺ lr−1 (a) .

Proof. If v = j0r+1 f , then irM (v)̺ lr−1 (a) = jx1 j0r f ◦ tf −1 (y) ◦ l(a) . On the other hand, irM vlr (a) = jx1 j0r f ◦ l(a) ◦ tl(a)−1 (f −1 (y)) . But tz ◦ l(a) = l(a) ◦ tl(a)−1 (z) , z ∈ Rm , is a well known relation from the affine geometry.

By Lemma 2, µ(Γ)(P 1 M ) is a reduction of P r+1 M to the subgroup lr (G1m ) ⊂ Gr+1 m . Indeed, using induction we obtain µ(Γ)(ua) = (irM )−1 Γ µ(Γ1 )(u) ̺ lr−1 (a)

= µ(Γ)(u)lr (a) .

On the other hand, every reduction Q ⊂ P r+1 M to the subgroup lr (G1m ) induces a map (denoted by the same symbol) Q : P 1 M → P r+1 M as follows. For every v ∈ Q we

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construct u = π1r+1 (v) and we set Q(u) = v. Any other v¯ in the same fiber of Q → M is of the form v¯ = vlr (a), a ∈ G1m . This implies π1r+1 (¯ v ) = ua, so that our definition is correct. Proposition 5. Proposition 4 establishes a bijection between torsion-free connections on P r M and reductions of P r+1 M to lr (G1m ). Proof. First we deduce that µ(Γ) : P 1 M → P r+1 M is a reduction to lr (G1m ). For every u ∈ P 1 M and a ∈ G1m we have µ(Γ)(ua) = (irM )−1 Γ µ(Γ1 )(ua)

= (irM )−1 Γ µ(Γ1 )(u)lr−1 (a)

= (irM )−1 Γ µ(Γ1 )(u) ̺ lr−1 (a)

= (irM )−1 (irM ) µ(Γ)(u) lr (a)

by definition, by the induction hypothesis, by right-invariance of Γ and by Lemma 2. Conversely, if Q : P 1 M → P r+1 M is a reduction to lr (G1m ), then Q1 = πrr+1 ◦ Q : P 1 M → P r M is a reduction to lr−1 (G1m ). We define Γ : Q1 (P 1 M ) → J 1 P r M by Γ Q1 (u) = irM Q(u). By Lemma 2, it holds Γ Q1 (ua) = irM Q(ua) = irM Q(u)lr (a) = r iM Q(u) ̺ lr−1 (a) = Γ Q1 (u) ̺ lr−1 (a) . Hence Γ is a right-invariant map, which is canonically extended into a connection on P r M .

4.

The r-th Order Exponential Prolongation

The following construction represents an interesting application of Proposition 5. Consider a torsion-free connection Λ on P 1 M . For every x ∈ M , Λ determines the exponential map expΛ x : Ux → M , where Ux ⊂ Tx M is a neighbourhood of the origin. Then we define a map Er (Λ) : P 1 M → P r+1 M by Er (Λ)(u) = j0r+1 (expΛ x ◦u) ,

u ∈ Px1 M ,

(9)

where u is interpreted as a map Rm → Tx M . Proposition 6. Er (Λ)(P r M ) is a reduction of P r+1 M to lr (G1m ). Proof. For all u ∈ P 1 M and a ∈ G1m , we have Er (Λ)(ua) = j0r+1 expΛ x ◦u ◦ l(a) = Er (Λ)(u)lr (a).

By Proposition 5, Er (Λ) is a torsion-free connection on P r M , that is called the r-th exponential prolongation of Λ. The rule Λ 7→ Er (Λ) is said to be the r-th order exponential operator on the bundle Qτ P 1 M of torsion-free connections on P 1 M . W. Mikulski invented another construction of the exponential prolongation, [27]. Every ˜ on Tx M by means of translations. The expoX ∈ Tx M is extended into a vector field X Λ ˜ locally into a vector field (expΛ ˜ nential map expx transforms X x )∗ (X) on M . Then we can construct r ˜ εr (Λ)(X) = j0r (expΛ (10) x )∗ (X) ∈ Jx T M .

In Section 5 we deduce that εr (Λ) is the algebroid form of Er (Λ).


173

This result enables us to describe another geometrically interesting construction of r ˜ Er (Λ). The flow prolongation P r (expΛ x )∗ (X) is a vector field on P M . By Section 1, r r the lifting map P M ×M T M → T P M of Er (Λ) is ˜ Er (Λ)(u, X) = P r (expΛ x )∗ (X) (u) ,

u ∈ Pxr M .

Further, consider an r-th order natural bundle F over m-manifolds, [18]. So F M is a fiber bundle associated to P r M with standard fiber F0 Rm . Every principal connection Γ on P r M induces a general connection ΓF on P r M . We shall use the construction of ΓF by means of lifting vector fields. In general, every right-invariant vector field Z on P r M r induces a vector field ZF on F M as follows. If Z(u) = dc(0) dt , c : R → P M , u = c(0), then d ZF {u, a} = 0 c(t), a , a ∈ F0 Rm . dt Since Z is right-invariant, this definition is correct. Then the ΓF -lift of a vector field X on M is prescribed by ΓF (X) = (ΓX)F . Clearly, the flow prolongation FX of a vector field X on M with respect to F satisfies FX = (P r X)F . Thus we have deduced Proposition 7. For every r-th order natural bundle F , the rule ˜ Er (Λ)F (v, X) = F (expΛ x )∗ (X) (v) ,

v ∈ Fx M, X ∈ Tx M

transforms every torsion-free connection Λ on P 1 M into general connection Er (Λ)F on FM.

5.

The Exponential Prolongation in the Algebroid Form

Consider an arbitrary linear splitting γ : T M → J r T M . For a linear frame u ∈ Px1 M , u = (A1 , . . . , Am ), Ai ∈ Tx M , we take vector fields Xi satisfying jxr Xi = γ(Ai ), i = 1, . . . , m. Then F ltX1 1 ◦ · · · ◦ F ltXmm (x)

is a local map Rm → M and we define

σ(γ)(u) = j0r+1 F ltX1 1 ◦ · · · ◦ F ltXmm (x) ∈ Pxr+1 M .

(11)

One verifies easily that σ(γ)(u) depends on u and γ only. Proposition 8. If γ is torsion-free, then σ(γ)(P 1 M ) is a reduction of P r+1 M to lr (G1m ). Proof is based, in a very instructive way, on the definition (4) of τ γ. We shall use the following two lemmas from [15]. Consider two vector fields X and Y on M . Then F ltX ◦ F lτY (x) is a local map r+1 R2 → M , so that j0,0 F ltX ◦ F lτY (x) ∈ (T2r+1 M ) is a (2, r + 1)-velocity on M . Lemma 3. If jxr−1 [X, Y ] = 0, then

r+1 r+1 j0,0 F ltX ◦ F lτY (x) = j0,0 F lτY ◦ F ltX (x) .

(12)

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174

Further, F ltX ◦ F lt (x) is a local map R → M , so that j0r+1 F ltX ◦ F ltY (x) ∈ (T1r+1 M )x is a (1, r + 1)-velocity on M . Y

Lemma 4. If jxr−1 [X, Y ] = 0, then j0r+1 F ltX ◦ F ltY (x) = j0r+1 F ltX+Y (x) .

We shall also apply the well known formula X F lct = F ltcX ,

(13)

c ∈ R.

(14)

Take a = (aij ) ∈ G1m and consider ua = (aji Aj ). Since γ is torsion-free, by (13), (14) and (12) we obtain gradually a1 X1 +···+am 1 Xm

σ(γ)(ua) = j0r+1 F lt11

a 1 X1

= j0r+1 F lt11

a1 X1 +···+am m Xm

◦ · · · ◦ F ltmm

a m Xm

◦ · · · ◦ F lt11

a 1 X1

◦ · · · ◦ F ltmm

a m Xm

◦ · · · ◦ F ltmm

m = j0r+1 F laX11t1 ◦ · · · ◦ F laXmmt1 ◦ · · · ◦ F laX11 tm ◦ · · · ◦ F laXm m mt

=

m

1

1

j0r+1

F laX11t1 +···+a1 tm m 1

◦ ··· ◦

F laXmmt1 +···+am tm . m 1

This proves Proposition 8. To clarify the relation of σ(γ) to the reduction µ(Γ) from Section 3, we need the folr M . Clearly, lowing form of the injection irM : P r+1 M → J 1 P r M . We have P r M ⊂ Tm r r m j0 f ∈ Tm M , f : R → M , can be expressed in the form r j0r f = (Tm f )(er ) ,

er = j0r id Rm .

(15)

∂ r i r Rm , where τ i : Rm → Rm is the translation t¯1 = j τ ∈ Ter Tm Write Ei = ∂t t 0 0 t t1 , . . . , t¯i = ti + t, . . . , t¯m = tm . If we consider j0r+1 ψ ∈ P r+1 M , then

r (T Tm ψ)(Ei )

(16)

is an m-tuple of tangent vectors at j0r ψ ∈ P r M . The linear span of these vectors defines r+1 r iM j0 ψ ∈ J 1 P r M .

Proposition 9. If γ is torsion-free and Γ is the corresponding connection on P r M , then σ(γ) = µ(Γ). Proof. We proceed by introduction. If γ1 and Γ1 are the underlying connections in the order r − 1, then σ(γ1 ) = µ(Γ1 ) by the induction hypothesis. Consider u = (A1 , . . . , Am ) ∈ Px1 M and write v = σ(γ1 )(u) = µ(Γ1 )(u) . By (16), irM j0r+1 F ltX1 1 ◦ · · · ◦ F ltXmm (x) is the linear span of the vectors

r T Tm F ltX1 1 ◦ · · · ◦ F ltXmm (Ei ) ,

i = 1, . . . , m .

(17)


175

Using the basic properties of flows, Lemma 3 and (15), we deduce that (17) is equal to ∂ r Xi Xm Tm F ltX1 1 ◦ · · · ◦ F lt+t (er ) i ◦ · · · ◦ F ltm 0 ∂t ∂ T rX T rX T rX = 0 F lt m i ◦ F lt1m 1 ◦ · · · ◦ F ltmm m (er ) ∂t r = Tmr Xi Tm (F ltX1 1 ◦ · · · ◦ F ltXmm )(er ) = Tmr Xi (v) .

By (2) and by the induction hypothesis, this m-tuple spans µ(Γ)(v).

From the proof of Proposition 8 we obtain easily that the construction (10) of εr (Λ) by W. Mikulski is the algebroid form of the exponential prolongation Er (Λ) introduced in Section 4.

6.

Splittings T ∗ M → T r+1∗ M

The space T r+1∗ M = J r+1 (M, R)0 of all (1, r + 1)-covelocities on M is a vector bundle, [18]. By a splitting s : T ∗ M → T r+1∗ M we mean a linear morphism satisfying π1r+1 ◦ s = id T ∗ M . We remark that such splittings play an interesting role in the construction of Poincaré-Cartan morphisms in the higher order variational calculus, [11]. Proposition 10. There is a canonical bijection between reductions Q ⊂ P r+1 M to subgroup lr (G1m ) and splittings s : T ∗ M → T r+1∗ M . Proof. Every b ∈ Tx∗ M determines a linear map λ(b) : Tx M → R. Let v = j0r+1 f ∈ Qx , so that u = π1r+1 (v) ∈ Px1 M can be interpreted as a map u : Rm → Tx M . Then we set s(b) = jxr+1 λ(b) ◦ u ◦ f −1 ∈ Txr+1∗ M .

(18)

Since Q is a reduction to lr (G1m ), (18) does not depend on the choice of v ∈ Qx , The fact that s is a splitting follows directly from (18). Conversely, let s : T ∗ M → T r+1∗ M be a splitting. A frame u ∈ Px1 M is a basis (e1 , . . . , em ) of Tx M . Consider the dual basis u∗ = (e1 , . . . , em ) of Tx∗ M . Then s(e1 ), . . . s(em ) are the components of an (r + 1)-jet −1 s(u∗ ) ∈ Jxr+1 (M, R)0 . Write Q(u) = s(u∗ ) ∈ Pxr+1 M for the inverse jet. If we j ∗ i j i take ua = (ai ej ), then (ua) = (˜ aj e ), where a ˜j is the inverse matrix to aij . Hence s (ua)∗ = a ˜ij s(ej ) = lr (a−1 ) ◦ s(u∗ ), which implies Q(ua) = Q(u)lr (a). Finally, one verifies easily that the maps Q 7→ s and s 7→ Q are inverse each other. Taking into account Proposition 5, we obtain a canonical bijection between torsion-free connections on P r M and splittings T ∗ M → T r+1∗ M . We remark that Proposition 10 yields another proof of Proposition 3.

7.

