Symmetry And Perturbation Theory: Proceedings of the International Conference SPT 2007 Otranto, Italy 2-9 June 2007

Proceedings of the International Conference SPT 2007 Symmetry and Perturbation Theory Symmetry and Perturbation Theo...

Author: Giuseppe Gaeta | Raffaele Vitolo | Sebastian Walcher

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Proceedings of the International Conference

SPT 2007 Symmetry and Perturbation Theory

Symmetry and Perturbation Theory SPT2007 Otranto, 2

-

9 June 2007

Edited by: Giuseppe Gaetaa, Raffaele Vitolob, Sebastian WalcherC a

Dipartimento di Matematica, Universitb di Milano, Milano, Italy; Dipartimento di Matematica, Universith del Salento, Lecce, Italy; Lehrstuhl A fur Mathematik, RWTH-Aachen, Aachen, Germany

The Conference SPT2007 received the financial support of 0 0 0 0

Gruppo Nazionale di Fisica Matematica - INdAM; Dipartimento di Matematica, Universith di Milano; Ufficio Centrale Ricerca, Universita di Milano; Dipartimento di Matematica, Universith del Salento.

Proceedings of the International Conference

SPT 2007 Symmetry and Perturbation Theory 2 - 9 June 2007

Otranto, Italy

Edited by

Giuseppe Gaeta Universita di Milano, Italy

Raffaele Vitolo Universita del Salento, Italy

Sebastian Walcher RWTH-Aachen, Germany

N E W JERSEY

-

LONDON

-

vp world Scientific SINGAPORE

*

BElJlNG * S H A N G H A I * HONG KONG * TAIPEI

-

CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224

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SYMMETRY AND PERTURBATION THEORY 2007 Proceedings of the International Conference on SPT2007 Copyright 0 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereox may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written pertnission from the Publisher.

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V

FOREWORD

This volume collects papers related to the conference SPT2007, where S P T stands for “Symmetry and Perturbation Theory” - albeit by now this traditional title is not exhaustive of the scope of this series of conferences. Previous conferences on “Symmetry and Perturbation Theory” were held in Torino [l](1996), Roma [2] (1998), and in Cala Gonone, on the eastern coast of Sardinia (2001 [3,4], 2002 [5] and 2004 [6,7]). The success of the conference - at least in terms of the number of participants and proposed communications (as for the scientific success, we leave it to be judged by the reader of these proceedings) - pushed us to resort again to a mix of plenary and parallel sessions; the latter were organized by members of the Scientific Committee, which we warmly thank for this help - which in many cases actually meant going beyond their originally intended tasks and diving concrete help to the organization committee. These parallel sessions were devoted to (and organized by): 0 0

0 0

0 0

0

Geometry of Differential Equations (M.E. Fels), Symmetry of Differential Equations (P.J. Olver), Normal Forms Theory (T. Gramchev), Perturbation of ODES (F. Verhulst), Perturbation of PDEs (M. Procesi), Integrable Systems - Analytic Aspects (A. Degasperis), Integrable Systems - Geometric Aspects (S. Rauch), N-Body Dynamics (S.Terracini).

The papers collected here should give some flavor of the many topics discussed, and results presented, at the conference. We were again taken by surprise by the large number of proposed contributions to these Proceedings; thus in order to stay within the planned length of this volume, we selected - again with the precious help of the Scientific Committee - some featured contributions, and unfortunately had to require other authors

vi

to limit their contribution to extended abstracts. This subdivision does not reflect the subdivision into plenary and parallel sessions, and was actually to a large extent influenced by the strict time limits we set for the preparation of this volume, which forced several possible contributors to drop off the project. We preferred, in line with the interdisciplinary nature of the whole SPT conference series and as in previous SPT conference volumes, t o present all of the contributions together, i.e. without a separation in different topics; this, by the way and to our satisfaction, would have been rather arbitrary in many cases. We would like to most warmly thank all of the authors (and referees) for accepting to work under extremely strict - and draconianly enforced time limits: this is even more remarkable considering this took place at a rather full time of the year. We hope the reader will enjoy the papers collected here and will find them useful to gather a picture of the recent progress in the fields our conference touched upon. If this is the case, the merit is of course not ours but of the authors.

References 1. D. Bambusi and G. Gaeta eds., S y m m e t r y and perturbation theory (SPT96), Quaderni GNFM-CNR, Firenze 1997. 2. A. Degasperis and G. Gaeta eds., S y m m e t r y and perturbation theory - SPT98, World Scientific, Singapore 1999. 3. D. Bambusi, M. Cadoni and G. Gaeta eds., S y m m e t r y and perturbation theory - SPT 2001, World Scientific, Singapore 2001. 4. G. Gaeta ed., Special issue on “Symmetry and Perturbation Theory”, Acta Applicandae Mathematicae vol. 70 1:3 (2002). 5. S. Abenda, G. Gaeta and S. Walcher eds., S y m m e t r y and perturbation theory - SPT 2002, World Scientific, Singapore 2002. 6. G. Gaeta, B. Prinari, S. Rauch-Wojciechowski and S. Terracini eds., “Symmetry and Perturbation Theory - SPT 2004”, World Scientific, Singapore 2005. 7. G. Gaeta ed., Special issue on “Symmetry and Perturbation Theory”, Acta Applicandae Mathematicae vol. 87 1:3 (2005).

vi i

ACKNOWLEDGMENTS

A number of people and Institutions also helped us in the organization and running of the conference, and we would like t o thank all of them here. First of all, the Scientific Committee, consisting (beyond ourselves) of: Antonio Degasperis (Roma), Mark E. Fels (Logan), Todor Gramchev (Cagliari), Peter J. Olver (Minneapolis), Michela Procesi (Roma), Stefan Rauch-Woijechowski (Linkoeping), Susanna Terracini (Milano), Ferdinand Verhulst (Utrecht) . Special thanks should also go to persons invoIved in non-scientific aspects of the conference: all the personnel of the Hotel Daniela, where the conference took place, as well as the staff of TIVIGEST (the society running Hotel Daniela); with special thanks to Dr. Enrico Belli. Last but by no means least, the conference received substantial financial support - which made it possible - by several Institutions, which we warmly thank here: GNFM-INdAM (Gruppo Nazionale di Fisica Matematica, Istituto Nazionale di Alta Matematica), by the Dipartimento di Matematica and by the Ufficio Centrale Ricerca of Universitk di Milano, and by the Dipartimento di Matematica of Universita del Salento (Lecce). The latter also provided precious and substantial logistics help.

Giuseppe Gaeta (Milano) Raffaele Vitolo (Lecce) Sebastian Walcher (Aachen)

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ix

CONTENTS

Foreword .............................................................. v Acknowledgments .................................................... vii

FEATURED CONTRIBUTIONS D. Alekseevsky & A. Spiro Homogeneous bi-lagrangian manifolds and invariant Monge- Ampere equations .............................................................. 3 I.M. Anderson, M.E. Fels & P.J. Vassiliou O n Darboux integrability ..........................................

.13

E. Asadi & J.A. Sanders Integrable systems in symplectic geometry .........................

.21

V. Barutello & S. Terracini Sangularitaes and collisions in N-body type problems . . . . . . . . . . . . . . ..29 S. Benenti Computing curvature without Christoflel symbols

. . . . . . . . . . . . . . . . . ..37

H. Broer, R. van Dijk & R. Vitolo Survey of strong nomnal-internal Ic : e resonances in quasiperiodically driven oscillators for != 1 , 2 , 3 ...........................

.45

A. Celletti & L. Chierchia Quasi-periodic attractors and spin-orbit resonances ................56 A. Degasperis & S. Lombard0 Darboux construction of solutions of integrable PDEs with nonvanishing boundary values ........................................

.64

X

D. Ferrario Transitive decomposition of n-body symmetry groups . . . . . . . . . . . . . .. 7 3 T. Gramchev & M. Yoshino Normal f o r m s for commuting vector fields near a c o m m o n fixed point ............................................................

81

H. Hanssmann A monkey saddle in rigid body dynamics ..........................

.92

J. Krashil’shchik Nonlocal geometry of P D E s and integrability .....................

,100

B. Kruglikov A n o m a l y of linearization and auxiliary integrals . . . . . . . . . . . . . . . . . .108 D. Krupka Natural variational principles ....................................

.116

0. Krupkova Variational exterior differential systems ..........................

124

S. Marmi & P. Tempesta O n the relation between formal groups, Appell polynomials and hyperfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.132

P. Morando Variational A-symmetries and deformed Lie derivatives . . . . . . . . . . .140 0 . 1 . Morozov Maurer-Cartan forms for symmetry pseudo-groups and coverings of differential equations ...................................

.148

N.N. Nekhoroshev Fuzzy fractional monodromy .....................................

.156

F. Oliveri, G . Manno & R. Vitolo O n the correspondence between differential equations and s y m m e t r y algebras ..................................................

.164

P.J. Olver & J. Pohjanpelto Moving frames and differential invariants for Lie pseudo-groups . . 172 M. Procesi Periodic solutions for a class of non-linear Schroedinger equations in D > 1 spatial dimensions ...............................

181

xi

M.A. Teixeira & P.R. Da Silva Singular perturbation for discontinuous ordinary differential equations ............................................................

189

A. Vanderbauwhede Continuation of periodic solutions in conservative and reversible systems ...................................................

.198

F. Verhulst Emergence of slow manifolds in nonlinear wave equations . . . . . . . ..206

P. Winternitz & i. Yurdqen Superintegrable systems with spin in two- and three-dimensional euclidean spaces .....................................................

215

B . Zhilinskii Generalization of Hamiltonian monodromy. Quantum manifestations ......................................................

.223

EXTENDED ABSTRACTS M.B. Abd-el-Malek & H.S. Hassan Internal flow through a conducting thin duct via symmetry analysis .............................................................

233

R. Alonso Blanco, G . Manno & F. Pugliese Contact geometry of parabolic Monge-Ampdre equations . . . . . . . . . ..235 K. Andriopoulos Complete symmetry groups and Lie remarkability . . . . . . . . . . . . . . . ..237 M. Arminjon Quantum wave equations in curved space-time wave mechanics , . . 239 D. Catalan0 Ferraioli Nonlocal interpretation of X-symmetries ..........................

241

C. Chanu, M. Chanachowicz & R.G. McLenaghan R-separation for the conformal Laplacian ........................

.243

A.M. Cherubini, G. Metafune & F. Paparella Asymptotic behavior of a bouncing ball ...........................

245

xii

G.M. Coclite & K.H. Karlsen Discontinuous solutions for the Degasperis-Procesi equation . . . . . .. 2 4 7 V. Golovko Variational Poisson-Nijenhuis structures for evolution P D E s . . . . . 249

A. Gonzalez-Enriquez & R. de la Llave Analytic approximations of geometric maps and applications t o K A M theory .....................................................

.251

M. Iwasa A method t o construct asymptotic solutions invariant under the renormalization group ...............................................

253

J. JanySka Utiyama’s reduction method and infinitesimal symmetries of invariant Lagrangians ...............................................

255

1.1. Kosenko & S.Ya. Stepanov Stability of the tethered satellite system relative equilibria. Unrestricted problem ................................................

,257

E. Maderna & A. Venturelli Globally minimizing parabolic motions in the Newtonian N-body problem .............................................................

.259

K. Marciniak Geodesically equivalent flat bi-cofactor systems . . . . . . . . . . . . . . . . . . ..261 L. Martina Symmetry group and symplectic structure for exotic particles in the plane .........................................................

264

D. Pinheiro & R.S. Mackay S o m e properties of the dynamics of two interacting particles in a uniform magnetic field .............................................. .267 A. Portaluri O n the dihedral n-body problem ..................................

.269

0. Rojas, P. van der Kamp & G.R.W. Quispel Lax representation for integrable OAEs ..........................

.271

A.R. Rutherford Pseudorotational spectra of molecules and isoparametric geometry ............................................................

273

xiii

F. Strazzullo Darboux integrable hyperbolic P D E i n the plane of generic type: a classification by means of Cartan tensor and Maple . . . . . . . . . . . . . . . .276

P. van der Kamp Towards global classifications: a diophantine approach . . . . , . . . . . . . 278 R. Vitolo, H. Broer & C. Sim6 The Hopf-saddle-node bifurcation for fixed points of 3Ddaffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .280

I. Yehorchenko Relative invariants f o r Lie algebras: construction and applications . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . .282 CONFERENCE INFORMATION List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .287 List of Communications . , , .. , . .. , , .. ,., . , , ... .... , , , . . , . , ... .. , , .. , , 291 Previous SPT Conferences , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .295


FEATURED CONTRIBUTIONS


3

HOMOGENEOUS BI-LAGRANGIAN MANIFOLDS AND INVARIANT MONGE-AMPERE EQUATIONS DMITRI V. ALEKSEEVSKY School of Mathematics and Maxwell Insitute for Mathematical Sciences, University of Edinburgh, Edinburgh, EH9 352, UNITED KINGDOM E-mail: D. [email protected] ANDREA SPIRO Dip. Matematica e Infonnatica, Universitri di Camerino, I- 62032 Camerino (Macerata), ITALY E-mail: [email protected] In this note, we give a description of the invariant bi-Lagrangian structures on a homogeneous symplectic manifold ( M = G / K , w ) of a semisimple Lie group G. We then use this description in order to obtain invariant generalized Monge-AmpBre equations, in the sense of V. Lychagin, on certain homogeneous contact manifolds, associated with homogeneous bi-Lagrangian manifolds. Keywords: Bi-Lagrangian structures; para-KahIer structures; Monge-AmpBre equations.

1. Bi-Lagrangian manifolds

A bi-Lagrangian structure on a symplectic manifold ( M , w ) is a decomposition T M = L+ L- of the tangent bundle into a direct sum of integrable Lagrangian distributions (see Ref. 3). Notice that, given a biLagrangian structure T M = L+ L- on ( M , w ) , the field of endomorphisms Iz E r ( E n d ( T z M ) ) , defined by ItL$ = &Id for z E M , is involutive ( I 2 = Id), skew-symmetric w.r.t. to the pseudo-Riemannian metric g = w ( I . , .) and parallel w.r.t. the Levi-Civita connection V of g . Conversely, a pair ( g , I ) formed by a metric g and a parallel field I of involutive skew-symmetric endomorphisms (called para-Kiihler structure) defines a symplectic structure w = g ( . , I . ) together with a bi-Lagrangian structure TM = L+ L - , where Lk are the (f1)-eigendistributions of I .

+

+

+

We list now some properties of a bi-Lagrangian manifold ( M ,w , T M =

4

L+

+ L-).

I ) For any pair of vector fields X + E I'(L+)and Y-E I'(L--),

vx+y-= (VX+Y-)L-= (VY-X++ [x+,Y-])L= [x+,y-],-(1) where ( . ) L - denotes the projection onto L - . We denote by N* = N * ( s ) the maximal integral leaves of the distributions T *M passing through a point J: E M . Then (1) implies that the connection induced by V on L - ( N + = T * N - is flat. The identity

g(vx+Y+, Z-)= -g(Y+, vx+z-) , for x+,Y+ E r ( L + ) ,z-E r ( L - ) , shows that also the induced connections on T N + = L + I N + and on

+

T M I N += L+IN+ L-IN+ = T N + -tT * N + are flat.

11) Assume that each maximal integral leaf N k ( z ) , s E M , is simply connected and that the flat linear connection VlrN*(,)is complete. Then, by (I), each leaf has the structure of an affine space and the exponential map exp : L*l, + N * ( z ) is an isomorphism of affine spaces. Moreover, if we fix a maximal integral leaf N + = N + ( x ) and we denote by exp : L - ) N + M the restriction of the exponential map of g a t the bundle L-IN+, then the M is a local diffeomorphism. map exp : L - / N + This last property follows from the fact that a vector field Jt = JI,, along the geodesic -yt = exp(tv-), for some w- E L - ~ N +is, a Jacobi field if and only if the components J h E Lk of J = J+ J - satisfy the equations ---f

--f

+

V+V,J +

=0

,

V+V+J -

+ R ( j ,J + ) . j

=0

.

(2)

From (2) it follows that if Jo = 0 and J $ 0, then either JZ or J F never vanishes on R \ (0). Hence exp, Iz) has trivial kernel for any w E L-IN+. In general, exp : L-IN+ + M is a local diffeomorphism and we use the map exp-l to identify a sufficiently small neighborhood ZA c M of N + with a tubular neighborhood V c L-IN+ = T * N + of the zero section. Consider a system of coordinates (4') on N + and the associated system of coordinates (q',p,) on T * N + . Then the integral leaves N' of the distributions Lh are identified with the submanifolds

N+

=

{ p1 = ...=p n = O } ,

N-={ql= ...=qn =O}

(3)

and the symplectic form w is of the form w = Z;dp, A dqJ. From dw = 0, we get that w = &dp,

A dq3 for some smooth function f ( q , p ) and the

5 def a coordinates (qi,p$ = $) are so that w = dp$ A dqi, i.e. they represent a system of symplectic coordinates. From (3), in such system of symplectic coordinates, the Lagrangian distribution L- is of the form L- = span{

&}.

111) Let M = G / K be a 2n-dimensional homogeneous space of a Lie group G and ( w , T M = L+ + L - ) a G-invariant bi-Lagrangian structure. Assume, for simplicity, that the maximal integral leaves N * ( z ) , z E M , are simply connected. As in (11) we can choose a diffeomeorphism cp : U c M Uc T*R" between an open neighborhood of U of a point x E M and an open set U of the cotangent bundle T*R"of Rn such that: ---f

i) cp*(w) = w,,, = dpi A dqi; ii) cp,(L-) is the vertical distribution on U c T*R"; iii) any transformation g E G induces (via the diffeomorphism cp) a local fiber preserving symplectic transformations ij of U c T*Rn.

2. Invariant Bi-Lagrangian structures on adjoint orbits of semisimple groups and gradations of semi-simple Lie algebras In the following, for any semisimple Lie algebra g, we will denote by B the Killing form of g. 2.1. Homogeneous symplectic manifolds of a semisimple

Lie group In this section, we describe the structure of homogeneous bi-Lagrangian manifolds of a semisimple Lie group.

Theorem 2.1 (Kirillov-Kostant-Souriou). Let ( M = G/K, w ) be a symplectic homogeneous space of a semisimple Lie group G. Then, up to a covering, M is the adjoint orbit M = AdG h = G / K C g, with K = Zc ' h ) , of some element h E g = Lie(G). Moreover, for any 2 E M c g, the symplectic form w is given by

w z ( X , Y ) = B ( Z , [ X , Y ] ),

for any X , Y E T z M

cg .

Theorem 2.2 (Hou-Deng-Kaneyuki-Nishiyama4). Let G be a semisimple (real or complex) Lie group and ( M = AdG h, w ) an adjoint orbit of an element h E g = Lie(G), equipped with invariant symplectic structure w. The manifold M admits a bi-Lagrangian structure if and only if h is a semisimple element.

6

A Z-gradation g = g-k

+ . . . + g-1 + go + g1 + . . . + g k ,

c gi+8j

[*",8j]

(4)

of a semi-simple Lie algebra g is called fundamental if the subalgebra g- = * - k

+ ...

+ g-l

is generated by g-'. The following theorem reduces the classification of homogeneous biLagrangian manifolds of a (complex or real) semisimple Lie group G to the description of fundamental gradations of its corresponding Lie algebra g.

Theorem 2.3 (Alekseevsky-Medori2). Let ( M = A d c h = G / K , w ) be as in the previous theorem and let t = Z,(h) = L i e ( K ) . There exists a natural one-to-one correspondence between

i) invariant hi-Lagrangian structures T M = L+ @ L-; ii) K -invariant decompositions (called bi-isotropic) of the Lie algebra (5)

g=n-+t+n+,

where n* are subalgebras such that Bin, = 0; iii) fundamental K-invariant Z-gradations (4) with go = t. More precisely, the bi-isotropic decomposition (5), which corresponds to the fundamental gradation (4), is given by n* = &,ogi and t = go, while the bi-Lagrangian decomposition T M = L+ L- associated with (5) is the natural invariant extension of the K-invariant decomposition TOM = n+ + n- of the tangent space of M = G / K , o = eK (under the standard identification T O M = g/t = n+ n-).

+

+

2.2. Fundamental gradations of a (complex or real)

semisimple Lie algebra g Let g = b+ColER go, be a root space decomposition of a complex semisimple Lie algebra g with respect to a Cartan subalgebra 0. We fix a system of simple roots II = { c Y ~ ,. . , a t } c R, that is a basis of b* such that any root CY E R has integer coefficients with respect to II of the same sign (20 or 5 0 ). Any disjoint decomposition II = no U II' of II defines a fundamental gradation of g as follows. First, define the function d : R -i Z by

dlno

=0

, dint = 1 , d ( a ) =

kid(ai) ,

for any

cy

=

kicq E R .

7

Then the fundamental gradation is given by a E R , d(a)=O

aER, d(a)=1

Notice that any fundamental gradation of g is conjugated to a unique gradation of such a form. Any real semisimple Lie algebra g is a real form of a complex semisimple Lie algebra 7 , that is 0 is the fixed point set 0 = g' of some antilinear involution CI of g, i.e. an antilinear involutive map CI : g --f g, which is an automorphism of g as a Lie algebra over R. We can always assume that CI preserves a Cartan subalgebra IJof g and induces an automorphism of the root system R. A root a E R is called compact (or black) if C I = ~ -a. I t is always possible to choose a system of simple roots II = {all ,a t } such that, for any non-compact root a , E II, the corresponding root c~a,is a sum of one non-compact root a3 E II and a linear combination of compact roots from n.The roots a, and a3 are called equivalent. +

.

9

Proposition 2.4 (Alekseevsky - Medori'). Let g be a complex semisimple Lie algebra g, cr : g + g an antilinear involution and g" the corresponding real form. The gradation of g, associated with a decomposition II = no U II', defines a gradation g" = C(gz)"of g' i f and only if 111 consists of non-compact roots and any two equivalent roots are either both in no or both in II*. 2 . 3 . Examples of fundamental gradations

2.3.1. Fundamental gradations of 5K(V)

+

+

Let V be a (complex or real) vector space and V = V 1 . . . V k a decomposition of V into a direct sum of subspaces. I t defines a fundamental gradation 51(V) = C:=-,gz of the Lie algebra 5I(V), where g i = { A E 5I(V), AVj

c V i + j ,j

= 1 , . . . , Ic }

.

Any fundamental gradation of sI(V) is of this form. 2.3.2. Fundamental gradations of g2

The root system of the complex exceptional Lie algebra g2 has the form R = { & ~ i , k ( ~-i ~ j ) i ,, j =, 2 , 3 } where the vectors ~i satisfy ~1

+ + EZ

~3 =

0, ~i 2 = 2/3,

( ~ i , ~= j ) -1/3,

i

#j .

8

Consider the system of simple roots I2 = { a 1= corresponding system of positive roots is

- E ~ , a2

=

E~ - E ~ } The .

R + = { 0 1 , ~ 2 , (+~a 12 , 2 a 1 + a 2 , 3 ~ 1 + ( ~ 2 , 3 a 1 $ 2 a 2 } . There are three fundamental gradations for the complex Lie algebra 82. For any of such gradations, we give below the subset 111 c II and the level sets Ri := { a E R, d ( a ) = i } of the grading function d : R + Z. 1) nl = II:

Ro = 0,

R1 = { ( * 1 1 , ~ 2 } , R2 = {a1 + Q},

R3 = {2a1+ a2}, R4 = { 3 a i + w } , R5 = {3ai 2)

R

+2 ~ ~ 2 ) ;

n1= {a1}:

0 -

-

2

{a2},R1= { a l , a l + a z } , R = {2al+a2},R3= (3oli+C~,3ai+2a2};

3) IIl = {az}:

Ro = { w } ,R 1 = (

~ 2a1 ,

+ a 2 , 2 0 1 -I-a2,3a1 + a2}, R2 =

{hi

+ 2~x2).

There are just two real forms of the complex Lie algebra 8 2 : the compact form, which has no non-trivial gradation, and the normal form g,; which has a diagonalizable Cartan subalgebra and no compact roots. The above listed gradations of the complex Lie algebra 8 2 define three gradations of the real Lie algebra 8.; 3. Generalized Monge-AmpBre equations and effective forms on contact manifolds

Let J' = be the space of 1-jets of real functions u : Rn -+ R, with standard coordinates ( q l , . . . , qn, u , p l , . . . , p n ) and natural contact form 19 = d u - pidqi. V. Lychagin' associates with any differential n-form 0 E P ( J 1 )a differential operator

A 0 : Cw(Rn;R)4 O n ( R n ) ,

ef

Ae(h)

0lj1(h)

, for any h E Cw(Rn;R)

called Monge-Ampdre operator determined by 0. Denote by F e ( z , j 2 ( h ) )=

F e (x,h, s, ah

the unique function such that

Ae(h)= F e ( x , j 2 ( h ) ) d x 1A . . . A d z n . The corresponding second order equation F e ( s , j 2 ( h ) )= 0 is called generalized Monge-Ampdre equation associated with 0.

9

Example 3.1. (i) Let n = 2 and 0 = dpl A dp2. The associated equation is the standard Monge-Ampkre equation F g ( x , j 2 ( h ) )= det = 0. (ii) Let 0 = C i ( - l ) i S 1 d p i A d z l A . . . i.. .Adz". The associated equation is the Laplace equation h

F@(z,j2(h)) = Ah = 0 . Two n-forms 0 and 0' that define the same Monge-AmpBre operators are called equivalent. Recall that the restriction w d 0 1 ~of d0 to the contact distribution V := ker 0 is non-degenerate. Denote by 7 be the Reeb vector field on J1, i.e. the vector field defined by

ef

6(7) 1 , 7J d 6 = 0 . A differential n-form 0 , on ~ J1 is called efSective if T_JOeff = O ,

(U)-'J

(0,fiI.O) =

(6)

o.