The Viewpoint of Higher Order G-Structures

A connection on P r M can be viewed as a kind of higher order G-structure on M . We recall that a k-th order G-structure on M is said to be integrable, if it is locally isomorphic to the

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product Rm ×G, where G ⊂ Gkm is the structure group. We are going to apply this approach to the algebroid form γ : T M → J r T M . (We remark that this kind of integrability plays an important role in our theory of the flow prolongation of some tangent valued forms, [1].) On Rm , there is a distinguished connection Cr : T Rm → J r T Rm defined by ˜, Cr (X) = jxr X

X ∈ Tx Rm ,

˜ is the constant vector field on Rm constructed from X by means of translations. where X Definition 1. We say that γ : T M → J r T M is integrable, if for every x ∈ M there exists a neighbourhood U and a diffeomorphism f : U → Rm satisfying Cr ◦T f = J r T f ◦(γ | U ). Clearly, every integrable connection γ is torsion-free. According to a classical result, a connection Λ on P 1 M is integrable, iff it is both torsion-free and curvature-free. Then every exponential prolongation εk (Λ) is also integrable. Thus, the torsion of γ is the first obstruction to its integrability. Consider the underlying connections γk = πkr ◦ γ, k = 1, . . . , r. Clearly, if γ is torsion-free or integrable, then each γk is so. Assume that γ is torsion-free. Then the curvature of γ1 is the second obstruction to the integrability of γ. If this curvature vanishes, each connection εk (γ1 ) is integrable. The difference γ2 − ε2 (γ1 ) is a tensor field of type T M ⊗ S 3 T ∗ M that is the third obstruction to the integrability of γ. Assume by induction that the first up to (k + 1)-st obstruction to the integrability of γ vanish. Then γk = εk (γ1 ) and the tensor field of type T M ⊗ S k+2 T ∗ M γk+1 − εk+1 (γ1 ) is the next obstruction to the integrability of γ. If all these r + 1 obstructions vanish, then γ = εr (γ1 ) is integrable. Thus, we have proved Proposition 11. γ is integrable if and only if all the following conditions are satisfied a) γ is torsion-free, b) γ1 is curvature-free, c) all the gradually defined tensor fields γk − εk (γ1 ), k = 2, . . . , r, vanish.

8.

Natural Operators C ∞ Qτ P 1 M → C ∞ QP r M

We write QP for the connection bundle of an arbitrary principal bundle P (M, G), [18]. The connections on P form the space C ∞ QP of all sections of QP → M . Further, we write Qτ P r M for the bundle of all torsion-free connections on P r M , [14]. So the r-th exponential operator on M is a natural operator C ∞ Qτ P 1 M → C ∞ Qτ P r M . Using Er , W. Mikulski solved a rather sophisticated problem of finding all natural operators C ∞ Qτ P 1 M → C ∞ QP r M and C ∞ Qτ P 1 M → C ∞ Qτ P r M , [27]. Every torsion-free connection Λ on P 1 M defines a vector bundle isomorphism ψΛ : J r T M →

r M

k=0

T M ⊗ Sk T ∗M

(19)

Connections on Higher Order Frame Bundles and Their Gauge Analogies as follows. Write I : J0r T Rm →

r M

k=0

177

T0 Rm ⊗ S k T0∗ Rm

for the standard identification. Let ϕ be a Λ-normal coordinate system on M with center x and B ∈ Jxr T M . We define ψΛ (B) =

r M

k=0

(T0 ϕ−1 ⊗ S k T0∗−1 ϕ−1 ) I J r T ϕ(B)

.

(20)

Since the identification I is G1m -equivariant, (20) is a correct definition. Proposition 12. Let D : C ∞ Qτ P 1 M → C ∞ QP r M be a natural operator. Then there exist uniquely determined natural operators Ak : C ∞ Qτ P 1 M → C ∞ (T M ⊗ S k T ∗ M ⊗ T ∗ M ) , k = 0, . . . , r, such that A0 = 0, A1 = 0 and

D(Λ) = Er (Λ) + 0, 0, A2 (Λ), . . . , Ar (Λ) in the sense of the identification (19).

Proof. The difference D(Λ) − Er (Λ) is decomposed into r + 1 natural operators by (19). The natural operators A0 and A1 vanish according to 25.3 and Lemma 33.4 in [18]. Now Proposition 3 yields directly Proposition 13. Let D : C ∞ Qτ P 1 M → C ∞ Qτ P r M be a natural operator. Then there exist uniquely determined natural operators Ak : C ∞ Qτ P 1 M → C ∞ T M ⊗ S k+1 T ∗ M , k = 0, . . . , r, such that A0 = 0, A1 = 0 and

D(Λ) = Er (Λ) + 0, 0, A2 (Λ), . . . , Ar (Λ) . The natural operators A2 , . . . , Ar in Proposition 12 or Proposition 13 can be prescribed arbitrarily. According to Lemma 33.4 of [18], all natural operators C ∞ Qτ P 1M → C ∞ T M ⊗ Nk ∗ T M are R-linearly generated by – the curvature tensor and its covariant derivatives, – constructing tensor products (including tensor products with invariant tensors) and contractions. In the case of some prescribed symmetries in the covariant part we add the corresponding symmetrizations of the operators in question.

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i i Example 1. Write Rjkl = −Rjlk for the curvature tensor of Λ. If we look for all natu∞ 1 ral operators D : C Qτ P M → C ∞ Qτ P 2 M , we have to determine all natural operators C ∞ Qτ P 1 M → C ∞ (T M ⊗ S 3 T ∗ M ). By the above mentioned procedure, all these operm Ri ators are the constant multiples of δ(j kl)m . Hence all D’s form the one-parameter family m i D(Λ) = E2 (Λ) + c(δ(j Rkl)m ) ,

c ∈ R.

Example 2. J. Janyˇska and the author determined all natural operators C ∞ Qτ P 1 M → C ∞ QP 2 M by using a direct approach, [7], [12]. The list of them is rather long. Using Proposition 12, we can interpret that list in a very clear geometric way.

9.

Semiholonomic 2-Jets

Some aspects of our problems are properly related with the theory of semiholonomic 2-jets. First we describe the general ideas, [3], [23]. Consider a fibered manifold p : Y → M . Its second nonholonomic prolongation J˜2 Y is defined by the iteration J˜2 Y = J 1 (J 1 Y → M ) . If xi , y p are some fiber coordinates on Y , the induced coordinates on J 1 Y are yrp = ∂i y p (x) and the coordinates further induced on J˜2 Y are p y0i = ∂i y p (x) and

p yij = ∂j yip (x) .

There are two canonical projections J˜2 Y → J 1 Y , namely the target jet projection β1 : J˜2 Y → J 1 Y and the jet prolongation J 1 β : J˜2 Y → J 1 Y of the target jet projection β : J 1 Y → Y . The second semiholonomic prolongation J¯2 Y is the set of all A ∈ J˜2 Y satisfying β1 (A) = (J 1 β)(A) . In coordinates, this condition means p y0i = yip .

(21)

The injection J 2 Y ֒→ J˜2 Y is defined by jx2 s 7→ jx1 (j 1 s) . So the subset J 2 Y ⊂ J˜2 Y is characterized by p yip = y0i

p p and yij = yji .

(22)

Hence J 2 Y ⊂ J¯2 Y . According to the general theory, both β1 : J˜2 Y → J 1 Y and J 1 β : J˜2 Y → J 1 Y are affine bundles. For two manifolds M and N , the space J˜2 (M, N ) or J¯2 (M, N ) of nonholonomic or semiholonomic 2-jets of M into N is the second nonholonomic or semiholonomic


179

prolongation of the product fibered manifold M × N → M , respectively. C. Ehresmann introduced the composition of nonholonomic jets, [3]. Consider another manifold Q and A ∈ J˜x2 (M, N )y , B ∈ J˜y2 (N, Q)z . So A = jx1 ϕ and B = jy1 ψ, where ϕ : M → J 1 (M, N ) and ψ : N → J 1 (N, Q) are sections of the source jet projection α. Hence αψ βϕ(u) = βϕ(u), u ∈ M , so that the composition of 1-jets ψ βϕ(u) and ϕ(u) is defined. Then we set B ◦ A = jx1 ψ βϕ(u) ◦ ϕ(u) ∈ J˜x2 (M, Q)z .

(23)

a , z a be the induced coordinates on J˜2 (N, Q) Let z a be some local coordinates on Q, zpa , z0p pq a a a and wi , w0i , wij be the induced coordinates on J˜2 (M, Q). Evaluating (23), we obtain the coordinate formula for the composition of nonholonomic 2-jets

wia = zpa yip ,

a a p w0i = z0p y0i ,

p a a p q wij = zpq yi y0j + zpa yij .

(24)

Clearly, the composition of two semiholonomic or holonomic 2-jets is semiholonomic or holonomic as well. A jet A ∈ J˜x2 (M, N )y is called regular, if there exists B ∈ J˜y2 (N, M )x such that B ◦ A = jx2 id M . By (24), A is regular iff both β1 (A) and (J 1 β)(A) are regular. In p coordinates this means that both yip and y0i are regular matrices. If dim M = dim N , then B ◦ A = jx2 id M implies A ◦ B = jy2 id N . In this case, regular is equivalent to invertible. Every ϕ(u) defines a linear map Tu M → T N , η p = yip (u)ξ i . This yields a local map T M → T N , whose tangent map at each point of Tx M is determined by jx1 ϕ. So the nonholonomic 2-jet A = jx1 ϕ(u) ∈ J˜x2 (M, N )y can be interpreted as a map (T T M )x → (T T N )y of the form η p = yip ξ i ,

p dy p = y0i dxi ,

p i dη p = yij ξ dxj + yip dξ i .

(25)

Consider the canonical involution ι of the iterated tangent functor, ιM (ξ i , dxi , dξ i ) = p (dxi , ξ i , dξ i ), and A ∈ J¯x2 (M, N )y in the form (25) with y0i = yip . Then the map ιN ◦A◦ιM is of the form η p = yip ξ i ,

dy p = yip dxi ,

p i dη p = yji ξ dxj + yip dξ i .

(26)

This map corresponds to another semiholonomic 2-jet κ(A) ∈ J¯x2 (M, N )y ,

p p κ(yip , yij ) = (yip , yji ).

(27)

Definition 2. The map κ is called the canonical involution of semiholonomic 2-jets. Since J¯2 (M, N ) → J 1 (M, N ) is an affine bundle, A and κ(A) determine a tensor ∆(A) = A − κ(A) ∈ Ty (N ) ⊗ Λ2 Tx∗ M

(28)

called the difference tensor of semiholonomic 2-jet A. (J. Pradines uses the name “dissymétrie”, [30].) Clearly, A is holonomic, iff ∆(A) = 0.

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Example 3. We present the first remarkable application of this concept. Consider a general connection Γ : Y → J 1 Y on an arbitrary fibered manifold Y → M , [18]. If Γ is viewed as a morphism over M , we can construct J 1 Γ : J 1 Y → J˜2 Y . Clearly, the values of the composition Γ′ = J 1 Γ ◦ Γ lie in J¯2 Y . The difference tensor ∆ ◦ Γ′ : Y → V Y ⊗ Λ 2 T ∗ M coincides with the curvature of Γ. The second order semiholonomic frame bundle P¯ 2 M of M is defined by P¯ 2 M = ¯ 2m ), where G ¯ 2m = reg J¯02 (Rm , M ). This is a principal bundle P¯ 2 M (M, G 2 m m inv J¯0 (R , R )0 is the second order semiholonomic jet group in dimension m. The in¯ 2m defines an injection (denoted by the same symbol) l1 : G1m → G ¯ 2m . One clusion G2m ⊂ G 1 1 1 2 ¯ verifies easily that (8) with r = 1 defines an identification iM : J P M ≈ P M . Consider a principal connection Γ : P 1 M → J 1 P 1 M with the coordinate expression xij,k = Γilk xlj .

(29)

Using (8), we verify directly the following result by P. Libermann, [23]. Proposition 14. The rule Γ 7→ i1M ◦ Γ defines a bijection between the connections on P 1 M ¯ 2m . and the reductions of P¯ 2 M to the subgroup l1 (G1m ) ⊂ G It is remarkable that the canonical involution κ yields a simple construction of the ˜ conjugate to Γ. Indeed, we have connection Γ ˜ = κ ◦ (i1M ◦ Γ) , i1M ◦ Γ

(30)

˜ i.e. κ transforms the reduction determined by Γ into the reduction determined by Γ. Example 4. Another interesting application of the concept of difference tensor is in the theory of G-structures. We recall that a (first order) G-structure on M is a reduction P of P 1 M to a subgroup G ⊂ G1m . Then J 1 P ⊂ J 1 P 1 M ≈ P¯ 2 M and ∆ suggests a very conceptual way to the construction of the structure function of P , [19], [23]. In particular, this approach clarifies, in an instructive way, the difference between the prolongability and the flatness of P , [19]. Finally we remark that the theory of the covariant differentation with respect to connections on P 2 M is systematically developed in [6].

10.

W r P as a Generalization of P r M

Consider a principal bundle π : P → M with structure group G, dim M = m. Its rr ϕ of local principal th order principal prolongation W r P is the bundle of all r-jets j(0,e) bundle isomorphisms ϕ : Rm × G → P ,

0 ∈ Rm ,

e = the unit of G.

(31)

r G := W r (Rm × G), whose This is a principal bundle over M with structure group Wm 0 action on W r P is given by the jet composition, [2], [18].


181

r P , we write Given A = j0r f ∈ Tm r πA = j0r (π ◦ f ) ∈ Tm M.

Further, we introduce r r reg π Tm P = {A ∈ Tm P;

r πA ∈ reg Tm M} .

Clearly, the local PB-isomorphism (31) is determined by its restriction ϕ | Rm × {e} : Rm → P . Hence r W r P = reg π Tm P. (32) ¯ , G) be another principal G-bundle, m = dim M ¯ . For every local principal bunLet P¯ (M r r r ¯ ¯ dle isomorphism f : P → P , Tm f : Tm P → Tm P restricts and corestricts into a map r P → reg T r P r r r¯ ¯ reg π Tm π m . This defines W f : W P → W P . r P , then πA ∈ T r M is invertible, so that A ◦ (πA)−1 satisfies π A ◦ If A∈ reg π Tm m −1 r (πA) = jx id M , x = π f (0) . This implies A ◦ (πA)−1 ∈ J r P . Hence W r P = P r M ×M J r P .