(7)

Theorem 3.2 (Lepage - Lychagin5). A n y n-form 0 E O n ( J 1 )is equivalent t o a unique effective n-form @,fit given by 0 ,= ~ 0 - .IyA (TJ 0 )

+ d6 A

0 1

where 01 is a uniquely defined ( n - 2)-form.In particular, the generalized Monge-Ampkre equations are in natural one-to-one correspondence with the effective n-forms on J1. Let now M be a (2nfl)-dimensional manifold endowed with a contact form 0 and denote by I and V the Reeb vector and the contact distribution ID = ker9 determined by 0. We call effective form any n-form 0 , E~R n ( M ) that satisfies (7). By Darboux's theorem, there are local coordinates (qi,u , p j ) on M such that 0 = d u - C p i d q i . In other words, we can always locally identify ( M ,0) with the space J1,equipped with the standard contact form. In particular, any such local identification allows to consider generalized Monge-Ampkre equations associated with effective n-forms on M . 4. Invariant Monge-AmpBre equations on homogeneous bi-Lagrangian manifolds

4.1. Invariant efSective f o r ms o n contactifications of homogeneous symplectic manifolds Let ( M ,w ) be a 2n-dimensional symplectic manifold. A differential n-form 0 on M is called effective if u-'J 0 = 0.

10

Consider a homogeneous symplectic manifold ( M = G / K , w ) . It is possible to associate with ( M , w ) a homogeneous contact manifold (fi = G/K,O), called contactification of ( M , w ) . It is defined as a homogeneous principal R-bundle over M = G / K , with an invariant connection form 8 : T M + R, whose curvature form d0 is equal to do = n*w (see e.g. Ref.l). By definition, 8 is a G-invariant contact form on 2. Such a homogeneous contactification is uniquely determined (up to a covering) and described as follows. The LieLgebra of is a central extension -git =isRZ g of g, with Lie brackets [.,.] defined by

+

I v

[ X , Y ]= w e K ( X , Y ) Z

+ [ X , Y ],

X , Y E g = Lie(G) ,

where X , Y E X ( M ) are the infinitesimal transformations associated with X , Y E g, while 0 is the unique &invariant contact l-form such that O,K(z) =

1,

BeKlg = 0

.

By construction, the Reeb vector 7 of 8 is the infinitesimal transformation 7 = 2associated with Z and the contact distribution D coincides with the unique G-invariant distribution D e K = Span{ X e K

7

X 6g

1

Proposition 4.1. Let (A4 = G / K , w ) be a homogeneous 2n-dimensional

symplectic manifold and (M = G / K ,0) the associated contactification. Then, there exists a natural 1-1 correspondence between G-invariant effective n-forms 0 on M and G-invariant effective n-forms O M on M . Furthermore, if the subgroup K is connected and the homogeneous manifold M = G / K is reductive, with a reductive decomposition g = t + m , then the invariant effective n-forms on M are in natural one to one correspondence with the ado-invariant exterior n-forms 0 , E hnm* that are effective, i e . so that w ; L J 0, = 0

where we denoted by w,

E

,

h2m* the value of w at the point o = e K

(8) E

GIK.

From this proposition, the problem of describing the invariant effective nforms on the homogeneous contact manifold (&' = G / K ,0) and associated invariant generalized Monge-AmpBre equations reduces to the classification of ado-invariant effective n-forms on the vector space m. In the next section, we describe such invariant effective forms on homogeneous bi-Lagrangian manifolds of the normal real form of the exceptional group Gz.

11

4.2. Invariant effective f o r m s o n homogeneous

bi-Lagmngian manifolds Consider the homogeneous bi-Lagrangian manifolds MI, M2 and M3 of the normal real form G; of the exceptional group Ga, associated with the fundamental gradations ( l ) ,(2) and (3) in s2.3. The manifolds Ma and M3 are both 10-dimensional. The space of invariant 5-forms on M3 is 2-dimensional. In order to give explicit expressions of two generators 0 and 0’ for this space, let us pass to the complexification and consider a basis (E:) of (mc)*, which is dual to a basis of root vectors (E,) of m@ = Czfogi. Let also

In this notation, one generator is

+

while the other 5-form 0’ is obtained from 0 by changing all signs into - and vice versa. Both forms 0 and 0’ are effective, i.e. satisfy (8) w.r.t. to the unique (up to a scalar multiple) invariant symplectic form w, of M 3 . The manifold M2 also has a 2-dimensional space of invariant 5-forms. A pair of generators is given by the 5-form

A

and the form

(E3*al+az

A

E+3al-Cxz

- E3*Cx1+2az A

E*3~l-2Cx2)

6’,obtained from 6 by replacing any element E ~ 1 , 1 + k 2 by ,2 6 nor 6’ is effective w.r.t. to the unique

E T k l a l - k z a zHowever, . neither

invariant symplectic form of M2. Finally, consider the 12-dimensional manifold M I . This manifold admits several invariant 6-forms and a 2-dimensional space of invariant symplectic forms. The following is an example of an invariant 6-form, which is effective w.r.t. any of the invariant symplectic forms:

-

0 = E3*a1+2azA E;al+az

E L , A E r a , A ETa,-a, A ET-3al -012

’

The complete classification of invariant symplectic forms and related effective forms can be obtained by direct computations.

12

References 1. D. V. Alekseevsky Contact homogeneous spaces Funktsional. Anal. i Prilozhen. 24 (4)(1990) - Engl. trans]. in Funct. Anal. Appl. 24 (4) (1991), 324-325. 2. D. V. Alekseevsky and C. Medori, Bi-isotropic decompositions of semisimple Lie algebras and homogeneous bi-Lagrangian manifolds, to appear in J. of Algebra (2007). 3 . R. L. Bryant, Bochner-Kihler metrics, J. Amer. Math. SOC.14 (2001), 623715. 4. Z. Hou, S. Deng, S. Kaneyuki, K. Nishiyama, Dipolarizations in semisimple Lie algebras and homogeneous para-Kahler manifolds, J. Lie Theory 9 (1999), 215-232. 5. V. Lychagin, Lectures on Geometry of Differential Equation - Part I, Quaderni del Consiglio Nazionale delle Ricerche, Roma, 1992.

13

ON DARBOUX INTEGRABILITY I.M. ANDERSON and M.E. FELS Department of Mathematics, Utah State University, Logan, Utah 84321, USA E-mail: Ian. AndersonOusu. edu, Mark. FelsOusu. edu P.J. VASSILIOU School of Information Sciences and Engineering, University of Canberra, Canberra, ACT 2601, Austrailia E-mail: Peter. VassiliouOcanberra. edu. au This article is an introduction t o Darboux integrability in terms of a recently developed theory of superposition.

Keywords: Darboux integrability; Superposition formula.

1. Introduction Certain non-linear differential equations have a geometric property known as Darboux integrability which allows one to write explicit formulas for their general solutions. One of the most well known Darboux integrable equations is the Liouville equation

uXY= -2e".

(1) This equation will be used throughout the article to demonstrate the theory. Other examples include the hyperbolic non Monge-Ampere equation

+

3 3uzxuyy 1 = 0,

the system of PDE UXY =

z',VY

- uxuy

2+2eu

I

uxy

=-

uxuy

+ uxuy

2+2eu ' which are the harmonic map equations for a map from Minkowski space to lR2 with metric tensor &(du2 d v 2 ) ,and the A,-Toda Molecule

+

P

14

where Ic,p is the Cartan matrix for the semi-simple Lie algebra A, = sl(n+ 1,R).(Liouville’s equation is the case A1.) In Anderson, Fels and Vassiliou (AFV)’ a notion of Darboux integrability is introduced which goes well beyond that for a scalar PDE’s in the plane. It is shown that this class of equations admit a superposition formula which ultimately produces their explicit solution. In rough terms a superposition formula for a system of differential equations A is a pair of differential equations A and A , together with a map C : A x A A which takes a pair of solutions to A and A and produces solutions to A. The wave equation uZy= 0 of course admits the superposition formula u = f (x) g(y). The PDE’s in this case are just vy = 0, w, = 0 , for the two functions v,w of the two variables x , y, and the superposition map C is just the sum of the solutions ‘u = f ( x ) , w = g(y). Let’s compare this to the general solution to the Liouville equation (1) given by --f

+

Thought of as a superposition formula, (2) combines solutions to the same two PDE ‘uy = O,w, = 0 for the two functions v , w of two variables x,y. However, this superposition formula involves not only the solutions u = f(x), w = g(y), but also the derivatives of the solutions u ‘, = f’(x), w y = g’(y)! Furthermore the superposition formula (2) is a highly non-linear combination of the solutions and their derivatives. I t is this level of generality that we have in mind when we define a superposition formula. The main result in AFV’ is then the following.

Theorem 1.1. If a system of PDE A is Darboux integrable, then there exists a superposition formula,

c:Ad+a which is (locally) surjective onto solutions of A . There are three issues in the statement of this theorem which require explanation. (1) What is Darboux integrability? (2) Given A , what are and A? ( 3 ) How is C constructed?

A

The first two of these issues are fairly easy to describe, while the last one is quite lengthy and is the central issue of AFV.l

15

2. Pfaffian Systems and Superposition

In order to give a precise definition of Darboux integrability and the notion of a superposition formula, we will represent differential equations as Pfaffian systems. More generally exterior differential systems can be used AFV' . A standard reference for differential systems i s Bryant et. aL2 A (constant rank) Pfaffian system is a constant rank sub-bundle I c T * M . An integral manifold of a Pfaffian system I is an immersion s : N 4 M where

s * ( I )= 0. A system of differential equations A can be geometrically encoded as a Pfaffian system I on a manifold M ,and solutions to A are integral manifolds of I . Example: Let (x,y, u,u,,u,,u,,,u,,, u,,) be the standard coordinates on the 8-dimensional manifold 5' (R2,EX). The rank 3 contact (Pfaffian) system on P ( R 2 , ~ )is,

C = { du - uxdz- UydY, dux - uxxdX - uxydy, du,

- UxydX- uyydy }. (3)

Liouville's equation (1) defines a 7 dimensional sub-manifold Ad7 with coordinates (z,y, u,u,,u y lu z x ,u,,),while the restriction of the contact system ( 3 ) to A47 defines the rank 3 Pfaffian system I = (0, Ox,S,} where

6'

=

du - u,dx

-

uydy, Qx

= du,

-

ux,dx+ 2e"dy, 19, = du,

+ 2e"dx - uyvdy.

(4) A solution u = F ( z , y ) to the Liouville equation (1) determines the 2dimensional integral manifold of I

(x,Y , U

= F ( x ,y ) , u X = d,F,u,

= ~,F,u,, = ~,"F,u,, = 8;F).

We now define a superposition formula for a Pfaffian system. Definition 2.1. Let I be Pfaffian system on a manifold M . A superposition formula for I is

(1) a pair of Pfaffian systems W on M ,and W on Ml (2) and a smooth map C : x -i M such that

C * ( I )c W e3 W . A superposition formula for integral manifolds then holds for C.

16

Lemma 2.1. Let 6 : N -+ M, and s : N -+ M be integral manifolds of W ,W respectively. Then the map s : N x N --+ M given by 4 x 1 Y) = =qi(x), d(Y)) satisfies s * ( I )= 0. Proof. Suppose B E I , then s*o =

(6, S)*

c*0.

The superposition principle (2) in Definition 2.1 implies that C*O = (8,8), and 9 E I&’. Therefore where 8 E

s*$ =

i*e + s*g = 0

because 6 and S are integral manifolds of (1)of Definition 2.1).

r/f/ and

respectively (property 0

Remark 2.1. Lemma 2.1 implies that the map s is an integral manifold of I provided that it is an immersion. This will always be the case if Annihilator(W @ L$’) n ker C, = 0. Example: To give a superposition principle for Liouville’s equations let

A2 = M

= J3(R,R),

and let W , W be the rank 3 contact structures W = { 6’” = dv

W = { 8”

-

vxdx, 0: = dux - vxxdx, Ozz

= dw - wydy,

= dux,

- vlxXdx},

13: = dwy - Wyydy, OFy = dWyy- wyyydy}.

The map C : J 3 ( R , R ) x J3(R,R) + M7 defined on the set {v 0, wy > 0 ) is

(5)

# w,wx >

C(x, 21, v x ,v x x ,v x x x ; Y, w, wy, wyy, WYYY) = v x wy ,uX= D X u , u y= D y u , u X x= DZU,U,, = D ~ u ) , (G Y, = 1% (v - w)2 (6) where for example Dxu= v,,v;~ - 2vx(v- w)-l. The first term in checking c * ( Ic ) W 8 W is

17

and the others are similar. Finally, if v = f(x) is a generic integral manifold of ,'$I and w = g(y) a generic integral manifold of W , then

is the general solution to Liouville's equation in ( 2 ) .