(33)

¯ is the base map of f . In Clearly, W r f is identified with P r f ×f J r f , where f : M → M particular, the structure group r r W0r (Rm × G) =: Wm G = Grm ⋊ Tm G

(34)

is the group semidirect product with the group composition

(g1 , C1 )(g2 , C2 ) = g1 ◦ g2 , (C1 ◦ g2 ) • C2 , r G, [18]. The first product projection where • denotes the induced group composition in Tm W r P → P r M is a principal bundle morphism with the associated group homomorphism r G → Gr determined by (34). Wm m W r P is a fundamental structure for the gauge theories of mathematical physics, [5]. In differential geometry, the main role of W r P is based on the fact that forevery associated bundle P [S], where S is a left G-space, the r-th jet prolongation J r P [S] is a fiber bundle associated to W r P , [18]. Further, we have a canonical injection P r M ֒→ W 1 P r−1 M , 1 P r−1 f , f : Rm → M . So W 1 P can play the role of a suitable recurrence j0r f 7→ j(0,e r−1 ) model for several geometric problems, [18]. In particular, the reductions of the principal bundle W 1 P are called generalized G-structures, [10]. Several properties of higher order G-structures are well reflected in the framework of this more general theory. If G = {e} is the one-element group, then M × {e} is identified with M and W r (M × {e}) = P r M . Hence many properties of W r P can be viewed as a generalization of the case r of P r M . In particular, we have the canonical one-form ϕr : T P r M → Rm ×gr−1 m on P M . On W r P , we introduce analogously a canonical one-form θr : T W r P → Rm × wr−1 m G= r−1 G, wr−1 G = Lie(W r−1 G). Consider T(0,Er−1 ) W r−1 (Rm × G), Er−1 = the unit of Wm m m r (u) ∈ W r−1 P , where π r u = j(0,e) ψ ∈ W r P and write u1 = πr−1 r−1 is the jet projection. The tangent map r−1 u ˜ = T(0,Er−1 ) W r−1 ψ : Rm × wr−1 P m G → Tu 1 W

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is a linear isomorphism depending on u only. For every Z ∈ Tu W r P , we define θr (Z) = u ˜−1 T πr−1 (Z) .

Clearly, the following diagram commutes T W rP

T P rM

θr

Rm × wr−1 m G

ϕr

Rm × gr−1 m

Analogously to Section 2, we introduced in [9] Definition 3. The torsion of a connection Φ on W r P is the covariant exterior differential DΦ θ r . The Lie algebroid LW r P of W r P coincides with the r-th jet prolongation J r (LP → M ), [17], [22]. Let ϕ : T M → J r LP be the algebroid form of Φ. Analogously to the case of J r T M , we have the truncated bracket [[ , ]]r−1 : J r LP ×M J r LP → J r−1 LP . The torsion τ ϕ of ϕ can be introduced as a morphism τ ϕ : T M ×M T M → J r−1 LP defined by (τ ϕ)(Z1 , Z2 ) = [[ϕ(Z1 ), ϕ(Z2 )]]r−1 ,

(Z1 , Z2 ) ∈ T M ×M T M .

In [17] it is deduced that DΦ θr and τ ϕ are naturally equivalent. The r-jets j0r gˆ, g ∈ G, of the constant maps gˆ : Rm → G, x 7→ g, define an injection r r G. Clearly, the direct group product l 1 r ν m : G → Tm r−1 (Gm ) × νm (G) is a subgroup of r Wm G. The following assertion, that is an analogy of Proposition 5, is proved in [17]. Proposition 15. The torsion-free connections on W r P are in bijection with the reductions r+1 (G) ⊂ W r+1 G. of W r+1 P to the subgroup lr (G1m ) × νm m

11.

Connections on W 1 P

We are going to discuss the connections on W 1 P in more details. By (33), W 1 P = P 1 M ×M J 1 P .

(35)

Write p1 : W 1 P → P 1 M and p2 : W 1 P → J 1 P for the product projections. Since p1 : W 1 P → P 1 M and the target jet projection β : W 1 P → P are principal bundle morphisms, every connection Φ : W 1 P → J 1 (W 1 P ) = J 1 P 1 M ×M J˜2 P

(36)


183

induces a pair of connections p1∗ Φ on P 1 M and β∗ Φ on P . Conversely, consider two connections Γ : P → J 1 P and Λ : P 1 M → J 1 P 1 M . Define W 1 P ⊃ R(Γ) =

u, Γ(v) ; (u, v) ∈ P 1 M ×M P .

(37)

Using the action ̺ of G on P from Section 3, one finds easily that R(Γ) is a reduction of 1 (G) ⊂ W 1 G. Since (37) identifies R(Γ) with P 1 M × P , W 1 P to the subgroup G1m × νm M m the product connection Λ × Γ on P 1 M ×M P is identified with a connection on R(Γ) and the latter connection is uniquely extended into a connection p(Γ, Λ) on W 1 P . Clearly, β∗ p(Γ, Λ) = Γ and p1∗ p(Γ, Λ) = Λ. Write L0 P for the adjoint bundle of P . (Our notation is motivated by the fact that L0 P is the subset of the Lie algebroid LP of all elements A satisfying q(A) = 0.) The projections β and p1 give rise to projections L0 W 1 P ⊗T ∗ M → L0 P ⊗T ∗ M and L0 W 1 P ⊗ T ∗ M → L0 P 1 M ⊗ T ∗ M . The common kernel of these projections is L0 P ⊗ [12].

2 N

T ∗M ,

Proposition 16. Connections on W 1 P are in bijection with triples (Γ, Λ, D), where Γ ∈

C ∞ (QP ), Λ ∈ C ∞ (QP 1 M ) and D ∈ C ∞ (L0 P ⊗

2 N

T ∗ M ).

Proof. For Φ ∈ C ∞ (QW 1 P ) we set Γ = β∗ Φ, Λ = p1∗ Φ and D = Φ−p(β∗ Φ, p1∗ Φ). 2 N

The factor T ∗ M ⊗ T ∗ M gives rise to an exchange map ex : L0 P ⊗

2 N

T ∗ M → L0 P ⊗

˜ and D by ex◦D, we T ∗ M . Thus, if we replace Λ by the classical conjugate connection Λ ˜ obtain a connection Φ said to be conjugate to Φ. In Section 13 we present a more geometric ˜ by using the canonical involution of semiholonomic 2-jets. construction of Φ There is another construction transforming the pair (Γ, Λ) into a connection on W 1 P . It is based on the general idea of flow prolongation of connections, [18]. Consider Γ in the lifting form Γ : P ×M T M → T P . For every vector field X on M , we first construct its Γ-lift ΓX : P → T P and then the flow prolongation W 1 (ΓX) : W 1 P → T W 1 P . This defines a map W 1 Γ : W 1 P ×M J 1 T M → T W 1 P . If we add Λ in its algebroid form T M → J 1 T M , we obtain the lifting map W 1 (Γ, Λ) : W 1 P ×M T M → T W 1 P of a principal connection on W 1 P . In [20] we deduced β∗ W 1 (Γ, Λ) = Γ, p1∗ W 1 (Γ, Λ) = 2

˜ So the difference p(Γ, Λ) − W 1 (Γ, Λ) ˜ is a section of L0 P ⊗ N T ∗ M . We recall that the Λ. curvature C(Γ) of Γ is a section of L0 P ⊗ Λ2 T ∗ M . In [20], we proved ˜ = C(Γ). Proposition 17. We have p(Γ, Λ) − W 1 (Γ, Λ)

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12.

Gauge-Natural Operators on Connections

Analogously to Section 8 one can pose the question of finding all geometric operators transforming a pair (Γ, Λ) of a connection Γ on P and a connection Λ on P 1 M into a connection on W 1 P . The precise meaning of “geometric” is “gauge-natural” in the sense of [2]. Roughly speaking, when passing from the classical natural operators to the gauge-natural ones, we meet the higher order principal prolongations W r P in the former role of the higher order frame bundles P r M , see [18] for a complete theory. In [20] we deduced the following list of all gauge-natural operators A : C ∞ (QP ) × C ∞ (Qτ P 1 M ) → C ∞ (QW 1 P ). (The assumption Λ is torsion-free is of technical character. If we replace Qτ P 1 M by QP 1 M , the list will be much longer with many further terms of less geometric interest.) First of all one proves that the underlying connections of A(Γ, Λ) are β∗ A(Γ, Λ) = Γ and p1∗ A(Γ, Λ) = Λ. So the difference A(Γ, Λ) − p(Γ, Λ) is 2 N

a section of L0 P ⊗ T ∗ M . We already know that the curvature C(Γ) is a section of that bundle. Let Z ⊂ Lin(g, g) be the subspace of all linear maps commuting with the adjoint action of G. Since every z ∈ Z is an equivariant map between the standard fibers, it induces a vector bundle morphism zP : L0 P → L0 P . Hence one can construct the modified curvature operator C(Γ)(z) = (zP ⊗ id ) ◦ C(Γ). On the other hand, by Example 28.7 of [18] all natural operators C ∞ (Qτ P 1 M ) → C ∞ (T ∗ M ⊗ T ∗ M ) are linearly generated by the contractions k ) and R (Λ) = (Rk ) of the curvature tensor (Ri ) of Λ. Let S ⊂ g be R1 (Λ) = (Rkij 2 ikj jkl the subspace of all vectors invariant with respect to the adjoint action. Since every B ∈ S is an invariant element of the standard fiber, it determines a section BP of L0 P . Our result from [20] reads Proposition 18. All gauge-natural operators C ∞ (QP ) × C ∞ (Qτ P 1 M ) → C ∞ (QW 1 P ) are of the form p(Γ, Λ) + C(Γ)(z) + B1P ⊗ R1 (Λ) + B2P ⊗ R2 (Λ) for all z ∈ Z and all B1 , B2 ∈ S. We underline that there exist many interesting open problems concerning the gaugenatural operators related with connections on W r P .

13.

¯ 2P Connections on W 1 P as Reductions of W

Finally we outline how the semiholonomic 2-jets can be used in the theory of connections on W 1 P . In general, the bundle of nonholonomic (n, 2)-velocities on a manifold M is defined by Tñ2 M = J˜02 (Rn , M ) . We shall frequently use a natural identification Tñ2 M ≈ Tn1 (Tn1 M ) .

(38)


185

Write tu : Rn → Rn for the translation x 7→ x + u. If ψ : Rn → J 1 (Rn , M ) is a section, then u 7→ ψ(u) ◦ j01 tu is a map Rn → Tn1 M . Passing to 1-jets defines (38). One verifies easily that (38) identifies reg Tñ2 M with reg Tn1 (reg Tn1 M ). The second order nonholonomic frame bundle of M is defined by P˜ 2 M = inv J˜02 (Rm , M ) . This is a principal bundle over M with structure group ˜ 2m = inv J˜02 (Rm , Rm )0 , G the right action of which on P˜ 2 M is determined by the composition of nonholonomic 2-jets. On the other hand, W 1 (P 1 M ) = P 1 M ×M J 1 P 1 M . Using (32) and (38), we obtain P˜ 2 M = W 1 (P 1 M ) .

(39)

1 G1 . ˜ 2m = G1m ⋊ Tm In particular, G m Further we introduce the second nonholonomic prolongation of P (M, G) by the iteration ˜ 2 P = W 1 (W 1 P ) . W (40) 2 G := W 1 (W 1 G). We have ˜m This is a principal bundle over M with structure group W m m

W 1 (W 1 P ) = P 1 M ×M J 1 P 1 M ×M J˜2 P = P˜ 2 M ×M J˜2 P and 1 1 1 1 1 1 2 ˜ 2m ⋊ T˜m Wm (Wm G) = G1m × Tm G m × Tm Tm G ≈ G G,

where the group semidirect product has an analogous meaning to (34). In the semiholonomic case, we have ˜ 2P ⊃ W ¯ 2 P := P¯ 2 M ×M J¯2 P . W (41)

2G = G 2 G. ¯ 2 P is W ¯m ¯ 2m ⋊ T¯m The structure group of W 1 1 1 Consider a connection Φ : W P → J (W P ) = J 1 P 1 M ×M J˜2 P . Using the identification J 1 P 1 M ≈ P¯ 2 M from Section 9 and the inclusion J¯2 P ⊂ J˜2 P , we obtain an ¯ 2 P ⊂ J 1 W 1 P . Let Γ and Λ be the underlying connections of Φ. In Secinclusion W 1 (G). One tion 11 we constructed the reduction R(Γ) ⊂ W 1 P to the subgroup G1m × νm 2 (G) ⊂ ¯ 2 P is a reduction to the subgroup l1 (G1m ) × νm verifies easily that Φ R(Γ) ⊂ W 2 2 2 2 ¯ ¯ Gm ⋊ Tm G ⊂ Gm ⋊ Tm G. The following assertion generalizes the result by P. Libermann mentioned in Section 9.

¯ 2 P to Proposition 19. The connections on W 1 P are in bijection with the reductions of W 2 2 2 1 ¯ m ⋊ T¯m G. the subgroup l1 (Gm ) × νm (G) ⊂ G

2 (G). Hence its projection Q into ¯ 2 P to l1 (G1m ) × νm Proof. Let Q be a reduction of W 1 1 1 1 W P is a reduction to the subgroup Gm × νm (G). Then Q can be interpreted as a map (denoted by the same symbol) Q : Q1 → J 1 W 1 P . This map is equivariant, so that Q can be uniquely extended into a connection on W 1 P .

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˜ conjugate to Φ by using the canonical inNow we can construct the connection Φ volution κ of semiholonomic 2-jets. Using Proposition 3 of [20], one proves that if ¯ 2 P is the reduction corresponding to Φ, then red(Φ) ⊂ W ˜ κ ◦ red(Φ) = red(Φ)

(42)

˜ is the reduction corresponding to Φ.

Acknowledgement The author was supported by the Ministry of Education of the Czech Republic under the project MSM 0021622409 and the grant GACR No. 201/05/0523.

References [1] A. Cabras and I. Koláˇr, Flow prolongations of some tangent valued forms, to appear in Czechoslovak Math. J. [2] D. Eck, Gauge-natural bundles and generalized gauge theories, Mem. Amer. Math. Soc. 247 (1981). [3] C. Ehresmann, Extension du calcul des jets aux jets non holonomes, CRAS Paris 239 (1954). [4] M. El´zanowski and S. Prishepionok, Connections on higher order frame bundles, In: New Developments in Differential Geometry (Proceedings, Kluwer, 1996) 131–142. [5] L. Fatibene and M. Francaviglia, Natural and Gauge Natural Formalism for Classical Fields Theories (Kluwer, 2003). [6] M. Ferraris, M. Francaviglia and F. Tubiello, Connections Over the Bundles of Second-Order Frames, In: Proceedings of Conference DGA (Brno 1989, WorldScientific, Singapore, 1990) 33–46. [7] J. Janyˇska, On natural operations with linear connections, Czechoslovak Math. J. 35 (1985) 106–115. [8] S. Kobayashi, Canonical forms on frame bundles of higher order contact, Proc. of Symposia in Pure Math. 3 (1961) 186–193. ˇ pˇest. mat. 100 (1975) 284–290. [9] I. Koláˇr, A generalization of the torsion form, Cas. [10] I. Koláˇr, Generalized G-structures and G-structures of higher order, Boll. Un. Math. Ital., Suppl. fasc. 3 (1975) 249–256. [11] I. Koláˇr, A geometric version of the higher order Hamilton formalism in fibered manifolds, J. of Geometry and Physics 1 (1984) 127–137. [12] I. Koláˇr, Some natural operators in differential geometry, In: Proc. Conf. Diff. Geom. and Its Applications (Brno, 1986, Dordrecht, 1987) 91–110.


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[13] I. Koláˇr, Torsion free connections on higher order frame bundles, In: New Developments in Differential Geometry (Proceedings, Kluwer, 1996) 233–241. [14] I. Koláˇr, On the torsion of linear higher order connections, Central European Journal of Mathematics 3 (2003) 360–366. [15] I. Koláˇr, On the torsion-free connections on higher order frame bundles, Publ. Math. Debrecen 67 (2005) 373–379. [16] I. Koláˇr, Weil Bundles as Generalized Jet Spaces, In: Handbook of Global Analysis (Elsevier, 2008) 625–664. [17] I. Koláˇr, Connections on principal prolongations of principal bundles, In: Proc. 10th Int. Conf. on Diff. Geom. and Appl. (Olomouc 2007, (O. Kowalski, D. Krupka, O. Krupková and J. Slovák, Eds.) World Scientific, Singapore, 2008) 279–291. [18] I. Koláˇr, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry (Springer Verlag, 1993). [19] I. Koláˇr and I. Vadoviˇcová, On the structure function of a G-structure, Math. Slovaca 35 (1985) 277–282. [20] I. Koláˇr and G. Virsik, Connections in first principal prolongations, In: Suppl. ai Rendiconti del Circolo Matematico di Palermo (Serie II, 43, 1996) 163–171. [21] D. Krupka and J. Janyˇska, Lectures on Differential Invariants (Univerzita JEP, Brno, 1990). [22] A. Kumpera and D. Spencer, Lie Equations I (Princeton University Press, 1972). [23] P. Libermann, Introduction to the theory of semi-holonomic jets, Arch. Math. (Brno) 33 (1997) 173–189. [24] P. Libermann, Sur les prolongements des fibrés principaux et des groupoides différentiables banachiques, In: Analyse globale (Sém. Math. Supérieures, No. 42, Presses Univ. Montreal, 1971) 7–108. [25] P. Libermann, Charles Ehresmann Concepts in Differential Geometry, In: Geometry and Topology of Manifolds (Banach Center Publications, Vol. 76, Institute of Mathematics PAN, Warszawa 2007) 35–50. [26] K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids (Cambridge University Press, Cambridge 2005). [27] W. M. Mikulski, Higher order linear connections from first order ones, to appear in Arch. Math. (Brno). [28] M. Paluszny and A. Zajtz, Foundations of Differential Geometry on Natural Bundles (Lecture Notes Univ. Caracas, 1984).

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[29] F. W. Pohl, Connections in differential geometry of higher order, Trans. Amer. Math. Soc. 125 (1966) 169–211. [30] J. Pradines, Fibrés vectoriels double symmétriques et jets holonomes d’ordre 2, CRAS Paris, série A 278 (1974) 1557–1560. [31] N. Que, Du prolongements des espaces fibrés et des structures infinitésimales, Ann. Inst. Fourier, Grenoble 17 (1967) 157–223. [32] D. J. Saunders, The Geometry of Jet Bundles (London Math. Society Lecture Notes Series 142, Cambridge, 1989). [33] P. C. Yuen, Higher order frames and linear connections, Cahiers Topol. Geom. Diff. 12 (1971) 333–371.


Chapter 11

N ATURAL L IFTS IN R IEMANNIAN G EOMETRY Oldˇrich Kowalski1∗and Masami Sekizawa2† 1 Faculty of Mathematics and Physics Charles University in Prague, Sokolovská 83, 186 75 Praha 8 Czech Republic 2 Department of Mathematics, Tokyo Gakugei University Koganei-shi Nukuikita-machi 4-1-1, Tokyo 184-8501, Japan

Abstract Using the concepts and the methods developed by D. Krupka, J. Janyˇska and V. Mikolásˇová, we fully classified (during 1986-88) naturally lifted metrics to tangent bundles, linear frame bundles and cotangent bundles. All classical constructions of metrics on such bundles are special examples of our lifted metrics. We shall survey our own earlier work and also the later development in the geometrical study of natural metrics.

2000 Mathematics Subject Classification. 53A55, 53C20, 53C25 Key words and phrases. Riemannian metric, natural lift, tangent bundle, linear frame bundle, cotangent bundle, sectional curvature, scalar curvature.

Introduction There are well-known classical examples of “lifted metrics” on the tangent bundle T M and on the linear frame bundle LM over a Riemannian manifold (M, g), and also of a lifted metric on the the cotangent bundle T ∗ M over an affine manifold (M, ∇). Namely, they are the Sasaki metric, the horizontal lift and the vertical lift on T M ; then the diagonal lift, the horizontal lift and the vertical lift on LM ; and finally the Riemann extension on T ∗ M . All these constructions have been studied extensively (see e.g. [12, 38, 30, 11, 14, 37, 43, 44, 47].) As we shall see, the classical constructions are examples of “natural transformations ∗ †

E-mail address: [email protected] E-mail address: [email protected]

190

Oldˇrich Kowalski and Masami Sekizawa

of the second order”. We shall survey our work [22, 21, 39] on the full classifications of (possibly degenerate) naturally lifted metrics on T M , LM and T ∗ M . We have proved that the complete family of naturally lifted metrics on T M and LM (for a fixed base metric) is a module over real functions generated by some generalizations of known classical lifts. In the case of T ∗ M , just two arbitrary parameters appear. Our idea of naturality is closely related to that of A. Nijenhuis [33], D. B. A. Epstein [13], P. Stredder [42] and others. Yet, we shall use for our purposes the concepts and methods developed by D. Krupka [26, 27] and D. Krupka and V. Mikolásˇová [28]. See also I. Koláˇr, P. W. Michor and J. Slovák [18, pp. 227–280], and D. Krupka and J. Janyˇska [29, pp. 160–166] for other presentations of our study and for the concept of naturality in general. “Rigidity” and “heredity” of general natural metrics on tangent bundles and linear frame bundles have been extensively studied. This study is usually a hard work. We shall survey our own work from the 1980s, and the later development in the geometrical study of natural metrics.

1.

Natural Transformations

Let us recall the general theory of natural transformations due to D. Krupka. We refer to [26, 27, 28, 29] for more details. Let r be any non-negative integer. Then the r-th order differential group Lrn of the n-dimensional Euclidean space Rn , n ≥ 2, is the Lie group of all r-jets of local diffeomorphisms of Rn with source and target at the origin o ∈ Rn . Let P and Q be smooth manifolds on which the group Lrn acts to the left. Then an r-th order differential invariant f : P −→ Q is an Lrn -equivariant map of the left Lrn -space P to the left Lrn -space Q. Further, let F r M denote the bundle of all frames of r-th order over M , which carries a natural structure of a principal Lrn -bundle F r (M, Lrn , πnr ). We get a natural functor from the category Dn of smooth n-manifolds and injective immersions into the category of principal Lrn -bundles and Lrn -bundle morphisms. For a left Lrn -space P , let FPr M denote the fiber bundle with fiber P , associated to the principal Lrn -bundle F r M . We obtain a natural functor FPr from the category Dn into the category of fiber bundles and their morphisms. For each manifold M and each differential invariant f : P −→ Q, we can define a morphism fM : FPr M −→ FQr M over the identity map id : M −→ M by fM ([y, p]) = [y, f (p)] for all [y, p] ∈ FPr M . This morphism fM is called the realization of a differential invariant f on the manifold M . An r-th order natural transformation T of the functor Fpr into the functor FQr is a collection of bundle morphisms TM : FPr M −→ FQr M over the identity map id : M −→ M , M ∈ Dn , such that FQr ϕ◦TM1 = TM2 ◦FPr ϕ holds for every morphism ϕ : M1 −→ M2 of Dn . The following Theorem by D. Krupka in [26] says that any concrete classification of all r-th order natural transformations of FPr to FQr can be reduced to a classification of all r-th order differential invariants f from P to Q. Theorem 1.1 ([26]). Let f : P −→ Q be an r-th order differential invariant. Then the correspondence Tf : M −→ fM , where M is an object of Dn , is a natural transformation

Natural Lifts in Riemannian Geometry

191

of the functor FPr to the functor FQr . Moreover, the correspondence f −→ Tf is a bijection between the set of all r-th order differential invariants from P to Q and the set of all r-th order natural transformations of FPr to FQr .

2.

Tangent Bundle

After recalling classical examples of lifts of a given metric on the base manifold to its tangent bundle, we shall present our full classification of naturally lifted metrics and related results. Then, we shall survey some geometric properties of such lifted metrics.

2.1.

Classical Examples of Metrics on the Tangent Bundle

The tangent bundle T M over an n-dimensional smooth manifold M , n ≥ 2, consists of all pairs (x, u), where x is a point of M and u is a vector from the tangent space Mx of M at x. We denote by p the natural projection of T M to M defined by p(x, u) = x. Let g be a Riemannian metric on the manifold M and ∇ its Levi-Civita connection. Then the tangent space (T M )(x,u) of T M at (x, u) ∈ T M splits into the horizontal and vertical subspace H(x,u) and V(x,u) with respect to ∇: (T M )(x,u) = H(x,u) ⊕ V(x,u) . If a point (x, u) ∈ T M and a vector X ∈ Mx are given, then there exists a unique vector X h ∈ H(x,u) such that p∗ (X h ) = X. We call X h the horizontal lift of X to T M at (x, u). The vertical lift of X to (x, u) is a unique vector X v ∈ V(x,u) such that X v (df ) = Xf for all smooth functions f on M . Here we consider a one-form df on M as a function on T M defined by (df )(x, u) = uf for all (x, u) ∈ T M . The map X 7−→ X h is an isomorphism between Mx and H(x,u) , and the map X 7−→ X v is an isomorphism between Mx and V(x,u) . In an obvious way we can define horizontal and vertical lifts of vector fields on M . These are uniquely defined vector fields on T M . The three classical constructions of metrics on tangent bundles T M which are derived from a Riemannian metric g on M are given as follows: (a) The metric g s constructed by Sasaki [38] is a (positive definite) Riemannian metric on T M given by s (X h , Y h ) = gx (X, Y ), g(x,u)

s (X h , Y v ) = 0, g(x,u)

s g(x,u) (X v , Y h ) = 0,

s (X v , Y v ) = gx (X, Y ) g(x,u)

for all X, Y ∈ Mx . (b) The horizontal lift g h of g is a pseudo-Riemannian metric on T M with signature (n, n) which is given by h g(x,u) (X h , Y h ) = 0,

h g(x,u) (X h , Y v ) = gx (X, Y ),

h g(x,u) (X v , Y h ) = gx (X, Y ),

h g(x,u) (X v , Y v ) = 0

for all X, Y ∈ Mx .