Remark 2.2. A superposition formula C : 2 x & -+ lM can sometimes be thought of as an integrable extension, a covering of M or a potential form. The map C in (6) with @ '&I is almost by definition an integrable extension of I on M T .It can be viewed as a covering (with three dimensional fibre) by taking D

=

{

a, + v,a, + . . . , a, + wya, + . . . }

as the Cartan distribution on the domain space, and checking that C,D is the Cartan distribution for the Liouville equation. The map C can also be thought of as giving a potential form for Liouville's equation. Let u = log(cb,/(a - b ) 2 ) , and suppose that the three functions ( a ,b, c) of (x,y) satisfy the system of PDE

a,

=

c , ay = 0 , b, = 0 , cy = 0.

To verify this is a potential form, we need to check this system is compatible with Liouville's equation. The only non-trivial compatibility condition is

a-b

2c2 (-by) ( a - b)2

--

cu,,

=

-c(2e"

+uZy).

To illustrate the nature of the superposition formula for Liouville's equation we point out that the function 'uxwy (v - w)2 whose logarithm appears in the solution to the Liouville equation, is the differential invariant of the prolongation of the (local) action of S L ( 2 , R ) on J'(R,R)x J'(R,R)given on J o ( R , R ) x J o ( R , R ) by

The map C in (6) is the quotient map for this SL(2,R)action on J 3 ( R ,R)x J3(R, R). Generalizing this observation is the following theorem.

Theorem 2.1. Let G be a symmetrg group of the two PfaBan systems W on & and l W on hi. Suppose that the quotient map IT : i$l x M -+

18

( M x M ) / G is a smooth submersion, then T is a superposition principle for the quotient (I&' I@)/G,

+

(I&'+ W ) / G = { e E ~ * ( ( h ;xr M ) / G ) 1 T * e E I&'+ W 1. For more details on quotients of PfatEan systems see Ref. [3]. 3. Darboux Integrability and Superposition

An intermediate integral for a partial differential equation in the plane

uzy = f(x1 Y l u, ux1 .y)

(7)

is a function

satisfying

dF

-= O

or

dF

-

=o

dx dY on solutions to the PDE (7). The PDE (7) is Darboux integrable if there exists two intermediate integrals F, G such that

-d _F dx

-

dG = 0 , 0, dY

and

dF dG f 0 , -# 01 dux, ...

auyy ...

when (7) is satisfied. Example: For Liouville's equation uxy = -2e" the functions

are intermediate integrals. This is easily checked by computing for example

Therefore Liouville's equation is Darboux integrable. The classical method of Darboux for integrating Liouville's equation proceeds by noting that any solution to Louville's equation may be obtained from the Frobenius system uxy = 2eu, u,, - zu9 1 = a(x), uyy -

pY 1 2 = b(y)

for some choice of a(.) and b ( y ) . However it does not appear to be possible to produce the solution ( 2 ) for arbitrary f and g by integrating this way.

19

Remark 3.1. In general, Darboux’s method for the PDE (7) requires the integration of the PDE (7) together with the (compatible) PDEs

F

= ~ ( x )G ,=

b(y)

where F and G are intermediate integrals. Vessiot4 attempted to provide a mechanism that would produce the explicit solutions to Darboux integrable equations. Vessiot’s idea were remarkably different than Darboux’s method and it is Vessiot’s ideas that we have extended in AFV.’

Definition 3.1. A Pfaffian system I c T*M is Darboua: Integrable if about each point x E M there exists an open set U and a local co-frame T*U = { P ,B”, T Z , w a } where I ( U ) = { P ,0.) with structure equations

d P = A f C w bA w‘, d9” = BYk,’ A T~ dub = O ,

mod 1

d d =O

(8)

where Agb,Bj”rc,E C*(U), and 1 5 i 5 m 1 , l 5 a 5

m2, rn1,mz

2 2.

Definition 3.1 is a generalization of the definition of Darboux integrability for a hyperbolic exterior differential system given by Bryant, Griffiths, and H s u . ~Intermediate integrals Fb,Gj can be identified by writing the closed forms wb,+ in (8) locally as wb = d F b ,

$

= dGj.

Example: The Pfaffian system of Liouville’s equation (4) satisfies

d6’ = 0, d%, = w’

A

w2

mod I

d8, = 7r1 A n2

where w1

= dx, w

2

= d(u,,

-

1 -u:), r1 = d y , 2

2

7r

ZU,).

= d(u,, - 1 2

Therefore I is a Darboux integrable Pfaffian system. The Pfaffian systems (or differential equations) in the superposition principle for Darboux integrable equations are given by restricting the original Pfaffian system to a level set of one set of the invariants (see [l]), A ~ = { ~ E M F ~I = F ; } , I @ = I I ~ & =I {PE M

I G’

=Gi},

I&’=

11~.

(9)

20

Example: For Liouville's equation (1) we have

Ms

= { p E M7

The restriction of I in (4) to =

&!

I x =0

1 2

u,, - -US = 0 }.

is the rank 3 system

+

{ du - uydy,du, - uyydyr dux 2e"dy }.

By making the change of variables on A& y = y, u = u z ,uy = -2e-",

zlyy

= -2e"uy, uyYy =

-2eu(u,,

+ ui),

it is easy to check that M with W can be identified with an open set of J3(R,R)with its standard contact structure. A similar result also holds for r/tr on A?f. By the superposition Theorem 1.1,any solution of Liouville's equation can be written as a superposition of a pair of integral manifolds of these contact systems. The truly remarkable fact about a Darboux integrable system I is that it admits a superposition principle constructed from the restricted Pfaffian systems r/i/ and r/tr in (9). In the construction of the superposition principle we have identified a group, called the Vessiot group, for which the superposition principle for I can be constructed along the lines of Theorem 2.1. The construction of the action of this group is rather lengthy. It should also be noted that the Vessiot group is generally not a subgroup of the symmetry group of the original Pfaffian system I . The details are given in AFV.'

Acknowledgments This work has been supported by NSF DMS-0410373.

References 1. I.M. Anderson, M.E. Fels, P.J. Vassilliou: A Generalization of Vessiot's Integration Method for Darboux Integrable Exterior Differential Systems, in preparation (2007). 2. R.L. Bryant, S. S.Chern, R. B. Gardner, H. Goldschmidt, P. A. Griffiths: Essays o n Exterior Differential Systems, Springer-Verlag,New York, 1991. 3. I.M. Anderson, M.E. Fels: Exterior Differential Systems with Symmetry, Acta. Appl. Math. 87, 3 - 31 (2005). 4. E. Vessiot: Sur les e'quations aux de'riue'es partielles du second ordre, F(x,y,z,p,q,r,s,t)=O, intkgrables par la me'thode d e Darboux, J. Math. Pure Appl. 18, 1 - 61 (1939). 5. R.L. Bryant, P.A. Griffiths, L. Hsu: Hyperbolic exterior differential systems and their conservation laws, Part I, Selecta Math., New series 1, 21 - 122 (1995).

21

INTEGRABLE SYSTEMS IN SYMPLECTIC GEOMETRY ESMAEEL ASADI and JAN A. SANDERS Vrije Universiteit Faculty of Sciences, Division of Mathematics De Boelelaan 1081a 1081 HV Amsterdam The Netherlands E-mail: asadi0few.vu.nl , jansa0cs.vu.nl In this article, we show that if one writes down the structure equations for the evolution of a curve embedded in an 4n-dimensional symplectic manifold with zero curvature, this leads to a Nijenhuis operator for an integrable scalar-vector evolution equation generalizing the known cases of the vmKdV equation and the noncommutative scalar KdV. The procedure also gives us the symplectic and Hamiltonian operators. Keywords: Integrable system; Cartan structure; Lie algebra; symplectic geometry.

1. Introduction We study the connection between the motion of a curve and the theory of integrable equations. It was shown recently in' that if one writes down the structure equations for the evolution of a curve embedded in an n-dimensional Riemannian manifold with constant curvature, this leads to a symplectic, a Hamiltonian and a Nijenhuis (or hereditary) operator. This gives us a natural connection among finite dimensional geometry] infinite dimensional geometry and integrable systems. The goal of the present paper is to generalize this analysis to symplectic geometry. We do this by replacing R by W,the skew field of the quaternions. The Riemannian and symplectic geometries are formally similar, in the sense that we can identify any quaternionic number with a 4 x 4 real orthogonal matrix, so that we can still work with a natural moving frame. Since we work over a skew field, we derive as a byproduct noncommutative integrable system. It turns out that a Nijenhuis operator, a symplectic op-

22

erator and a Hamiltonian operator naturally come out of the analysis just like in the Riemannian case. This confirms our working hypothesis that the approach is applicable to any Cartan geometry. The paper is organized as follows. In section 2 we introduce symplectic geometry. In section 3 we compute the structure equations in the symplectic case while we choose a connection and obtain a Hamiltonian operator, symplectic operator and a Nijenhuis operator, leading t o the integrable equations. In section 4 we express all the geometric operators in terms of Lie algebra bracket, Killing form and projections. In section 5 we compare our equation with some known results. Remark 1.1. Complete proofs of the results in this paper can be found in the PhD thesis which is being prepared by the first author (E.A.) under supervision of second author (J.A.S.).

2. Moving frame method in symplectic geometry We define the symplectic Lie group over a quaternionic algebra by

G = Sp(n

+ 1) = {A E G L ( n + 1,W)IA*A = I } ,

G,

in which A t = Aij E W. Then the Lie subgroup H = Sp(1) x Sp(n) of G will be closed so that the pair (G,H ) will define a Klein geometry. The Lie algebras of G and H are g = 5 ~ , + ~= { A E G L ( n

+ l,W)IA* + A = 0},

lj = SP, x SP,.

As is known in the literature, M = G / H is then a smooth manifold. In fact M is the quaternionic projective space WP"-l. For a comprehensive reference, see.' Similar to the situation in Riemannian manifold,' given a curve in M , we know its tangent vectors D , and want to compute all possible Dt. Let w be Cartan 1-form with its values in the Lie algebra g. We make a specific choice of ~ ( 0and %leave ) w ( D t ) as a general element of g. We see that the dimension of M is equal to the dimension of g/b which is easily computed to be 4n. With x taken to be the arc length parameter, the dimension of the space of differential invariants in w(D,) describing the curve must be one less than the dimension of the manifold, that is, 4n - 1. 3. Cartan structure equation in symplectic geometry

Now let us choose a Cartan matrix w(D,) similar to that of the parallel coframe in Riemannian geometry with proper dimension counting as fol-

23

lows:

Here u is purely imaginary, and u E Elnp1.

Remark 3.1. Other choices of coframe tend to destroy the scalar-vector character of the analysis and complicate matters tremendously, which seems to be one of the main reasons why the n-dimensional analysis using FrenGt frames never took of. We see that this matrix is parametrized by 4n - 1 real parameters. Notice that here we have taken the curvature and torsion part of Cartan form in one picture. Now w ( D t ) must be a typical element of g which we write as follows:

4Dt)=

):;-=y-

mll m12 m22 ml m2

(m21

In the Riemannian case, if we use a parallel frame and assume constant curvature N ,this can be taken zero and we still can derive all the geometric quantities. Therefore we have taken the curvature equal to zero. Hence the Cartan structure equation evaluated at the evolutionary vector fields D,, Dt is as follows:

D,w(Dt) - Dtw(D,)

+ [4Dt),w(Dz)l= 0.

Before we explore the Cartan structure equation, let us define some notation. Commutators of vectors and scalars are defined by

Cum22 := umg2 - m22u,

Cum2 := (u,m2) - (m2,u ) ,

where the inner product (., .) is the Hermitian inner product. Right multiplication by scalar u on vector h and left multiplication by vector u on scalar h are defined respectively by

R,h

=

hu,

L,h

=

uh.

On the other hand, the anti-commutator on vector and scalar are defined by

A,h = (u, h)

+ (h, u),

A,h = uh + hu.

Now we explicitly write the components of the Cartan structure equation. Among these equations, the four first equations are concerned with

24

the curvature and the last three with the torsion. These equations lead to evolution of the scalar invariant u and the vector invariant u as combination of geometric operators applied on the proper torsion variables of w(Dt) according to the proposition below, in which we have defined Ejl as the operator acting on vectors by Ejlh = (DG1(hut- uKt))u, where, for instance, hut is the outer product of a vector and a covector, that is, a matrix. Hence, for instance, we can write M u = fjlrn2

Dxmll - m 1 2 - mz1 = 0 DXm22 - Dtu - Cumzz Cuma m12 Dxm2 - Dtu RUm2 film:!- L,m22 D,M - m2ut+ u E Z t= 0

+ +

+

+

+ +

m21 ml

=0 =0

D X m l - m2 - umzl = 0 Dxm12 Dxm21

+ +

mll

- m22

mll

- m2z

+ m12u - (ml,u) = 0 umzl + (u, =0. ml)

-

Solving these equations we obtain

Proposition 3.1. The evolution of differential invariants can in the form

where

-Lu and

(

C, Dx+Ru+Ejl

12 D" - 14C, -+A,D;

J=

-;L,D;~;A,

-

) , .=( :Au

;L,

- C,)D;l -L,Dil

(20,

;Cu+ ;uD;lA, D,

+ ;L,D;~A,

Remark 3.2. If we subtract the equation (If) from (lg), then that m21- m12 =

0,'

In fact, the expression on the right is the Killing form of two proper matrices in the Lie algebra g and this indeed appeared in the Lie algebra form of J as described here and in the next section. Compare this equation with the equation (13.1) (the equation after (13) in that paper) in the

25

n-dimensional Riemannian case using the parallel coframe in' and equation (12) in 3-dimensional Riemannian geometry using the F'renGt frame in.3 The difference with the later paper is that we use 0;' instead of dividing by curvature /E in there.