192

Oldˇrich Kowalski and Masami Sekizawa (c) The vertical lift g v of g is a degenerate metric of rank n on T M given by v g(x,u) (X h , Y h ) = gx (X, Y ),

v g(x,u) (X h , Y v ) = 0,

v g(x,u) (X v , Y h ) = 0,

v g(x,u) (X v , Y v ) = 0

for all X, Y ∈ Mx .

2.2.

Naturally Lifted Metrics on the Tangent Bundle

Let us consider the symmetric tensor product E = Rn∗ ⊙ Rn∗ , i.e., the vector space of symmetric bilinear forms on Rn and let E+ ⊂ E denote the open subset of all positive inner products. Put P = Rn × Tn1 E+ and denote by (uk , gij , gij,k ), 1 ≤ i ≤ j ≤ n; k = 1, 2, . . . , n, the system of canonical coordinates of P . Also, put Q = Rn ⊕ (R2n∗ ⊙ R2n∗ ) and denote by (v k , GAB ), k = 1, 2, . . . , n; A, B = 1, 2, . . . , 2n, the system of canonical coordinates on Q. This can be also written in the form (v k , Gij , Gik∗ , Gi∗j∗ ), 1 ≤ i ≤ j ≤ n; k = 1, 2, . . . , n, where i∗ stands for n + i. We define the actions of L2n on P and Q, respectively, having in mind the transformation rules on Riemannian objects under changes of systems of natural local coordinates. We see easily that the corresponding associated L2n -bundles FP2 M and FQ2 M over a manifold M always have canonical bundle projections FP2 M −→ T M,

FQ2 M −→ T M.

(2.1)

We define the problem to find all second order natural transformations of a Riemannian metric on a manifold to a metric on its tangent bundle as the problem to find all those natural transformations of FP2 M to FQ2 M which, for each M and via the projections (2.1), induce the identity map id : T M −→ T M . Hence, by Theorem 1.1, this reduces to the problem to find all second order differential invariants f : (uk , gij , gij,k ) −→ (v k , GAB (uk , gij , gij,k )) from P to Q such that v k ◦ f = uk , k = 1, 2, . . . , n. We use the computational method proposed by Krupka and Mikolásˇová in [28]. We obtain Theorem 2.1 ([22]). Let αij , βij and γij be functions on P which are solutions of the following system of differential equations: ∂ζij = 0, ∂gpq,r 2

n X

a=1

gap

(2.2)

∂ζij ∂ζij − uq p = ζip δjq + ζpj δiq ∂gaq ∂u

(2.3)

for i, j, p, q, r = 1, 2, . . . , n. Then a necessary condition for a map f : P −→ Q such that v k ◦ f = uk , k = 1, 2, . . . , n, to be a differential invariant is that its representation by the canonical coordinates is of the form Gij =

n X

b,c,s,t=1

ub uc Γsbi Γtcj αst +

n X

b,s=1

ub Γsbi βsj +

n X

b,s=1

ub Γsbj βsi + γij ,

(2.4)

Natural Lifts in Riemannian Geometry Gik∗ =

n X

ub Γsbi αsk + βki ,

193 (2.5)

b,s=1

Gi∗j∗ = αij

(2.6)

for 1 ≤ i ≤ j ≤ n and k = 1, 2, . . . , n, where Γkij ’s are the formal Christoffel symbols derived from gij and gij,k . We have given in [22] a geometric meaning to the system of differential equations (2.3) and its solution. We have shown first which type of differential invariants of the form ζij = ζij (uk , gkl ) (and which type of natural transformations) belongs to the system of differential equations. Consider the left L1n -spaces V = Rn ×E+ and W = Rn∗ ⊗Rn∗ with the natural actions of L1n defined, again, having in mind transformation rules under changes of systems of natural local coordinates. Let (uk , gij ), 1 ≤ i ≤ j ≤ n; k = 1, 2, . . . , n, and (ζij ), i, j = 1, 2, . . . , n, be the systems of canonical coordinates of V and W , respectively. We can check easily that the system of differential equations (2.3) gives a necessary condition for a map ζij = ζij (uk , gkl ) to be (first order) differential invariant from V to W . Long but routine calculations show that the formulas (2.4)–(2.6) provide a differential invariant f from P to Q if and only if the functions αij , βij and γij defined on V describe some differential invariants α, β and γ from V to W . Thus our problem reduces to the 1 . study of the first order natural transformations of FV1 into FW We say that a bundle morphism of the form ζ : T M ⊕ T M ⊕ T M −→ M × R is an F-metric on M if it is linear in the second and the third argument (and smooth in the first argument). We also say that ζ is symmetric or skew-symmetric if it is symmetric or skewsymmetric with respect to the second and third argument, respectively. (We use here the letter “F” to recall the Finsler geometry.) Any Riemannian metric g on M is a symmetric F-metric which is independent on u. In our special case, letting g be a given Riemannian metric on M , we speak about natural F-metrics derived from g which are F-metrics ζ, for a fixed u ∈ T M , whose components ζ(u)ij = ζ(u, ∂/∂xi , ∂/∂xj ) with respect to a system of local coordinates (x1 , x2 , . . . , xn ) in M are solutions of the system of differential equations (2.3). We obtain that Theorem 2.2 ([22]). Let (M, g) be an n-dimensional oriented Riemannian manifold. Then all natural F-metrics ζ on M derived from g are given as follows: (1) For n = 2, all symmetric natural F-metrics are of the form ζ(u; X, Y ) = α(kuk2 )g(X, Y ) + β(kuk2 )g(X, u)g(Y, u) + γ(kuk2 ){g(X, u)g(Y, Ju) + g(X, Ju)g(Y, u)}, and all skew-symmetric natural F-metrics are of the form ζ(u; X, Y ) = δ(kuk2 ){g(X, u)g(Y, Ju) − g(X, Ju)g(Y, u)}, where α, β, γ and δ are arbitrary functions of kuk2 = g(u, u) and J is one of the two canonical almost complex structures of (M, g) (for which (M, g, J) is a Kähler manifold).

194

Oldˇrich Kowalski and Masami Sekizawa (2) For n = 3, all symmetric natural F-metrics are of the form ζ(u; X, Y ) = α(kuk2 )g(X, Y ) + β(kuk2 )g(X, u)g(Y, u)

(2.7)

and all skew-symmetric natural F-metrics are of the form ζ(u; X, Y ) = ϕ(kuk2 )g(X × Y, u),

where α, β and ϕ are arbitrary functions of kuk2 = g(u, u), and X × Y is the usual vector product of X and Y . (3) For n > 3, all natural F-metrics are symmetric and of the form (2.7). I. Koláˇr, P. W. Michor and J. Slovák have given in [18, Proposition 33.22] a new and elegant proof of the classification of natural F-metrics on non-oriented Riemannian manifolds. Theorem 2.3 ([18]). Let (M, g) be an n-dimensional non-oriented Riemannian manifold, n ≥ 1. Then all natural F-metrics ζ on M derived from g are symmetric and given by (2.7), where α and β are arbitrary smooth functions defined on the interval (0, ∞). In particular, β = 0 if n = 1. M. T. K. Abbassi has proved in [1] explicitly that all basic functions from Theorems 2.2 and 2.3 can be prolonged to smooth functions on the set of all non-negative real numbers. This result found many applications in the techniques used for the thorough investigation of g-natural metrics by M. T. K. Abbassi and M. Sarih. For a given Riemannian metric g on M , we define the classical lifts of F-metrics from M to T M , with respect to g, as follows: (a) The Sasaki lift ζ s,g of a symmetric F-metric ζ with respect to g is defined by s,g ζ(x,u) (X h , Y h ) = ζx (u; X, Y ),

s,g ζ(x,u) (X h , Y v ) = 0,

s,g ζ(x,u) (X v , Y h ) = 0,

s,g ζ(x,u) (X v , Y v ) = ζx (u; X, Y )

for all X, Y ∈ Mx . (b) The horizontal lift ζ h,g of an arbitrary F-metric ζ with respect to g is defined by h,g ζ(x,u) (X h , Y h ) = 0,

h,g ζ(x,u) (X h , Y v ) = ζx (u; Y, X),

h,g ζ(x,u) (X v , Y h ) = ζx (u; X, Y ),

h,g ζ(x,u) (X v , Y v ) = 0

for all X, Y ∈ Mx . (c) The vertical lift ζ v of a symmetric F-metric ζ with respect to g is defined by v (X h , Y h ) = ζx (u; X, Y ), ζ(x,u)

v ζ(x,u) (X h , Y v ) = 0,

v ζ(x,u) (X v , Y h ) = 0,

v ζ(x,u) (X v , Y v ) = 0

for all X, Y ∈ Mx . Obviously, the vertical lift does not depend on the choice of g. We note that ζ s,g , ζ h,g and ζ v are (not necessarily regular) pseudo-Riemannian metrics on T M . If we take ζ = g, then ζ s,g , ζ h,g and ζ v are just the classical lifts g s , g h and g v , respectively. Thus we have all metrics on T M which come from a second order natural transformation of a given Riemannian metric on M .


195

Theorem 2.4 ([22]). Let g be a Riemannian metric on an n-dimensional smooth manifold M , n ≥ 2, and let G be a (possibly degenerate) pseudo-Riemannian metric on the tangent bundle T M which comes from a second order natural transformation of g. Then there are natural F-metrics ζ1 , ζ2 and ζ3 derived from g, where ζ1 and ζ3 are symmetric, such that G = ζ1 s,g + ζ2 h,g + ζ3 v . Moreover, all natural F-metrics derived from g are given by Theorem 2.2. If n = 2, then the family of all natural metrics G on T M depends on 10 arbitrary functions of one variable, for n = 3 it depends on seven arbitrary functions of one variable, and for n > 3 on six arbitrary functions of one variable. Our concept of F-metric coincides with that of “M-tensor of type (0, 2)” introduced by Y. C. Wong and K. P. Mok in [46]. Further, our Theorem 2.2 is strongly related to Theorem 5.1 of the paper by K. P. Mok, E. M. Patterson and Y. C. Wong in [31]. Nevertheless, the authors above do not mention the naturality problem in their investigations, and they also do not give any nontrivial examples of M-tensor. In fact, Theorem 5.1 mentioned above only says that each metric on the tangent bundle T M is a sum of three classical lifts of three independent M-tensors of type (0, 2) with respect to a generalized connection (called M-connection). The metrics given by Theorem 2.4 are called g-natural metrics by M. T. K. Abbassi in [1]. He has formulated these metrics in the form: Theorem 2.5 ([1, 4]). Let (M, g) be an n-dimensional Riemannian manifold and G be a g-natural metric on the tangent bundle T M . Then there are real valued functions αi and βi , i = 1, 2, 3, defined on [0, ∞) such that G(x,u) (X h , Y h ) = (α1 + α3 )(r2 )gx (X, Y )

+ (β1 + β3 )(r2 )gx (X, u)gx (Y, u), G(x,u) (X h , Y v ) = α2 (r2 )gx (X, Y ) + β2 (r2 )gx (X, u)gx (Y, u), G(x,u) (X v , Y h ) = α2 (r2 )gx (X, Y ) + β2 (r2 )gx (X, u)gx (Y, u), G(x,u) (X v , Y v ) = α1 (r2 )gx (X, Y ) + β1 (r2 )gx (X, u)gx (Y, u) hold at each point (x, u) ∈ T M for all u, X, Y ∈ Mx , where r2 = gx (u, u). For n = 1, the same holds with βi = 0, i = 1, 2, 3.

2.3.

Riemannian Geometry of the Tangent Bundle

Let φ be a (local) transformation of a manifold M . Then we define a transformation Φ of T M by Φ(x, u) = (φx, φ∗x u) for all (x, u) ∈ T M . If φ is a (local) affine transformation with respect to the Levi-Civita connection ∇ of (M, g), then we have Φ∗ (X h ) = (φ∗ X)h ,

Φ∗ (X v ) = (φ∗ X)v

for all X ∈ X(M ). Using this fact we can easily see that all g-natural metrics are invariant:

196


Theorem 2.6 ([24]). Let φ be a (local) isometry of a Riemannian manifold (M, g). Then every g-natural metric G on the tangent bundle T M over (M, g) is invariant by the lift Φ of φ. In other words, Φ is a local isometry of (T M, G) whose projection on (M, g) is φ. Riemannian geometry of the tangent bundle T M with the metric g s defined by S. Sasaki in [38] has been studied by many authors (see for example, [12, 19, 32, 47]). The first author of the present paper has proved in [19] that (T M, g s ) is never locally symmetric unless the base metric g on M is locally Euclidean. Further, E. Musso and F. Tricerri have proved in [32] that g s is “extremely rigid” in the following sense: Theorem 2.7 ([32]). The tangent bundle (T M, g s ) with the Sasaki metric has constant scalar curvature if and only if the base manifold (M, g) is locally Euclidean. It seems that another metric nicely fitted to the tangent bundle is the so-called CheegerGromoll metric. Its construction has been suggested by J. Cheeger and D. Gromoll in [8] and expressed more explicitly by E. Musso and F. Tricerri in [32]. It is a “nonclassical” natural metric on the tangent bundle T M over a Riemannian manifold (M, g) defined at each point (x, u) ∈ T M by cg g(x,u) =