Hence the evolution in the proposition takes the following form:

(g;:)

= tri

(;) + (;)

If we make the specialization

(i)

=

,

% = AJU-'.

(z:)

(3)

, where u1, u1 are the deriva-

tives of u and u with respect to x, respectively, then we obtain the noncommutative evolution equations:

{

+ u2u) + $(u,U)Ul+ (u,u1)u + +u(u,u1) + $cuu2, + 2u1(u1 + + 2(u,u)).

1 3 ut = xu3 + ,(-uu1u

+2u(u1, u) Ut = u3

+ ;u2u

-

- uu2

i(U1,

u)u

$212

Definition 3.1. The pairing between

(L)

and

(i)

is defined by

in which cr is a section of g/b subject to the zero constant curvature condition into the subvector space of the Lie algebra g generated just as the Cartan matrix w(De). For instance, one can take

The adjoint of the operator P is defined by

Since the pairing is nondegenerate, P* is well-defined. Using the explicit formula for the Killing form, one can prove that the operators fi and J are

26

indeed skew-symmetric, that is, rj* = -4 and 3* = -3. The following lemma shows that in fact %rj is also a skew-symmetric operator.

Lemma 3.1. W e have that %rj = 4%'. Hence we can write the operator 8 in the form of

8 = (3%') (%-1*3%-'). Using the pairing we defined and the fact that the Killing form is invariant under the adjoint action:

+

K([XY , ] 2) , K ( Y ,[X, Z]) = 0 for X, Y ,Z E g, and that d 2 K = 0 where d is the boundary operator of forms on the Lie algebra, one can, although not easily, prove:

Theorem 3.1. The operators 4U*,Q-1*3U-1 and 3 are Hamiltonian, symplectic and Nijenhuis operator, respectively. Furthermore, the operators U and 2-l are Nzjenhuis operators, that is, their Nijenhuis tensor is zero. 4. The geometric operators in terms of Lie bracket, Killing

form and projections In the method we are using, the only tools we have are the Lie algebra and the Cartan geometry, hence we expect to be able to write the geometric operators rj and 3 in terms of the Lie bracket, the Killing form and proper projections. Let us define the projections 7r0 and 7r1 as follows:

and

; y;;) (-;):

mll

Tl(

0

0

0 0

0

=

Let C and m 2 be the projection of w ( D z ) and w ( D t ) over the Lie subalgebra 9, respectively and r i z l , the projections of w ( D t ) over the vector space g/g which itself is indeed the dual orthogonal of Q with respect to the Killing form. Then we simply find that

+

= 6(7rl??~)' h o ,

f i ~ g= TladA??Il,

27

where

fi = Dx- nlada - adaD;'7roadfi. This is exactly the Poisson operator (1.131 in4 which in general is defined on symmetric spaces. Now the torsion part gives in fact the following matrix equation:

Since ad: # X I for any X E R,we can not solve equation (4) in the usual way. Therefore the existence of the Nijenhuis operator B plays a crucial rule in the symplectic case. Notice that in the Riemannian case we do have ad; = -I. In order to get rid of this difficulty, we define two projections p1,po which in fact split off the scalar and vector part:

(; -") (em;. 0) 0 0

PI

% ?;

0 0 0

=

(00;-); 0 0

,

Po

(y2-+) 0 0

0 =

Now one can show that Gt = fi%?Zo

+ Qih2.0,

in which the Lie algebra form ! I of geometric operator J and % ! of Nijenhuis operator Q appears as

5. Reduction to scalar part

If we take vector part u = 0, the equation becomes: 1 4

ut = --u3

3 + -(uu1u 8

-

uuz

+ uzu),

which is the noncommutative mKdV equation and the Nijenhuis operator becomes:

28

as found by V.V. Sokolov e.a.5 using the Lax method. From our results it also follows that the Hamiltonian operator is:

nR1 = ( 2 0 , - CU)D,l(D,- CU), and the symplectic operator nU-1*R#r1

=

(20, - CJIDZ(DZ

-

L U ) ( 2 D ,- C J 1 ( D Z

+ R,)D,(2DX - c p .

Remark 5.1. In6 the quaternion version of the reduced scalar equation is given in equation (3.17). If we identify u E fi with a vector in ii E R3, then 1 R = - ( D z - 2 i i x . ) ( D , - ii x . + iiD-yii, .))(1- ii x D ; y . 4

The operator (1- G x D;'.)-', when applied t o symmetries of the equation, only contains on expansion as many terms as the order of the symmetry.

Remark 5.2. The Lax operator in the symplectic case is

L = D,

+ AG + 6.

Then the Sokolov-Drinfeld method can be applied to the Kac-Moody algebra, as a Z-graded Lie algebra, based on the symplectic Lie algebra

B = *P,+1

.

Acknowledgments Thanks go to the Netherlands Organization for Scientific Research (NWO) for their financial support of the project Geometry and classification of integrable systems (project number: 613.000.315). We thank Dr Jing Ping Wang (University of Kent at Canterbury) for very useful discussions. References 1. J. A. Sanders and J. P. Wang, Mosc. Math. J . 3,1369 (2003). 2. R. W. Sharpe, Differential geometry, Graduate Texts in Mathematics, Vol. 166 (Springer-Verlag,New York, 1997). Cartan's generalization of Klein's Erlangen program, With a foreword by S. S. Chern. 3. J. A. G. Mari Beffa, Sanders and J. P. Wang, J . Nonlinear. sci. 3,143 (2002). 4. C.-L. Terng and G. Thorbergsson, Results Math. 40, 286 (2001), Dedicated to Shiing-Shen Chern on his 90th birthday. 5 . M. Gurses, A. Karasu and V. V. Sokolov, J . Math. Phys. 40, 6473 (1999). 6. P. J. Olver and V. V. Sokolov, Comm. Math. Phys. 193,245 (1998).

29

SINGULARITIES AND COLLISIONS IN N-BODY TYPE PROBLEMS V. BARUTELLO and S. TERRACINI Dipartamento di Matematica e Applicationi, Universitd di Milano Bicocca, Milano, 20125, Italy E-mail: vivina. [email protected] : Susanna. [email protected] This expository paper gathers the contents of talks given by the authors in the conference SPT2007. It is focused on the recent results on the structure of the singular set of generalized solutions to n-body type problems extending the classical Von Zeipel Theorem and Sundman asymptotic estimates. Keywords: Singularities in the N-body problem; locally minimizing trajectories; collisionless solutions; logarithmic potentials.

We are concerned with dynamical systems of the form

where the forces are induced by a positive potential U , possibly timedepending, undefined on an attractive singular set A, that is (UO) lim U ( t , x )= fm, uniformly in t. x-A The set A is assumed to be a cone. This type of systems play a main rSle in Celestial Mechanics and in other areas of Classical Mechanics, and the singular set (that is for example the set of collisions between two or more particles in the n-body problem) plays a fundamental r61e in the phase p ~ r t r a i t ' ~and > ' ~strongly influence the global orbit structure. Two are the major steps in the analysis of the impact of the singularities in the n-body problem: the first consists in performing the asymptotic analysis along a single collision trajectory (see S ~ n d m a nWintner37 ,~~ and, in more recent years, Sperling, Pollard, Saari, Diacu and other aut h o r ~ ~ ~The~second ' ~ step ~ consists ~ ~ ~in blowing-up ~ ~ ~ ~the ~singularity ~ ~ ~

)

.

30

by a suitable change of coordinates introduced by McGeheeZ5 and replacing it by an invariant boundary - the collision manifold - where the flow can be extended in a smooth manner. In many interesting applications it turns out that the flow on the collision manifold has a simple structure: it is a gradient-like, Morse-Smale flow featuring a few stationary points and heteroclinic connection^.^^^'^ The analysis of the extended flow allows us to obtain a full picture of the behavior of solutions near the singularity, despite the flow fails to be fully regularizable (except in a few cases). The geometric approach, via the McGehee coordinates and the collision manifold, can be successfully applied also to obtain asymptotic estimates in some cases, such as the collinear threebody problemIz5the anisotropic Kepler p r ~ b l e m , ' ~ ~ ' the ~ J t~hJr e~e b o d y problem both in the planar isosceles case" and the full perturbed three-body.l3>l5Besides the quoted cases, however, one needs to establish the asymptotic estimates before blowingup the singularity, in order to prove existence of the blow-up through the convergence of a family of rescaled trajectories. The reason is quite technical and mainly rests in the fact that a singularity of the n-body problem needs not be isolated, for the possible occurrence of partial collisions in a neighborhood of the total collision. In the literature, this problem has been usually overcame by extending the flow on partial collisions via some regularization technique (such as Sundman's,12 or Levi-Civita's''). Such a device works well only when partial collisions are binary, for these are the only globally removable singularities. Thus, the extension of the geometrical analysis to the full n-body problems finds a strong theoretical obstruction: partial collisions must be regularizable, which is known to hold true only in few cases. Other interesting cases in which the geometric method is not effective are that of quasi-homogeneous potentials (where there is a lack of regularity for the extended flow) and that of logarithmic potentials (for the failure of the blow-up technique). In this paper we summarize the main results contained in Ref.5, where the authors extend the classical asymptotic estimates near collisions in three main directions. First of all, they take into account a very general notion of solution, the so called generalized solution, for the dynamical system (1), which fits particularly well to solutions found by variational techniques. Second, they extend the analysis to a wide class of potentials including not only homogeneous and quasi-homogeneous potentials, but also those with weaker singularities of logarithmic type. Finally, potentials are allowed to depend on time in the following way

31

in order to take into account models where masses vary in time. The theory we expose can be applied to the following potentials:

Homogeneous isotropic potentials:

U,(t, x) =

c

mi ( 1x2

i<j

t b j(t)

- xjp

and in particular to the n-body problem (with time depending masses) potential ( a = 1). n

Logarithmic potentials: i,j=l

which arises in the study of systems of n almost-parallel vortex filaments.

a-p homogeneous potentials: (where 0 < p < a and A E potentials.

+ XU~(x),

U,,p(z) = Ua(x)

EX) as Manev, Lenard-Jones, Van der Waals

c n

Anisotropic n-body potentials:

U ( t ,x) =

U i , j ( t ,xi - xj)

i<j i,j=l

as the Gutzwiller potential (electrons with time-depending masses in Coulomb field spherically symmetric). We can also consider C1 perturbations of the previous potentials. To proceed we need the following definitions.

Definition 0.1. We say that the (generalized) solution % for the dynamical system (1) defined on (a,to) has a singularity at t o < +m if it is not possible to extend % as a solution to some interval ( a , t l ) with tl > t o . If t o is a singularity for Z and lim,-,; 3 ( t ) ,exists, then we say that has a collision at t o . We proceed step by step keeping in mind the following questions. ( Q l ) Under which assumptions a singularity is a collision? (Q2) How can collisions occur on a generalized solution? ( A priori our solutions can have huge sets of singularities, we just know that is a zero-measure set). ( Q 3 ) When can we exclude the occurrence of collisions?

32

Locally minimal and generalized solutions We will deal with the following class of trajectories.

Definition 0.2. A path x E H/o, ( ( a ,b ) , R k )is a locally minimal solution (or minimal in the sense of Morse) for (1)if, for every t o E ( a ,b), there exists 60 > 0 such that the restriction of x to the interval 10 = [ t o - SO,to SO], is a local minimizer for the lagrangian action functional on the interval 10 with respect to compactly supported variations (fixed-ends). A path x is a generalized solution if there exists a sequence x, of locally minimal solutions such that zn -+ x uniformly on compact subsets of ( a ,b) and for almost all t E ( a ,b) the associated total energy converges.

+

The fundamental property enjoyed by a generalized solution is (see Ref.5, Proposition 2.9) that there exists a sequence of solutions 5, of smooth problems (i.e. with regular potentials) such that

0 0

x, -+ 3 uniformly; xe + 2 in L2 and a.e.

The main motivation for the study of generalized solutions comes from the variational approach to the study of selected trajectories to the nbody problem. Indeed the exclusion of collisions is a major problem in the application of variational techniques as it results from the recent literature, where many different arguments have been introduced to prove that the trajectories found in such a way are collisionless.’~2~4i6~7~g~20~27~28i33.3” As a first application we shall be able to extend some of these techniques in order to prove that action minimizing trajectories are free of collisions for a wider class of interaction potentials (as we will see in the last section of the present paper; see also Ref.5, section 5). Besides the direct method, other variational techniques - Morse and minimax theory - have been applied to the search of periodic solutions in singular problem^.^^^^^^^^^ In the quoted papers, however, only the case of strong force interaction2’ has been treated and solutions are understood as limits of sequence of solutions to penalized problems where an infinitesimal sequence of strong force terms is added to the potential. Since these solutions are generalized solution (according to Definition 0.2) our main results apply also to this class of trajectories.