1 kuk2 gxv − kxv + gxs,g + kxs,g , 2 1 + kuk

where k is the natural F-metric given by k(u; X, Y ) = g(X, u)g(Y, u). In the other form, g cg is given by cg g(x,u) (X h , Y h ) = gx (X, Y ), cg g(x,u) (X v , Y v ) =

cg cg g(x,u) (X h , Y v ) = g(x,u) (X v , Y h ) = 0,

1 + g (X, u)g (Y, u) g (X, Y ) x x x 1 + kuk2

for all X, Y ∈ Mx . The curvatures of g cg have been studied in detail by the second author of the present paper in [41]. In particular, it has been proved that the scalar curvature of g cg is never constant if g has constant sectional curvature. (Unfortunately there are computational errors in the paper [41], which have been pointed and corrected precisely by S. Gudmandsson and E. Kappose in [16]). It should be interesting to find “non-rigid” metrics on T M over a Riemannian manifold (M, g). The first example of such metrics has been given by V. Oproiu in [36]. The family of Oproiu metrics depends on two arbitrary functions of one variable, and belong to a family of metrics on T M which come from a second order natural transformation of g given by Theorem 2.4. In terms of Theorem 2.5, The Oproiu metrics are g-natural metrics on T M such that (α1 + α3 )(t) = v(t/2), 1 , v(t/2) α2 = β2 = 0, α1 (t) =

(β1 + β3 )(t) = w(t/2), β1 (t) =

w(t/2) , v(t/2) v(t/2) + tw(t/2)

where v and w are real valued smooth functions defined on (0, ∞) such that v > 0 and v(t) + 2tw(t) > 0 for all t ∈ (0, ∞). (see [5]). The main result of [36] is the following


197

Theorem 2.8. If (M, g) is an n-dimensional space of negative constant sectional curvature, n > 2, then T M equipped with any Oproiu metric is a Kähler Einstein manifold with positive constant scalar curvature. M. T. K. Abbassi and M. Sarih have proved in [6] the following (highly nontrivial) “heredity” theorem: Theorem 2.9. Let G be any Riemannian g-natural metric on T M , where (M, g) is an arbitrary Riemannian manifold. Then, if (T M, G) is flat, or locally symmetric, or of constant sectional curvature, or of constant scalar curvature, or an Einstein manifold, respectively, then (M, g) possesses the same property, respectively. As an application of study on the Oproiu metric, they have proved that Theorem 2.10 ([6]). Let (M, g) be an n-dimensional space of negative constant sectional curvature, n ≥ 3. Then there is a one-parameter family F of g-natural metrics on T M with nonconstant defining functions αi and βi , i = 1, 2, 3, such that, for every G ∈ F, (T M, G) is a space of positive constant scalar curvature. Moreover, for each (M, g) as above, and each prescribed positive constant S, there is a metric G ∈ F with the constant scalar curvature S. Moreover they have obtained examples of Riemannian g-natural metrics with constant defining functions αi and βi , i = 1, 2, 3, which can have an arbitrary constant scalar curvature (not necessarily positive as in the above Theorem). Theorem 2.11 ([6]). Let (M, g) be a space of constant sectional curvature K 6= 0, and let G = a g s + b g h + c g v a metric on the √ tangent bundle T M , where a, b and c are constant such that a > 0 and b2 = (−1 + 13 )a(a + c)/6. Then (T M, G) is a space of nonzero constant scalar curvature with the same sign as K. Furthermore, we can choose the constants a, b and c so that the scalar curvature of G has any prescribed nonzero constant value with the same sign as K.

3.

Linear Frame Bundle

After recalling classical examples of lifts of a given metric on the base manifold to its linear frame bundle, we shall present our full classification of naturally lifted metrics. Then, we shall remind some rigidity results for special cases of these metrics.

3.1.

Classical Examples of Metrics on the Linear Frame Bundle

The linear frame bundle LM over an n-dimensional smooth manifold M , n ≥ 2, consists of all pairs (x, u), where x is a point of M and u is a basis for the tangent space Mx of M at x. We denote by p the natural projection of LM to M defined by p(x, u) = x. Let g be a Riemannian metric on the manifold M and ∇ its Levi-Civita connection. Then the tangent space (LM )(x,u) of LM at (x, u) ∈ LM splits into the horizontal and vertical subspace H(x,u) and V(x,u) with respect to ∇: (LM )(x,u) = H(x,u) ⊕ V(x,u) .

198


If a point (x, u) ∈ LM and a vector X ∈ Mx are given, then there exists a unique vector X h ∈ H(x,u) such that p∗ (X h ) = X. We call X h the horizontal lift of X to LM at (x, u). We define naturally n different vertical lifts of X ∈ Mx . If ω is a one-form on M , then ιµ ω, µ = 1, 2, . . . , n, are functions on LM defined by (ιµ ω)(x, u) = ω(uµ ) for all (x, u) = (x, u1 , u2 , . . . , un ) ∈ LM . The vertical lifts X v,λ , λ = 1, 2, . . . , n, of X ∈ Mx to LM at (x, u) are the n vectors such that X v,λ (ιµ ω) = ω(X)δµλ , λ, µ = 1, 2, . . . , n, hold for all one-forms ω on M , where δµλ denotes the Kronecker’s delta. The n vertical lifts are always uniquely determined, and they are linearly independent if X 6= 0. In an obvious way we can define horizontal and vertical lifts of vector fields on M . These are uniquely defined vector fields on T M . The three classical constructions of metrics on linear frame bundles LM which are derived from a Riemannian metric g on M are given as follows: (a) The diagonal lift g d of g defined by K. P. Mok in [30] is a (positive definite) Riemannian metric on LM given by d g(x,u) (X h , Y h ) = gx (X, Y ),

d g(x,u) (X h , Y v,µ ) = 0,

d g(x,u) (X v,λ , Y h ) = 0,

d g(x,u) (X v,λ , Y v,µ ) = gx (X, Y )δ λµ

for all X, Y ∈ Mx and λ, µ = 1, 2, . . . , n. This metric is called also the Sasaki-Mok metric because it resembles the Sasaki metric of the tangent bundle over a Riemannian manifold. (b) The horizontal lift g h of g is a degenerate metric on LM of rank 2n and signature (n, n) which is given by h (X h , Y h ) = 0, g(x,u)

h (X h , Y v,µ ) = gx (X, Y ), g(x,u)

h (X v,λ , Y h ) = gx (X, Y ), g(x,u)

h (X v,λ , Y v,µ ) = 0 g(x,u)

for all X, Y ∈ Mx and λ, µ = 1, 2, . . . , n. (c) The vertical lift g v of g is a degenerate metric of rank n on LM given by v g(x,u) (X h , Y h ) = gx (X, Y ),

v g(x,u) (X h , Y v,µ ) = 0,

v g(x,u) (X v,λ , Y h ) = 0,

v g(x,u) (X v,λ , Y v,µ ) = 0

for all X, Y ∈ Mx and λ, µ = 1, 2, . . . , n.

3.2.

Naturally Lifted Metrics on the Linear Frame Bundle

Put P = GL(n, R) × Tn1 E+ and denote by (ukλ , gij , gij,k ), 1 ≤ i ≤ j ≤ n; k = 1, 2, . . . , n, the system of canonical coordinates of P . Also, put Q = GL(n, R) × {(Rn∗ ⊙ Rn∗ ) ⊕ (Rn∗ ⊗ Rn∗ ⊗ Rn ) ⊕ [(Rn∗ ⊗ Rn ) ⊙ (Rn∗ ⊗ Rn )]} and denote by (vνk , Gij , Gi νk , Gλµ ij ) the system of canonical coordinates on Q. Here 1 ≤ i ≤ j ≤ n; k, ν = 1, 2, . . . n and (1, 1) ≤ (i, λ) ≤ (j, µ) ≤ (n, n) in the lexicographic arrangement. We define the actions of L2n on P and Q, respectively, having in mind the transformation rules on Riemannian objects under changes of systems of natural local coordinates. Discussing as in the previous section, we have


199

λµ Theorem 3.1 ([21]). Let αij , βi µj and γij be functions on P which are solutions of the following system of differential equations:

∂ζij = 0, ∂gpq,r 2

n X

a=1

gap

(3.1)

n X ∂ζij ∂ζij − uqα p = ζip δjq + ζpj δiq ∂gaq α=1 ∂uα

(3.2)

for i, j, p, q, r = 1, 2, . . . , n. Then a necessary condition for a map f : P −→ Q such that v k ◦ f = uk , k = 1, 2, . . . , n, to be a differential invariant is that its representation by the canonical coordinates is of the form Gij =

n X

νω uaν ubω Γsai Γtbj αst +

a,b,s,t,ν,ω=1

n X

ubω Γsbi βs ωj +

b,s,ω=1

Gi νk =

n X

n X

ubω Γsbj βs ωi + γij , (3.3)

b,s,ω=1

µν uaµ Γsai αsk + βi νk ,

(3.4)

a,s,µ=1 λµ Gλµ ij = αij

(3.5)

for 1 ≤ i ≤ j ≤ n; k, ν = 1, 2, . . . n and (1, 1) ≤ (i, λ) ≤ (j, µ) ≤ (n, n) in the lexicographic arrangement, where Γkij ’s are the formal Christoffel symbols derived from gij and gij,k . We have given in [21] a geometric meaning to the system of differential equations (3.2) and its solution. Consider the left L1n -spaces V = GL(n, R) × E+ and W = Rn∗ ⊗ Rn∗ with the natural actions of L1n defined, again, having in mind transformation rules under changes of systems of natural local coordinates. Let (ukλ , gij ), 1 ≤ i ≤ j ≤ n; k, λ = 1, 2, . . . , n, and (ζij ), i, j = 1, 2, . . . , n, be the systems of canonical coordinates of V and W , respectively. We can check easily that the system of differential equations (3.2) gives a necessary condition for a map ζij = ζij (uk , gkl ) to be (the first order) differential invariant from V to W . Long but routine calculations show that the formulas (3.3)–(3.5) provide a differential λµ invariant f from P to Q if and only if the functions αij , βi µj and γij defined on V describe some differential invariants α, β and γ from V to W . We say that a bundle morphism of the form ζ : LM ⊕ T M ⊕ T M −→ M × R is an L-metric on M if it is linear in the second and the third argument (and smooth in the first argument). We also say that ζ is symmetric or skew-symmetric if it is symmetric or skew-symmetric with respect to the second and third argument, respectively. Any Riemannian metric g on M is a symmetric L-metric which is independent on u. In our special case, letting g be a given Riemannian metric on M , we speak about natural L-metrics derived from g which are L-metrics ζ, for a fixed u ∈ LM , whose components ζ(u)ij = ζ(u, ∂/∂xi , ∂/∂xj ) with respect to a system of local coordinates (x1 , x2 , . . . , xn ) in M are solutions of the system of differential equations (3.2). Solving this system of differential equations, we obtain

200


Theorem 3.2 ([21]). Let (M, g) be an n-dimensional Riemannian manifold. Then all natural L-metrics ζ on M derived from g are given by ζ(u; X, Y ) =

n X

ϕαβ ω α (X)ω β (Y )

(3.6)

α,β=1

where {ω 1 , ω 2 , . . . , ω n } are the dual frame to a linear frame u = {u1 , u2 , . . . , un } and ϕαβ , α, β = 1, 2, . . . , n, are arbitrary smooth functions of n(n + 1)/2 variables wλµ = g(uλ , uµ ), 1 ≤ λ ≤ µ ≤ n. For a given Riemannian metric g on M , we define the classical lifts of L-metrics from M to LM with respect to g as symmetric (0, 2)-tensor fields on LM which are constructed as follows: (a) Let ζ be a symmetric L-metric and (ζ λµ ), 1 ≤ λ ≤ µ ≤ n, a family of arbitrary L-metrics. The diagonal lift ξ d,g of the family ξ = (ζ, ζ λµ ) with with respect to g is defined by d,g ξ(x,u) (X h , Y h ) = ζx (u; X, Y ), d,g ζ(x,u) (X h , Y v,µ ) = 0,

µ = 1, 2, . . . , n,

d,g ξ(x,u) (X v,λ , Y v,µ ) = ζxλµ (u; X, Y ),

1 ≤ λ ≤ µ ≤ n,

for all X, Y ∈ Mx . (b) The horizontal lift ξ h,g of an n-tuple ξ = (ζ µ ) of L-metrics with respect to g is defined by h,g ξ(x,u) (X h , Y h ) = 0, h,g ξ(x,u) (X h , Y v,µ ) = ζxµ (u; Y, X),

µ = 1, 2, . . . , n,

h,g ξ(x,u) (X v,λ , Y v,µ ) = 0,

λ, µ = 1, 2, . . . , n,

for all X, Y ∈ Mx . If we take ζ = g and ζ λµ = g δ λµ in (a), and ζ µ = g in (b), then ξ d,g and ξ h,g are just the classical lifts g d and g h , respectively. Also, if we take ζ = g and ζ λµ = 0 in (a), then ξ d,g is the vertical lift g v of g. Thus we have all metrics on LM which come from a second order natural transformation of a given Riemannian metric on M : Theorem 3.3 ([21]). Let g be a Riemannian metric on an n-dimensional smooth manifold M , n ≥ 2, and let G be a (possibly degenerate) pseudo-Riemannian metric on the linear frame bundle LM which comes from a second order natural transformation of g. Then there are families ξ1 = (ζ, ζ λµ ) and ξ2 = (ζ ν ) of natural L-metrics derived from g, where 1 ≤ λ ≤ µ ≤ n, ν = 1, 2, . . . , n and ζ is symmetric, such that G = ξ1 d,g + ξ2 h,g . Moreover, all natural L-metrics derived from g are given by Theorem 3.2.