33

Asymptotic analysis along a total collision trajectory The first step consists in a detailed study of the motion of a total collision trajectory near the collision (this means that Iz(t)J-+ 0 as t tends to the collision time, to).First of all we prove the following conservation laws (both to be intended in the distributional sense)

dU

h(t) = -(t,%) at 1 .. 0

(conservation of the energy);

+

- I ( % ( t )2 ) 2h(t) ( 2 - & ) U ( t ,% ( t )) CZ (Lagrange-Jacobi). 2

In particular from the Lagrange-Jacobi inequality we deduce that total collisions are isolated among total collisions. In order to exclude also the accumulation of partial collisions near a total one, we prove some asymptotic estimates near total collisions (we refer to Theorems 2,3,4 & 5 in Ref.5). To do that we need some further assumptions on the potential U ; we will not enter in the details of such hypotheses, anyway they are satisfied by the potentials we choose as main examples. With A := (2 + a ) - ' , the behavior of the radial part of the motion is proved to be

r(t)

-

[K(to- t)IzA, f ( t )

-

-2KA[K(to- t ) ] - a A

K>O

while the angular part tends to the set of central configurations (critical points of the potential U constrained on the configuration with moment of inertia equal to 1). The asymptotic analysis along a total collision trajectory allows us not only to exclude partial collisions for neighboring times (since we have lim,,,o r"U(t,rs) = b > 0) but it also implies that a total collision solution, near the total collision, satisfies the dynamical system (1). Partial collisions Our aim now is to extend the asymptotic estimates to partial collisions (that is when a generalized solution 3 approaches x* E A, x* # 0). We suppose that the singular set A is the union of linear subspaces and that the potential U verifies some further assumptions; then we prove a generalization of Von Zeipel's The~rem,~' that gives the answer to (Ql). Theorem 0.1. If the moment of inertia is bounded, then every singularity

of a generalized solution admits a limiting configuration; hence all singularities are collisions.

34

Then we reduce from partial (even simultaneous) collisions to total ones by decomposing the system in colliding clusters. Next we have the following result, that explains how collisions can take place in a generalized solution and answers to (Q2).

Theorem 0. 2. A bounded generalized solution a: o n a bounded interval has at most a finite number of collisions. Furthermore, the asymptotic estimates proved for total collisions hold for each (maximal) colliding cluster.

Blow-up and averaging estimates The last step consists in the exclusion of the occurrence of collisions in some classes of generalized solutions. The results concerning this topics require some homogeneity properties of the potentials which are satisfied by our first three examples, but not by the last one (the anisotropic problem). For quasi-homogeneous potential the proof develops in the following way: first of all we construct a blow-up solution (of a limiting autonomous problem) which has the same minimality properties of the starting collision solution; then we prove that such blow-up can not be minimal exhibiting a variation that makes the action functional decrease. In order to do that we can use the method of averaged variations of a family of variations parameterized on a sphere introduced by M a r ~ h a l . When ~ ? ~ ~dealing ? ~ ~ with logarithmic-type potentials, the blow-up technique is not available since converging blow-up sequences do not exists; we can anyway prove that the average over all possible variations is negative by taking advantage of the harmonicity of the function logIx1 in EX2. With this result we can then extend to quasi-homogeneous and logarithmic potentials all the analysis of the (equivariant) minimal trajectories carried out in the paper by Ferrario and the second author.20

References 1. A. Ambrosetti and V. Coti Zelati. Periodic solutions of singular Lagrangian systems. Birkhauser Boston Inc., Boston, MA, 1993. 2. G. Arioli, V. Barutello, and S. Terracini. A new branch of mountain pass solutions for the choreographical 3-body problem. Comm. Math. Phys., 268:439463, 2006. 3. A. Bahri and P.H. Rabinowitz. Periodic solutions of Hamiltonian systems of 3-body type. Ann. Inst. H. Poincare' Anal. Non Line'aire, 8:561-649, 1991. 4. V. Barutello, D.L. Ferrario, and S. Terracini. Symmetry groups of the planar

3-body problem and action-minimizing trajectories. Arch. Rational Mech. Anal. To appear.

35

5. V. Barutello, D.L. Ferrario, and S. Terracini. On the singularities of generalized solutions to n-body type problems. Ara7iv:rnath/O701174, preprint (2006). 6. V. Barutello and S. Terracini. Action minimizing orbits in the n-body problem with simple choreography constraint. Nonlinearity, 17:2015-2039, 2004. 7. U. Bessi and V. Coti Zelati. Symmetries and noncollision closed orbits for planar N-body-type problems. Nonlinear Anal., 16:587-598, 1991. 8. A. Chenciner. Action minimizing solutions of the newtonian n-body problem: from homology to symmetry. In Proceedings of the ICM, Peking, 2002. 9. A. Chenciner and R. Montgomery. A remarkable periodic solution of the three body problem in the case of equal masses. Ann. of Math., 1522381-901, 2000. 10. R. L. Devaney. Collision orbits in the anisotropic Kepler problem. Invent. Math., 45:221-251, 1978. 11. R. L. Devaney. Nonregularizability of the anisotropic Kepler problem. J . Differential Equations, 29:252-268, 1978. 12. R. L. Devaney. Triple collision in the planar isosceles three-body problem. Invent. Math., 60:249-267, 1980. 13. R. L. Devaney. Singularities in classical mechanical systems. Progr. Math. 10,Birkhauser Boston, Mass., 1981, 211-333. 14. F. Diacu. Regularization of partial collisions in the N-body problem. Differential Integral Equations, 5:103-136, 1992. 15. F. Diacu. Near-collision dynamics for particle systems with quasihomogeneous potentials. J . Differential Equatzons, 128:58-77, 1996. 16. F. Diacu. Singularities of the N-body problem. In Classical and celestial mechanics (Recife, 1993/1999), pages 35-62. Princeton Univ. Press, Princeton, NJ, 2002. 17. F . Diacu and M. Santoprete. On the global dynamics of the anisotropic Manev problem. Phys. D, 194:75-94, 2004. 18. F. Diacu, E. PBrez-Chavela, and M. Santoprete. The Kepler problem with anisotropic perturbations. J . Math. Phys., 46:072701, 21, 2005. 19. M.S. ElBialy. Collision singularities in celestial mechanics. S I A M J . Math. Anal., 21:1563-1593, 1990. 20. D.L. Ferrario and S. Terracini. On the existence of collisionless equivariant minimizers for the classical n-body problem. Invent. Math., 155:305-362, 2004. 21. W.B. Gordon. A minimizing property of Keplerian orbits. Amer. J . Math., 99:961-971, 1977. 22. T. Levi Civita. Sur la regularisation du probkme des trois corps. Acta Math., 42:39-42, 1920. 23. P. Majer and S. Terracini. On the existence of infinitely many periodic solutions to some problems of n-body type. Comm. Pure A p p l . Math., 48:449-470, 1995. 24. C. Marchal. How the method of minimization of action avoids singularities. Celestial Mech. Dynam. Astronom., 83:325-353, 2002. 25. R. McGehee. Triple collision in the collinear three-body problem. Invent.

36

Math., 27:191-227, 1974. 26. R. Moeckel. Some qualitative features of the three-body problem. In Hamiltonian dynamical systems (Boulder, CO, 1987), volume 81 of Contemp. Math., pages 1-22. Amer. Math. SOC.,Providence, RI, 1988. 27. R. Montgomery. The geometric phase of the three-body problem. Nonlinearity, 9:1341-1360, 1996. 28. R. Montgomery. Action spectrum and collisions in the planar three-body problem. In Celestial mechanics (Evanston, IL, 1999), volume 292 of Contemp. Math., pages 173-184. Amer. Math. SOC.,Providence, RI, 2002. 29. H. Pollard and D.G. Saari. Singularities of the n-body problem. I. Arch. Rational Mech. Anal., 30:263-269, 1968. 30. H. Pollard and D.G. Saari. Singularities of the n-body problem. 11. In Inequalities, 11 (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), pages 255-259. Academic Press, New York, 1970. 31. H. Riahi. Study of the critical points at infinity arising from the failure of the Palais-Smale condition for n-body type problems. Mem. Amer. Math. SOC., 138(658), 1999. 32. D. G. Saari. Singularities and collisions of Newtonian gravitational systems. Arch. Rational Mech. Anal., 49:311-320, 1972173. 33. E. Serra and S. Terracini. Collisionless periodic solutions to some three-body problems. Arch. Rational Mech. Anal., 120:305-325, 1992. 34. H.J. Sperling. On the real singularities of the N-body problem. J. Reine Angew. Math., 245:15-40, 1970. 35. K. F. Sundman. Memoire sur le problbme des trois corps. Acta Math., 36:105179, 1913. 36. S. Terracini and A. Venturelli. Symmetric trajectories for the 2n-body problem with equal masses. Arch. Rational Mech. Anal., 2006. To appear. 37. A. Wintner. The Analytical Foundations of Celestial Mechanics. Princeton Mathematical Series, v. 5. Princeton University Press, Princeton, N. J., 1941. 38. H. von Zeipel. Sur les singularitks du problhme des n corps. Ark. Math. Astr. F ~ s . ,4:l-4,1908.

37

COMPUTING CURVATURE WITHOUT CHRISTOFFEL SYMBOLS S. BENENTI Dipartimento d i Matematica, Universitd d i Torino, Via Carlo Alberto 10, 10123 Torino, Italy E-mail: sergio. benentiOunito.it The Riemann tensor of a given metric, of any dimension and signature, can be computed ’by hand calculation’, avoiding the explicit calculation of the ( 1 / 2 )n2(n+1) Christoffel symbols. The algorithm presented here works with n quadratic form & a in the velocity-variables coming from the Lagrange geodesic equations, and with 2 n cubic forms R6 and R; generated by them. An example of this method is illustrated: it concerns the application of Geodesic Equivalence theory to General Relativity.

Keywords: Riemannian geometry; Geodesic equivalence

1. The six steps of the aIgorithm We start by describing the algorithm; comments will be given later on. Step 1. 0 Take the covariant components g . . of the given metric tensor, and write a? the “Kinetic energy” K = gij vz v3. 0 Compute the Lagrange binomials d dK Li:= _-

d t dvi

-

dK dqi’

_ .

Step 2. 0 Compute the inverse matrix [gzj] of

0

Take the quadratic form in

Li:Qi

[ g i j ] .0

Do not extract the the n quadratic forms Qi

:= I’kkuhvk. 0

$ n2(n+ 1) Christoffel symbols! We work with

only. Step 3.

Rise the index of Li:

0 Compute the total formal derivative of Qi w.r.to a “new time” dQi dqi dvi = i j a and =Qi. t, Ri:= -=-, by setting dt df dt

38

+

Step 4. 0 Split R2 into the sum R2 = Rh RZ,:R: is the part of R2 which does not contain the terms Q'. In RZ, replace all Q' by Q2 = - Q a ( ( v ) i.e., put v2 H 3 in Q 2 and change the sign. Result: Rh are 3rd-degree homogeneous polynomials in vhvaZIJ,and RZ, are 3rd-degree homogeneous polynomials in vhijafi~. Step 5.

Compute the A-symbols: A'[,,,

d3Rh

:= 1

damduedun d3RZ, Compute the B-symbols: B2emn:= - 1 4 dun afie Step 6. 0 Compute the Riemann tensor:

R2emn = A'emn -

iB\mn

- B\nm.

2. Explanation (1) La have the form L2 = (2) RLhave the form

$ + rZ,,vh vk.

R' = d m q n vm ve vn + 2 rimQ k = d m q n vm ve vn - 2 rim

ve vn

(3) The Riemann tensor is defined as (see Eisenhart,' p.19)

Raemn:= amI?in- dnFjm + rim

-

rinr!m.

If we introduce the symbols A",,, := dmrjn and B'e,, then the Riemann tensor can be written

R2emn = Aaemn - A'tnm

:=

FZ,mr~nl

+ B'emn - Bhnm.

(1)

(4) The expression R' = dmr$,vmvevn - 21'imI'!n;;,eiPvmin item 2 shows that these symbols can be obtained by three partial derivatives, as in the definitions of Step 5. Proof. After the splitting Ra = dmrjn am ve vn + 2 I'Le Q k ve it follows that

= Rh

+ Ri ,

39

+

Remark 2.1. The splitting Ri= Rb Rf is useful for shortening the "by hand" calculations. It is not necessary when using a software. Indeed, d3Rh aum due avn

d 3Ri aum due dun'

d2 R;

avn w a v m

d2 Ri

-

avn

3. An example

Let us apply the algorithm to a kinetic energy of the kind

K = L2

gaa

(va)'

+ i f hap

V"

v'.

(2)

The coordinates are divided into two subsets: (qi) = ( q a , q a ) . The Roman indices a , b, c , . . . assume values from 1 to nR.The Greek indices a , p, y,. . . assume values from nR 1 to nR+n, = n.Moreover, the components gaa and the conformal factor f depend on the Roman coordinates only, while the components of the metric tensor hap depend on the Greek coordinates only. The interest of such a metric will be explained below.

+

3.1. The computation of the quadratic f o r m s Q i From ( 2 ) we get:

d dK

--

dt dva

= f hap

dvp dt + f d,hap

The Lagrangian binomials are dva La = gaa - &gaa V b V a dt

+

dV'

La = f hap dt

+ f d,h,,

v p V'

+ daf hap V'

-

f dagbb (Vb)'

V'

V'

-

(3)

va.

f d a f hap V"

f da f hap v p va -

Up,

4 f dahp, v p vy.

Then, the quadratic forms Qi are given by

Q"

=

gaa

(dbgaa V b V a -

Q" = h a p (d,h,p

-

4 dagbb (Vb)'

f d,hp,)

-

$ 8 ,f hap V a U p ) ,

+ V" da log 1f 1 va.

V P vUy

(4)

Since, for the moment, we do not specify the expressions of gaarf and ha', we can continue our calculation by observing that the Q Z assume the form A

Q a = I ' ~ c v b ~ c f-afh , p v a v P

Q a = F & ~ p ~ Y + ~ " ~ a(5) Fa,

40

where ?gc and F;, are the Christoffel symbols of the metric respectively, and f a := gab a b f = gaa a a f , Fa := a a log I f [ . 3 . 2 . The computation of the cubic forms

gab

and hap,

Ri

According to Step 3, from (5) we get

R" =

R" =

dQ"

dt

h

= aeF&ubuCiie -

8, f a hap ua up f i e

- fa

a,h,,

ua up ij7

+ 2 ?tcub Qc - f a hap v" Q p.

dQ* = a,i-.g, + a,Fa dt + Fa ( Q a + ua Q"). vy iip

V a v a iie

+ 2 ~ g p, , 87

V"

According to Step 4, we have

3.3. The computation of the A-symbols

According to Step 5, we have to compute the partial derivatives of Rb with respect to the variables ui and iji. In doing this 6-symbols will arise systematically, since (avi/avj) = (aiji/aiij)= 6;. However, the calculation

41

can be shortened if we consider that

CaX , 6; -rim= 0, h

= Xe,

CaYa6,"

= Ye.

when one of the two indices (e, m) is Greek. hem = 0 when one of the two indices (t?, m) is Roman. f a = 0, Fa = 0. - f a , Fa depend on the Roman coordinates only. r;,, hap depend on the Greek coordinates only.

r&,= 0,

Then, from (6) we derive

aR,a = d,fgCvbvC dii" = d,f$&

dmf a hapva up - f a amhap va u p ,

upvy + d,F,

va va.

Hence, according to Step 5, the A-symbols are

3.4. The computation of the B-symbols We have to compute the partial derivatives of Rq. By a calculation similar to that done for the A-symbols, and still taking into account the rules ( 8 ) , we get

42

Hence, according to Step 5 , the B-symbols are

3.5. The Ricci tensor Following Eisenhartl (see p.21), the Ricci tensor is defined by Rem = Riemi. Hence, according to ( l ) ,

In our example, the Ricci tensor components are the sum of A-terms and B-terms, Rem = ARem BRem,with

+

Starting from (9) we get

We then consider the three essential cases (b,c), (A, p ) , (A, b) of the pair (elm ) ,and the result is the following

h

-

being Rbc and Rxp the Ricci tensors associated with the metrics g a a and hap , respectively.

43

4. How to use this example

Assume n = 4, n, = n, = 2:

Since n,

= 2,

the components (13) become

Now we assume that gaa and

f have the form

gaa = e, (c - ua)(u2- ul),

f = (c - u ' ) ( c - u 2 ) ,

(16)

where ua is a function of qa only, c is a constant and e, = *l, according to the signature. As shown in my Tutorial Paper2 for SPT-2004 (Appendix A, Theorem A.4.1), a metric of this kind admits an equivalent metric. Two metrics g and g on a manifold Qn are said to be equivalent if they have the same unparametrized geodesics. A metric g admits an equivalent metric ij iff it admits a (non-singular) J-tensor J. A J-tensor is a torsionless trace-type conformal Killing tensor. If there exists a J-tensor J with eigenvalues (ui), then there exist standard coordinates ( q i ) such that the components of g and J assume a certain standard form and such that ui(qi).This form depends on the multiplicity of the eigenvalues. The metric (14)-(16) is just one of the four possible cases, for n = 4, corresponding to the case in which two eigenvalues are simple and one is double. The equivalent metric g i j can be computed by means of J , following the rule (described in Ref.2): 1 A,. g . .- -

P2

p := det[Jk],

[A;] = cof [Jk],

23'

where cof[-] is the cofactor-matrix operator. A crucial property of the geodesic equivalence is that the aerie parameters t and f of two corresponding geodesics are related by the equation d t l d t = p. Hence, a spontaneous question arises: can Metric Equivalence Theory find any application in General Relativity? In other words, can an empty (i.e., Ricci-flat) space-time admit an equivalent metric?

44

We can give a first partial answer to this question by considering the metric (14)-(16) and by imposing the condition Ri, = 0: this metric is Ricci-flat if and only if h

Rbc =

-

i (Fb Fc + 2 ?bFc> ,

RxP

+

eaf

a

hxP = 0.

(17)

It can be shown that for an orthogonal metric of the kind (16), kbc = 0 for b # c. Then the first equation (17) is equivalent to A

vaFb = --+FaFb, The second equation (17) has the form

a # b.

-

RAP = i n h x p ,

t~ := -

(18)

Oafa,

where fix, and hxP are functions of the Greek coordinates, while n is a function of the Roman coordinates. It follows that n is a constant and the two-dimensional manifolds qa = constant are Einstein manifolds. It can be shown that, while Eq. (18) is identically satisfied, the second equation (17) is satisfied iff (i) ua = constant and (ii) n = 0. This means that, (i) gaa and f are constant and, (ii) the Ricci tensor is zero and the submanifolds qa = constant are flat (since they have dimension 2). Hence: the metric (14)-(16) is Ricci-flat and admits an equivalent metric if and only if it is flat. This shows that geodesic equivalence does not occur in General Relativity, at least in the case considered here - one of the four possible cases. But it can be conjectured that this happens also for the remaining three cases. A complete discussion of this matter, with detailed calculation, will be available on my personal web-site.

Credits. Bibliographical references and more recent results about the Geodesic Equivalence Theory can be found in Ref.2 . I wish to thank L. Fatibene, of my Department, for testing this method of computing the curvature tensors by a software. In spite of several simplification routines of the output sheet, the final formulae were really cumbersome. On the contrary, the “by-hand” calculation has shown the advantage of several step-by-step significant simplifications, associated with a better understanding of the meaning of the written formulae. References 1. L.P. Eisenhart, Riemannian Geometry, Princeton University Press, Fifth

printing (1964). 2. S. Benenti, “Special Symmetric Two-Tensors,Equivalent Dynamical Systems, Cofactor and Bi-cofactor Systems”, Acta Appl. Math. 67 (2005), 33-91.

45

SURVEY OF STRONG NORMAL-INTERNAL k : t, RESONANCES IN QUASI-PERIODICALLY DRIVEN OSCILLATORS FOR .t = 1 , 2 , 3 H.W. BROER* and R. VAN DIJK Institute for mathematics and computing sciences, University of Groningen, The Netherlands *E-mail: [email protected] http://www.math.rug.nl/lbroer R. VITOLO

Dipartimento da Matematica e Informatica, Universiti d i Camerino, Italy E-mail: renato.vitoloQunicam.it Recently, semi-global results have been reported by Wagener6 for the k : C resonance where C = 1 , 2 . In this work we add the e = 3 strong resonance case and give an overview for C = 1 , 2 , 3 . For a n introduction t o the topic, as well as results on the non-resonant and weakly-resonant cases, see Refs. 1,2,6.

Keywords: Quasi-periodically driven oscillators; DufSlng - Van der Pol oscillator; 'Stoker's response problem'; strong resonance.

1. Introduction to the problem We consider the forced Dufing - Van der Pol oscillator, given by

i

xj=

w3,

j = l ,. . . ,m ,

Y l = y2, $2 = - ( ~ + C $ ) Y Z

(1) - by1 - d y ~ + & f ( 1 ~ 1 , . . . , ~ m ~ ~ 1 , ~ / 2 , a , b , c , d , & ) ,

which is a vector field defined on T m x JR2 = { (z1, . . . , x,), (y1, ~ 2 ) ) .Here f is a smooth function which is 2~-periodicin its first m-arguments. We take a = ( a , b ) as parameter, E as perturbation parameter and ( c , d ) as coefficients. The internal frequency vector w = ( ~ 1 ,. .. ,w,) is to be nonresonant, or quasi-periodic. The search is for quasi-periodic response solutions, i.e. invariant m-tori of (l),with frequency vector w , that can be represented as graphs y = y ( z ) over I = T" x (0). For large values of la1 we can use a contraction argument. However, if la/ g; (b)-(c) Blowups of the indicated rectangles. Phase portraits for the regions labelled 1 , .. . ,9 are given in Fig. 6.

55

:

o

:

-1.5

-1

i

,yl

r

1

I

-1.5 -0.5

0.5

0

1

I d

I5

-

2

0

2

4

(b) Region 2

(a) Region I 4/

20.

2.

d. 1

-

4

-

2

0

2

4

-

(c) Region 3

-3

-2

-1

0

1

2

,

-3

-

2

0

2

4

(f) Region 6

I

Magnification

I

3

(e) Region 5

I

4

(d) Region 4

.\ -2

-1

0

1

2

3

(g) Region 7 3

3

2

2

1

1

0

0

1

1

2

2

3

3 -3

-2

-1

0

1

2

(h) Region 8

3

-3

-2

-1

0

1

2

3

(i) Region 9

Fig. 6. (a)-(i) Phase portraits of the ‘principal part’ for the k : 3 resonance (7). The labels “Region 1 . . . 9” correspond to the regions in the bifurcation diagram in Fig. 5 . Note that two limit cycles coexist in region 6, see the magnification.

56

QUASI-PERIODIC ATTRACTORS AND SPIN-ORBIT RESONANCES A. CELLETTI Dipartimento di Matematica, Universith di Roma TOTVergata Via della Ricerca Scientijka 1, 1-00133 Roma (Italy) ‘E-mail: cellettiOmat.uniroma2.it www.mat. uniroma2. it/- cel letti L. CHIERCHIA Dipartimento di Matematica, Universith “Roma Tre” Largo S. L. Murialdo 1, 1-00146 Roma (Italy) * E-mail: luigiOmat.uniroma3.it www.mat. uniroma3. it/users/chierchia Small dissipation limits for nearly-integrable systems are considered; theorems concerning the existence of quasi-periodic attractors smoothly approaching KAM tori are presented and an application to the capture in 3:2 resonance of Mercury is discussed. Keywords: Quasi-periodic attractors; small divisors; celestial mechanics; spinorbit problem; spin-orbit resonances; nearly Hamiltonian systems; dissipative systems.

1. Introduction

Mechanical systems, in real life, are typically dissipative, and perfectly conservative systems arise as mathematical abstractions. In this note, we shall consider nearly-conservative mechanical systems having in mind applications to celestial mechanics. In particular we are interested in the spin-orbit model for an oblate planet (satellite) whose center of mass revolves around a “fixed” star; the planet is not completely rigid and averaged effects of tides, which bring in dissipation, are taken into account. We shall see that a mathematical theory of such systems is consistent with the strange case of Mercury, which is the only planet or satellite in the Solar system being stacked in a 3:2 spin/orbit resonance (i.e., it turns three times around its rotational spin axis, while it makes one revolution around the Sun).

57

2. The spin-orbit model

Let us consider the dynamics of a triaxial nearly-rigid body (planet or satellite), having its center of mass revolving on a given (fixed) Keplerian ellipse, and subject to the gravitational attraction of a major body sitting on a focus of the ellipse. For simplicity, we consider vanishing obliquity, i.e., we assume that the satellite is symmetric with respect to an “equatorial plane” and study motions having the equatorial plane coinciding with the Keplerian orbital plane (such motions belong to the invariant submanifold of vertical spin axis). Under such hypotheses, the motions of the satellite may be described by the angle z formed by, say, the direction of the major physical axis of the satellite (assumed to lie in the equatorial plane) with a fixed axis of the Keplerian orbit plane (say the direction of the semimajor axis of the ellipse; see figure).