201

The family of all natural metrics G on LM over an n-dimensional Riemannian manifold (M, g) depends on n(n3 + 3n2 + n + 1)/2 arbitrary functions of n(n + 1)/2 variables. Equivalently, the metrics G in Theorem 3.3 can be expressed in the following form: G(x,u) (X h , Y h ) =

n X

ϕαβ ωxα (X)ωxβ (Y ),

α,β=1

G(x,u) (X h , Y v,µ ) =

n X

ϕα µβ ωxα (X)ωxβ (Y ),

(3.7)

α,β=1

G(x,u) (X v,λ , Y v,µ ) =

n X

α β ϕλµ αβ ωx (X)ωx (Y )

α,β=1

for all X, Y ∈ Mx , where ϕαβ , ϕα µβ and ϕλµ αβ , α, β, λ, µ = 1, 2, . . . , n, are arbitrary smooth functions of n(n + 1)/2 variables wρσ = g(uρ , uσ ), 1 ≤ ρ ≤ σ ≤ n. In the following we shall call G a g-natural metric on LM . Remark. The case of symmetric affine connection given on the base manifold gives a completely analogous classifications of metrics on LM as described for the metric case in Theorems 3.2 and 3.3, where just two lifts of “metric type” of L-metrics are replaced by the lifts of “affine type”. (See [40].)

3.3.

Riemannian Geometry of the Linear Frame Bundle

Let φ be a (local) transformation of a manifold M . Then we define a transformation Φ of LM by Φ(x, u) = (φx, φ∗x u1 , φ∗x u2 , . . . , φ∗x un ) for all (x, u) = (x, u1 , u2 , . . . , un ) ∈ LM . If φ is a (local) affine transformation with respect to the Levi-Civita connection ∇ of (M, g), then we have Φ∗ (X h ) = (φ∗ X)h ,

Φ∗ (X v,λ ) = (φ∗ X)v,λ

for all X ∈ X(M ) and λ = 1, 2, . . . , n. Using this fact we can easily see that all g-natural metrics are invariant: Theorem 3.4 ([25]). Let φ be a (local) isometry of a Riemannian manifold (M, g). Then every g-natural metric G on the linear frame bundle LM over (M, g) is invariant by the lift Φ of φ. In other words, Φ is a local isometry of (LM, G) whose projection on (M, g) is φ. Let (M, g) be an n-dimensional Riemannian manifold, n ≥ 2, and LM its linear frame bundle. It seems that phenomenon similar to geometry of T M with the Sasaki metric g s happens about geometry of LM with the diagonal lift g d . L. A. Cordero and M. de León have shown in [10] that g d is also “rigid” in some sense. Namely, they have proved Theorem 3.5 ([10]). The Riemannian manifold (LM, g d ) is never locally symmetric unless (M, g) is locally Euclidean.

202


Theorem 3.6 ([10]). If (LM, g d ) is an Einstein manifold, then (M, g) is flat. Theorem 3.7 ([10]). The Riemannian manifolds (LM, g d ) and (M, g) have the same constant scalar curvature if and only if both are flat. It should be interesting to find “non-rigid” metrics on LM . We study in this section specific Riemannian metrics from our list given by Theorem 3.3, which are a bit more general ones than the diagonal lift g d . The horizontal and vertical distributions are no more orthogonal in general with respect to the metrics g¯ treated in this section. Usually, calculations for getting geometric objects of LM are not short. They are complicated sometimes. We have given in [23] some simpler procedure in order to calculate them. The metrics which we pick in the following are still rigid when the base manifold is of constant curvature. Let g¯ be a metric on LM defined by taking the coefficient functions in (3.7) as ϕα µβ = cµ wαβ

ϕαβ = wαβ ,

λµ and ϕλµ αβ = c wαβ ,

where cµ and cλµ = cµλ are constants. That is, g¯ is given by g¯(x,u) (X h , Y h ) = gx (X, Y ), g¯(x,u) (X h , Y v,µ ) = cµ gx (X, Y ),

(3.8)

g¯(x,u) (X v,λ , Y v,µ ) = cλµ gx (X, Y ) for all X, Y ∈ Mx and λ, µ = 1, 2, . . . , n. The metric g¯ is positive definite if and only if all principal minor determinants of 1

c1

c2

 1 c   2 c   

c11

c12



c21 ...

...

cn



. . . c1n    22 2n c ... c    ... ...  

cn cn1 cn2 . . . cnn

are positive. In particular, the matrix [cλµ ] is positive definite. If cλ = 0 and cλµ = δ λµ for λ, µ = 1, 2, . . . , n, then the metric g¯ is the diagonal lift g d of g. We have shown in [23] that the metric g¯ is rigid in the following sense: Theorem 3.8 ([23]). Let (M, g) be a space of constant sectional curvature K, and g¯ the positive definite metric on LM defined by (3.8). If the scalar curvature of the frame bundle (LM, g¯) is constant then both manifolds are flat. This can be applied to the diagonal metric and hence we obtain a new result in the spirit of Theorems 3.5–3.7.

4.

Cotangent Bundle

If the base manifold M has a Riemannian metric g, then its cotangent bundle T ∗ M is dual to the tangent bundle T M , and hence all natural lifts of g to T ∗ M are settled from the those


203

of g to T M through this duality. But there is a situation when only T ∗ M comes in the game, namely if the base manifold is an affine manifold (M, ∇). Here a new operation, so-called Riemann extension is known [37]. After recalling this notion, we shall present the full classification of naturally lifted (pseudo-Riemannian) metrics from a given symmetric affine connection ∇ on M to T ∗ M .

4.1.

Classical Examples of Metrics on the Cotangent Bundle

The cotangent bundle T ∗ M over an n-dimensional smooth manifold M , n ≥ 2, consists of all pairs (x, w), where x is a point of M and w is a covector from the cotangent space Mx∗ of M at x. We denote by p the natural projection of T ∗ M to M defined by p(x, w) = x. Let p∗ : T (T ∗ M ) −→ T M be the differential of the natural projection p, and q : T (T ∗ M ) −→ T ∗ M be the natural projection of the tangent bundle T (T ∗ M ) (over T ∗ M ) to T ∗ M . The canonical one-form on T ∗ M is a one-form θ defined by θ(X) = q(X)(p∗ X) for all X ∈ T (T ∗ M ), and the canonical two-form on T ∗ M is the exterior derivative dθ of θ. Let ∇ be an affine connection on the base manifold M . Then the tangent space ∗ (T M )(x,w) of T ∗ M at (x, w) ∈ T ∗ M splits into the horizontal and vertical subspace H(x,w) and V(x,w) with respect to ∇: (T ∗ M )(x,w) = H(x,w) ⊕ V(x,w) . Let X = hX + vX be the decomposition of a vector filed X on T ∗ M into the horizontal and vertical part. The Riemann extension g¯ of the affine connection ∇ on M to T ∗ M is a pseudo-Riemannian metric of the signature (n, n) defined by g¯(X, Y) = (dθ)(vX, hY) + (dθ)(vY, hX) for all X, Y ∈ T (T ∗ M ). This metric does not depend on the skew-symmetric part of ∇.

4.2.

Naturally Lifted Metrics on the Cotangent Bundle

Let us consider the space P = Rn∗ ⊕ Rn ⊗ (Rn∗ ⊙ Rn∗ ) , and denote by (wh , Γhij ), h = 1, 2, . . . , n; 1 ≤ i ≤ j ≤ n, the system of canonical coordinates of P . Also, put Q = Rn∗ ⊕ (Rn ⊕ Rn∗ ) ⊙ (Rn ⊕ Rn∗ ) and denote by (zh , Gij , Ghi , Gij ), h = 1, 2, . . . , n; 1 ≤ i ≤ j ≤ n, the system of canonical coordinates on Q. We define the actions of L2n on P and Q, respectively, having in mind the transformation rules on Riemannian objects under changes of systems of natural local coordinates. Discussing as in the previous section, we have

Theorem 4.1 ([39]). All differential invariants f : P −→ Q such that, in the canonical coordinates, zh ◦ f = wh , h = 1, 2, . . . , n, are given by Gij = −2a

n X

ws Γsij + b wi wj ,

s=1

Ghi = a δih , Gij = 0 for 1 ≤ i ≤ j ≤ n and h = 1, 2, . . . , n, where a and b are constants.

204

Oldˇrich Kowalski and Masami Sekizawa Thus we have

Theorem 4.2 ([39]). Let ∇ be a symmetric affine connection on an n-dimensional manifold M , n ≥ 2. Then a (possibly degenerate) pseudo-Riemannian metric G on the cotangent bundle T ∗ M over M comes from a second order natural transformation of ∇ if and only if G = a g¯ + b θ2 , where g¯ is the Riemann extension of ∇, θ2 is the tensor square of the canonical one-form θ of T ∗ M , and a, b are constants. This metric G is non-degenerate (of signature (n, n)) if and only if a 6= 0.

Acknowledgement ˇ 201/05/2707 and by the project The first author was supported by the grant GA CR MSM 0021620839.

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Chapter 12

I NVARIANT VARIATIONAL P ROBLEMS AND I NVARIANT F LOWS VIA M OVING F RAMES Peter J. Olver∗ School of Mathematics, University of Minnesota Minneapolis 55455, USA

Abstract This paper reviews the moving frame approach to the construction of the invariant variational bicomplex. Applications include explicit formulae for the Euler-Lagrange equations of an invariant variational problem, and for the equations governing the evolution of differential invariants under invariant submanifold flows.

1.

Introduction

This survey paper describes some aspects of the author’s recent research, done partly in collaboration with Irina Kogan, [31, 53], into moving frames, the invariant variational bicomplex, and invariant submanifold flows. These results are based on combining two powerful ideas in the modern, geometric approach to differential equations and the variational calculus. The first is the variational bicomplex, which is of fundamental importance in the study of the geometry of jet bundles, differential equations and the calculus of variations. Its origins can be found in the work of Dedecker, [15], then developed in full detail by Tulczyjew, [67], and Vinogradov, [68, 69]. Later contributions of Tsujishita, [66], Anderson, [2, 3], and Krupka and Janyˇska, [32, 33], have amply demonstrated the power of the bicomplex formalism for both local and global problems in the geometric theory of differential equations and the calculus of variations. The second ingredient is a reformulation of Cartan’s method of moving frames, [17, 51]. For a general finite-dimensional transformation group G, a moving frame is defined as an equivariant map from an open subset of jet space to the Lie group G. Moving frames are constructed by the process of normalization based on the choice of cross-section to the ∗

E-mail address: [email protected],www.math.umn.edu/ olver

210

Peter J. Olver

group orbits. The moving frame then provides a canonical mechanism, called invariantization, that allows us to systematically construct the invariant counterparts of all objects of interest in the usual variational bicomplex, including differential invariants, invariant differential forms, invariant differential operators, etc. The key recurrence formulae relate the differentials of ordinary functions and forms to the invariant differentials of invariant functions and forms, and thereby lead to the complete structure of the algebra1 of differential invariants, including the syzygies and commutation formulae, [26, 27, 52]. The equivariant moving frame method has impacted a remarkable range of subjects, including symmetry methods for partial differential equations, the calculus of variations, classical invariant theory, computer vision, numerical analysis, Hamiltonian systems, integrable soliton equations, materials and micromagnetics, joint invariants, relativity, quantum mechanics, invariants of Lie algebras, Lie pseudo-groups, symbolic methods, and (non-commutative) differential algebra; see [50, 51] for recent surveys of developments in the field. A key application of the invariant variational bicomplex is the general solution to an outstanding problem in the calculus of variations. Every group-invariant variational problem can be written in terms of the differential invariants. The associated Euler-Lagrange equations inherit the symmetry group, and so can also be written in terms of the differential invariants. The problem is to directly construct the invariant form of the Euler–Lagrange equations from the invariant form of the variational problem. Before the general solution to this problem appeared in [31], only a few specific examples were known, [3, 22]. A striking recent application of these techniques is the work of Starostin and van der Heijden, [65], on equilibrium configurations of flexible Möbius bands. A second application is to the evolution of differential invariants under invariant submanifold flows. Invariant curve flows and surface flows arise in an impressive range of applications, including geometric optics, [7], elastodynamics, [37], computer vision, [55, 56, 60, 62, 64], visual tracking and control, [45], vortex dynamics, [25, 36], interface motions, [64], thermal grooving, [9], and elsewhere. A celebrated example is the Euclidean invariant curve shortening flow, [18, 20], in which a plane curve moves in its normal direction in proportion to its curvature. In computer vision, Euclidean curve shortening and its equi-affine counterpart have been successfully applied to image denoising and segmentation, [55, 61, 62]. In three dimensional space, Euclidean-invariant curve flows include the integrable vortex filament flow, [25, 36], while mean curvature and Willmore flows of surfaces have been the subject of extensive analysis and applications, [6, 14]. Given an invariant submanifold flow, a key issue is to track the induced evolution of its basic geometric invariants — curvature, torsion and the like. While a number of particular examples have been worked out by direct computation, e.g., in [18, 43], many cases of interest have yet to appear in the literature, owing to their computational complexity. Therefore, it is worth developing general, practical tools to ameliorate this often tedious task. Mansfield and van der Kamp, [39], have developed a method based on the differential invariant syzygies. Here we present a direct approach, applying the invariant variational bicomplex calculus discussed above. As we will see, the same basic invariant differential operators appearing in the construction of invariant Euler–Lagrange equations also play a 1

Technically, because differential invariants may only be locally defined, we should speak of the “sheaf of differential invariants”. However, as we work locally on suitable open subsets, this extra level of abstraction is not required; moreover, experts can readily translate our constructions into sheaf-theoretic language, [70].

Invariant Variational Problems and Invariant Flows via Moving Frames

211

key role in this context.

2.

The Invariant Variational Bicomplex

In this section, we review the basics of prolonged group actions on submanifold jets, moving frames, and the induced invariant variational bicomplex. Basic references include [48, 49] for jets, contact forms, and prolonged Lie group actions, [3, 66] for the variational bicomplex, [17, 51, 52] for the equivariant approach to moving frames, and [31] for the moving frame construction of the invariant variational bicomplex. For simplicity, we will only deal with finite-dimensional Lie group actions in this paper, although the general ideas can be straightforwardly adapted to infinite-dimensional pseudo-group actions using more recent extensions of the moving frame technology, [54]. Let G be an r-dimensional Lie group, acting smoothly on a m-dimensional manifold M . We will study the induced action on p-dimensional submanifolds S ⊂ M . For 0 ≤ n ≤ ∞, let Jn = Jn (M, p) denote the n-th order (extended) jet bundle for such submanifolds, [49]. The action of G on M naturally prolongs to an action on Jn . Since the prolonged group actions are all mutually compatible under projection Jn → Jk , we will avoid explicit reference to the order of prolongation, and just use g · z (n) for the action of g ∈ G on the jet z (n) ∈ Jn , rather than the more traditional notation g (n) · z (n) . By definition, a moving frame is right-equivariant2 map3 ρ : Jn → G, meaning that ρ(g · z (n) ) = ρ(z (n) ) · g −1 for all g ∈ G and all z (n) ∈ Jn where defined. The existence of a moving frame requires that the prolonged group action be free, meaning the isotropy subgroups of each individual jet are trivial, and regular, meaning the prolonged group orbits form a regular foliation, on an open subset V ⊂ Jn . Under these conditions, a moving frame can be algorithmically constructed by a normalization process based on the choice of a compatible cross-section K n ⊂ Jn to the group orbits. Specifically, given z (n) ∈ Jn , we set g = ρ(z (n) ) to be the unique group element such that g · z (n) ∈ K n , when defined. Compatibility of moving frames under the jet space projections allows us to also suppress the order in the notation of ρ. We use ι to denote the invariantization process induced by the moving frame. The invariantization of a differential form Ω is the unique invariant differential form ι(Ω) that agrees with Ω when restricted to the cross-section. In particular, if Ω is an invariant differential form or function, then ι(Ω) = Ω. Invariantization defines an (exterior) algebra morphism that projects differential functions and forms on Jn to invariant differential functions and forms. Let (x, u) = (x1 , . . . , xp , u1 , . . . , uq ) be local coordinates on M . Viewing the x’s as independent variables and the u’s as dependent variables, we let uαJ = ∂ #J u/∂xJ be the usual induced local coordinates on Jn . Separating the local coordinates (x, u) on M into independent and dependent variables naturally splits the differential one-forms on J∞ into horizontal forms, spanned by dx1 , . . . , dxp , and vertical forms, spanned by the basic 2 All classical moving frames, [23], are left-equivariant, and can be obtained by composing ρ with the group inversion g 7→ g −1 . We choose to concentrate on the right-equivariant version to (slightly) simplify some of the calculations. 3 All maps, differential forms, differential functions, etc., need only be locally defined; thus, the domain of ρ is typically a suitable open subset of Jn .

212

Peter J. Olver

contact one-forms θJα = duαJ −

p X

uαJ,i dxi ,

α = 1, . . . , q,

i=1

#J ≥ 0.

(1)

Let πH and πV denote the projections mapping one-forms on J∞ to their horizontal and vertical (contact) components, respectively. The induced splitting d = dH + dV of the differential into horizontal and vertical components results in the variational bicomplex4 . In particular, if F (x, u(n) ) is any differential function, its horizontal and vertical differentials are dH F =

p X

(Di F ) dxi ,

dV F = DF (θ) =

i=1

X ∂F X ∂F DJ θ α = θα , α ∂uJ ∂uαJ J

(2)

α,J

α,J

in which Di = Dxi denote the total derivative operators with respect to the independent variables, DJ = Dj1 · · · Djk are the higher order total derivatives, θ = (θ1 , . . . , θq )T is the column vector containing the order zero contact forms, while DF = (DF,1 , . . . , DF,q ) is the Fréchet derivative or formal linearization of the differential function F . We will employ our moving frame to invariantize the variational bicomplex as follows. First, invariantization of the jet coordinate functions produces the fundamental differential invariants: H i = ι(xi ),

IJα = ι(uαJ ),

α = 1, . . . , q,

#J ≥ 0.

(3)

These naturally split into two classes: The r = dim G combinations defining the crosssection equations will be constant, and are known as the phantom differential invariants. The remainder, called the basic differential invariants, form a complete system of functionally independent differential invariants. Next, let ̟i = ω i + η i = ι(dxi ),

where

ω i = πH (̟i ), η i = πV (̟i ),

(4)

denote the invariantized horizontal one-forms. Their horizontal components ω 1 , . . . , ω p form, in the language of [49], a contact-invariant coframe for the prolonged group action, while η 1 , . . . , η p supply “contact corrections” that make the one-forms ̟1 , . . . , ̟p fully G-invariant. The corresponding dual invariant total differential operators D1 , . . . , Dp are defined so that H i = ι(xi ),

IJα = ι(uαJ ),

α = 1, . . . , q,

#J ≥ 0.

(5)

for any differential function F and, more generally, differential form Ω, on which the Di act via Lie differentiation. Finally, let ϑαJ = ι(θJα ),

α = 1, . . . , q,

#J ≥ 0.

(6)

4 Since the splitting depends on a choice of independent variables on M , the variational bicomplex is not intrinsic. A more refined version of this construction, known as the C spectral sequence, [68, 69], relies on the contact filtration of the algebra of differential forms. However, since all our calculations take place in local coordinates, we will avoid all the extra complications inherent in this more sophisticated machinery. Experts will be able to readily translate our results as desired.


213

be the invariantized basis contact forms. As in the usual, non-invariant bicomplex construction, the decomposition of invariant one-forms on J∞ into invariant horizontal and invariant contact components induces a decomposition of the differential. However, now d = dH + dV + dW splits into three constituents, where dH adds an invariant horizontal form, dV adds a invariant contact form, while dW replaces an invariant horizontal one-form with a combination of wedge e r,s denote the space products of two invariant contact forms. In other words, if we let Ω of differential forms of degree r + s spanned by wedge products of r invariant horizontal one-forms (4) and s invariant contact one-forms (6), then e r,s −→ Ω e r+1,s , dH : Ω

e r,s −→ Ω e r,s+1 , dV : Ω

e r,s −→ Ω e r−1,s+2 . (7) dW : Ω

The resulting invariant variational quasi-tricomplex is characterized by the formulae d2H = 0,

dH dV + dV dH = 0,

d2W

dV dW + dW dV = 0,

= 0,

d2V + dH dW + dW dH = 0.

(8)

Fortunately, the third, anomalous component dW plays no role in the applications; in particular, dW F = 0 for any differential function F . The most important fact underlying the moving frame construction is that the invariantization map ι does not respect the exterior derivative operator. Thus, in general, d ι(Ω) 6= ι(dΩ). The recurrence formulae, [17, 31], which we now review, provide the missing “correction terms” dι(Ω) − ι(dΩ). Remarkably, these formulas can be explicitly and algorithmically constructed using only linear differential algebra — without knowing the explicit formulas for either the differential invariants or invariant differential forms, the invariant differential operators, or even the moving frame! The only required ingredients are the cross-section equations and the formulae for the prolonged infinitesimal generators of the group action. Let v1 , . . . , vr be a basis for the infinitesimal generators of our transformation group. We prolong each infinitesimal generator to Jn . For conciseness, we will retain the same notation vκ for the prolonged vector fields on any Jn which, in local coordinates, take the form p q n X X X ∂ ∂ i ξκ (x, u) i + vκ = ϕαJ,κ (x, u(j) ) α , κ = 1, . . . , r. (9) ∂x ∂uJ α=1 j=#J=0

i=1

The coefficients ϕαJ,κ = vκ (uαJ ) can be successively constructed by Lie’s recursive prolongation formula, [48, 49]: ϕαJi,κ = Di ϕαJ,κ −

p X

uαJj Di ξκj .

(10)

j=1

A straightforward induction establishes the explicit prolongation formula, first written down by the author in [47]: ϕαJ,κ

=

DJ Qακ

+

p X

ξκi uαJ,i ,

where

i=1

are the components of the characteristic of vκ .

Qακ

=

ϕακ

−

p X i=1

ξκi uαi

(11)

214

Peter J. Olver

Strikingly, all the recurrence relations are consequences of a single universal recurrence formula that prescribes the differential of an invariantized differential function or form. Theorem 1. If Ω is any differential form on J∞ , then d ι(Ω) = ι(d Ω) +

r X κ=1

ν κ ∧ ι [vκ (Ω)],

(12)

where ν 1 , . . . , ν r

are the invariantized Maurer–Cartan forms dual to the infinitesimal generators v1 , . . . , vr , while vκ (Ω) denotes the Lie derivative of Ω with respect to the prolonged infinitesimal generator vκ . The invariantized Maurer–Cartan forms ν 1 , . . . , ν r are obtained by pulling back the usual dual Maurer–Cartan forms µ1 , . . . , µr on G by the moving frame map: ν κ = ρ∗ µκ . Details would take us too far afield, [31], but, fortunately, are not required thanks to the following marvelous result that allows us to compute them directly without reference to their underlying definition: Lemma 2. Let I1 = ι(z1 ), . . . , Ir = ι(zr ) be the phantom differential invariants stemming from our cross-section. Then the corresponding phantom recurrence formulae 0 = dIς = d ι(zς ) = ι(dzς ) +

r X κ=1

ν κ ∧ ι [vκ (zς )],

ς = 1, . . . , r,

(13)

can be uniquely solved for the invariantized Maurer–Cartan forms ν 1 , . . . , ν r . Having solved the linear system (13) for ν 1 , . . . , ν r , we then decompose the resulting invariantized Maurer–Cartan forms into their invariant horizontal and contact components: κ

κ

κ

ν =γ +ε ,

κ

γ =

where

p X

Riκ ̟i ,

εκ =

i=1

X

Sακ,J ϑαJ ,

(14)

α,J

where Riκ , Sακ,J are certain differential invariants. The Riκ will be called the Maurer–Cartan invariants, [26, 27, 52]. In the case of curves, the Riκ appear as the entries of the Frenet– Serret matrix Dρ(x, u(n) ) · ρ(x, u(n) )−1 , in the case G ⊂ GL(N ) is a matrix Lie group, [23]. Substituting (14) back into the universal formula (12) produces a complete system of explicit recurrence relations for all the differentiated invariants and invariant differential forms. In particular, taking Ω to be any one of the individual jet coordinate functions xi , uαJ , results in the recurrence formulae for the fundamental differential invariants (3): dH i = ι(dxi ) + dIJα

=

ι(duαJ )

=

p X

r X

κ=1 r X

+

ν κ ι [vκ (xi )] = ̟i + ν

κ

ι [vκ (uαJ )]

κ=1

i=1

α IJi ̟i + ϑαJ +

=ι

r X

κ=1 p X

uαJi dxi

i=1

r X κ=1

ι(ξκi ) ν κ ,

ι(ϕαJ,κ ) ν κ .

+

θJα

!

+

r X κ=1

ι(ϕαJ,κ ) ν κ

(15)


215

In view of (14), the coefficient of ̟i in (15) yields the recurrence relations j

Di H =

δij

+

r X

Riκ ι(ξκi ),

Di IJα

κ=1

=

α IJi

+

r X

Riκ ι(ϕαJ,κ ),

(16)

κ=1

where δij is the usual Kronecker delta. Owing to the functional independence of the basic (non-phantom) differential invariants, these formulae, in fact, serve to completely characterize the structure of the non-commutative differential invariant algebra, [17, 26, 52]. Similarly, the contact components in (15) yield the vertical recurrence formulae dV H i =

r X

ι(ξκi ) εκ ,

dV IJα = ϑαJ +

κ=1

r X

ι(ϕαKκ ) εκ ,

(17)

κ=1

while, as noted earlier, dW H i = dW IJα = 0. The recurrence formulae (12) for the derivatives of the invariant horizontal forms are i

i

2 i

d̟ = d[ι(dx )] = ι(d x ) +

r X κ=1

p X

ν κ ∧ ι [vκ (dxi )]

! q i X ∂ξ κ α = νκ ∧ ι Dk ξκi dxk + θ ∂uα κ=1 α=1 k=1 i p q r X r X X X κ ∂ξκ i k = ι Dk ξκ ν ∧ ̟ + ν κ ∧ ϑα . ι ∂uα r X

κ=1 k=1

(18)

κ=1 α=1

The resulting two-form can be decomposed into three basic constituents, belonging, ree 2,0 ⊕ Ω e 1,1 ⊕ Ω e 0,2 . In view of (14), the terms in (18) spectively, to the invariant summands Ω e 2,0 , yield involving wedge products of two horizontal forms, i.e., in Ω X i dH ̟ i = − Yjk ̟j ∧ ̟k , j

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