We shall assume that the non-rigidity of the planet (meant to reflect the averaged effect of tides) is modeled by the averaged MacDonald’s torque [8]. Then, the differential equation governing the motion of the satellite, in suitable units, is given by

wherea “The conservative equation ( K = 0) is derived and discussed, e.g., in Ref. 4;compare, in = 1. The dissipative term particular, Eq. (2.2) with the normalization n := ( K # 0) is derived, e.g., in Ref. 9; compare, in particular, Eq. (21), where (as above)

d

w

58

a

a

K 2 0 (the “dissipation parameter”) is a physical constant depending on the internal (non-rigid) structure of the satellite; 0, > 0 and N, > 0 are known functions of the eccentricity e E [0, 1) of the Keplerian orbit and are given by:

5 1 + -e4 + -e16 8 ”> (1-e2)6 a €=--

a

‘

(2)

3B-A , where 0 < A < B < C are the principal moments of 2 c

inertia of the satellite; p e ( t ) and fe(t)are, respectively, the (normalized) orbital radius and the true anomaly of the Keplerian motion, which (because of the assumed normalizations) are 2~-periodic function of time t. The explicit expression for pe and f, may be described as follows. Let u = u e ( t )be the 2n-periodic function obtained by inverting

t = u - esinu ,

(“Kepler’s equation”) ;

(3)

then

p e ( t ) = 1 - ecosu,(t)

Remark 2.1. (i) For K = 0 the equation (1) corresponds to the Hamiltonian flow associated to the one-and-a-half degree-of-freedom Hamiltonian

(y, x) being standard symplectic variables. Such a Hamiltonian system (the “spin-orbit Hamiltonian model”), whose phase space is R x T2 (T2being the standard flat torus R2/(27rZ2)), is non-integrable if E > 0 andb e > 0.

(ii) For K > 0 the equation (1) is dissipative and, for solution is given by

E

= 0, the general

n = 1 and in view of our assumption about the spin axis being vertical, one has to take vanishing ex and e y components, i = 0 and Gm = I ;K is the constant in front of the curly brackets in Eq. (21); the functions n, and N , are denoted in Ref. 9, respectively, f i ( e ) and fi(e). bWhen e = 0, uo(t) = t = fo(t), po = 1 so that H = (1/2)[y2 - E cos(21 - 2 t ) ] ,which is easily seen to be integrable.

59

showing that the periodic (remember that z is an angle) solution z = cost vet, x = v, is a global attractor for the dynamics on the cylinder (phase space) R x 9, B denoting the circle Rl(27rZ). The limiting frequencyC 1 $e4 5 6 3 Ne .v, := = 1 6e2 -e4 O(e6) (7) 8 Re (1 - el312

+

+ Fe2+

+

+

+

+

will play an important role in the sequel; we notice, in particular, that it is a ) ; denote real-analytic invertible function of e mapping ( 0 , l ) onto ( 1 , ~ we by Y-' : (1,co) -+ ( 0 , l ) the inverse map (which is also real-analytic). (iii) In many examples taken from the Solar system, both E and K are small. For example, for the Earth-Moon system and for the Sun-Mercury system E is of the order of lop4, while K is of the order of (iv) A quasi-periodic solution z(t) with frequency w E R a solution of the form

\ Q of Eq.

z ( t )= w t + u ( w t , t )

(1) is (8)

where u(8) = u(81,82) is a C2 function defined on T2 (i.e., 27r-periodic in the variables 81 and 82). Notice that time-derivative for z(t) corresponds to the directional derivative

+

for the function ~ ( 8 )since ; the flow 8 E T2+ 8 (wt, t ) is dense in T2, one sees immediately that z ( t ) is a quasi-periodic solution of (1) if and only if u solves the following quasi-linear PDE on T2:

82u + 7 a,u

+ p e ("W 3 sin (2(e1 + u)- 2 fe(e2)) = q(u, - w ) , ~

(10)

where, as above 7 := KR, and v, := N,/R,. (v) The frequency w and the function V, are not independent: it is not difficult to check that if satisfies (10) then one has necessarily

where (.) denotes average over 'IT2. Eq. (11) may be interpreted as a compatibility condition. (vi) The above spin-orbit model is relatively simple (since a lot of approximations have been done), nevertheless it is rather well accepted in the =As usual, f = O ( x k )means that f is a smooth function of x having equal to zero the first k derivatives at x = 0.

60

astronomical community: for example, it has recently been used by Correia and Laskar [7] to discuss Mercury's capture in resonance. 3. Results

Standard KAM theory (see e.g. Ref. 1) implies that, when K = 0 and E > 0 is small enough, (1) admits many quasi-periodic solutions as in (8) with w Diophantine, i.e., satisfying

for some K , T > 0. Furthermore, such solutions are analytic in E and are Whitney smooth ind w . In the following, V K , T denotes the set of Diophantine numbers in R satisfyinge (12).

Theorem 3.1. Fix K , T E ( 0 , l ) and T 2 1. There exists EO > 0 such that n [l+r, 1/r], there exist for any E E [0,E O ] , any K E [0,1] and any w E VK,T unique functionsf e,

= e,(K,w) = u-'(w)

+ O ( E ~, )

u = u,(8; K, w)

= O ( E ),

with ST,ud6' = 0 , satisfying (10) with e = e,. The functions e, and u, are smooth in the sense of Whitney in all their variables and are real-analytic in 8 E T2 and E , C" in K and Whitney C" in w. Remark 3.1. (i) Theorem 3.1 implies that the 2-torus

z , ~ ( w ):= { ( z , t )= (el + u , ( ~ ; K , w ) , Q :~ ) = (e1,e2)E T ~ ,}

(13)

is a quasi-periodic attractor for the dynamics on the phase space R x T2 associated to (1) with e = e,(K,w) and that the dynamics on ?;,K(w) is analytically conjugated to the linear flow 8 4 8 (w t, t ) .

+

(ii) The result is perturbative in E but it is uniform in K. Indeed, one could replace the parameter range for K into any compact interval of R. It is particularly noticeable the smooth dependence of us on K as K + 0, dA function f : A c Rn + W is Whitney C k or C&, if it is the restriction on A of a Ck(Wn) function; for a more formal definition and for relevance in dynamical system, see, e.g., Ref. 2. eObserve that if (12) holds, then 0 < K. < 1 and T 2 1. In fact, taking n1 = 1 and 122 = -[u] ([z] = integer part of z) in (12) shows that n < 1, while the fact that T 2 1 comes from Liouville's theorem on rational approximations ("For any w E R\Q and for any N 2 1 there exist integers p and q with IqI 5 N such that Iwq -pl < l/N"). Finally, o, 'Dn,7 is a set of full Lebesgue measure. we recall that, when T > 1, U fThe map u-' is the inverse map of e + u, defined in (7).

61

which shows that the invariant KAM torus ?;,o(w) smoothly bifurcates into the attractor (13) as K # 0. (iii) The invertibility of the frequency map v, associated to the unperturbed attractor ~ , K ( w may ) be interpreted as a nondegeneracy condition, allowing to fix the eccentricities for which quasi-periodic attractors exist in the full dynamics. Notice that the parameter values e = 0 and w 5 1 are excluded. (iv) The proof of the above theorem is based upon a “Nash-Moser” method. That is, equation (10) is rewritten as F(u;e) = 0, where .F is a functional acting on functions on T2 and numbers e E (0,l); the unknowns are u and e, while E , K and w (which is taken in the Cantor set DT,-, with y and r fixed) are regarded as parameters. Then, the equation F(u;e) is solved iteratively starting by the trivial approximate solution u = 0 and e = u- ’ ( w ) . To cope with the small divisors introduced by inverting the linearized functional d 3 one introduces a scale of larger and larger Banach spaces and uses a Newton method: the speed of convergence of the method is enough to beat the divergences introduced by the small divisors. With this approach the most delicate part concerns the discussion of the solution of the linearized equation, which is a degenerate linear PDE on T2 with nonconstant coefficients. Full details are given in Ref. 6. Actually, the above Nash-Moser approach is rather robust and general; indeed it can be easily adapted to cover dissipative maps such as the “fattened Arnold family” studied in Ref. 3 or it could be extended to systems with more degrees of freedom.

A simple consequence of Theorem 3.1 is the following Theorem 3.2. For small enough oblateness E and any rigidity parameter K E [0,1] there exists a (Cantor) set of positive measure & c (0,l), which depends smoothly o n E and K , such that for any e E & there exists a unique 2-dimensional torus, which i s a quasi-periodic attractor for the dynamics governed by ( l ) , and o n which the ftow i s analytically conjugated ). to (61,62) -+ (61 w t , 62 t ) with Diophantine w = ue O ( E ~ Finally, the Lebesgue measure of € tends t o 1 as E 4 0.

+

+

+

62

4. Mercury’s capture

Many satellites in the solar system are observed in a 1:l spin/orbit resonance, i.e., while making a revolution around their primary body, they make one turn around their rotational internal axis: in this way they show always the same “face” to their primary body. The most familiar example is our Moon; other examples are: Deimos, Phobos, 10, Europa, Ganymede, Callisto, Mimas, Enceladus, Tethys, Dione, Rhea, Titan, Janus, Epimetheus, Ariel, Umbriel, ... On the other hand only one celestial body is observed in a different spinorbit resonance: namely, Mercury, which is observed in a 3:2 resonance. Explaining this anomaly is a very intriguing and actual problem; compare Ref. 7 and references therein. It is a fact that all the satellites observed in the 1:l resonance have small eccentricity (the largest is that of the Moon, which is about 0.055), while the eccentricity of Mercury’s orbit is about 0.205. Now, a look at the graph of the function e -+ v e

0.1

0.205

0.3

0.4

shows that the resonance 3:2 (corresponding to a frequency w = 1.5) is above the graph of u in correspondence to Mercury’s eccentricity e = 0.205. Now, the phase space of our system is three dimensional and twodimensional tori that are graphs over the angles ( z , t ) (such as the tori

63

arising in Theorem 3.2) separate the phase space in two invariant regions. Thus a periodic or nearly-periodic orbit with frequency 1.5 would remain trapped forever above the invariant torus. In view of the fact that, according to Theorem 3.2, the measure of the set € of eccentricities corresponding to quasi-periodic attractors is close t o 1 for small E (which for Mercury is we see that the above analytical results might indicate a rigorous explanation to the spin-orbit trapping of Mercuryg. References 1. V. I. Arnold (editor), Encyclopedia of Mathematical Sciences, Dynamical Systems 111, Springer-Verlag 3,1988. 2. H.W. Broer, G.B. Huitema, M.B. Sevryuk, Quasi-periodic motions in families of dynamical systems. Order amidst chaos, Lecture Notes in Mathematics, 1645. Springer-Verlag, Berlin, 1996. 3. H.W. Broer, C. Sim6, J.C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity 11,(1998) 667-770. 4. A. Celletti, Analysis of resonances in the spin-orbit problem in Celestial M e chanics: T h e synchronous resonance (Part I), Journal of Applied Mathematics and Physics (ZAMP) 41 (1990), 174-204. 5. A. Celletti, L. Chierchia, O n the stability of realistic three-body problems, Commun. in Math. Physics 186 (1997), 413-449. 6. A. Celletti, L. Chierchia, Quasi-periodic attractors in celestial mechanics, preprint April 2007, downloadable at -.mat .uniroma3.it/users/chierchia 7. A. C. M. Correia, J. Laskar: Mercury’s capture into the 3/2 spin-orbit resonance as a result of its chaotic dynamics, Nature 429 (24 June 2004), 848-850. 8. G.J.F. MacDonald, Tidal friction, Rev. Geophys. 2 (1964), 467-541. 9. S.J. Peale, The free precession and libration of Mercury, Icarus 178 (2005), 4-18.

gThis mechanism seems also compatible with evolutive models, which study slow changes of orbital parameters; compare Ref. 7 where it is given numerical evidence that during million of years the eccentricity of Mercury might have changed quite a bit reaching values beyond 0.3.

64

DARBOUX CONSTRUCTION OF SOLUTIONS O F INTEGRABLE PDES WITH NONVANISHING BOUNDARY VALUES

A. DEGASPERIS’, s. LOMBARDO~ Dipartamento di Fisica, Universitd di Roma “La Sapienza” and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Rome, Italy. E-mail: antonio.degasperisOroma1.infn.it Department of Mathematics, Vrije Universiteit, Amsterdam, NL. E-mail: [email protected] In this short note we review the role of the Darboux matrix transformation in soliton theory; in particular we present a Darboux construction of solutions of integrable PDEs with nonvanishing boundary values.

1. Introduction

A broad class of nonlinear evolution partial differential equations in 1 + 1 dimensions can be spectrally analysed, indeed linearised, by the Inverse Spectral Transform (IST). Their key property is to be equivalent to the compatibility condition of two linear, generally matrix, ordinary differential equations, called Lax pair1i2y3. One of the main advantages of the linear structure of Lax equations is the possibility to transform them by an important class of transformations, the Darboux transformations, first introduced by Darboux4 in 1882. These transformations, originally applied to second order scalar differential equations, such as the (stationary) Schrodinger equation, play the role of a similarity transformation of a linear differential operator and, therefore, allow one to solve a new differential equation in terms of solutions of the old one. When applied to the Lax pair, these transformations eventually lead to mapping a solution of a nonlinear partial differential equation (PDE) into a new solution of the same equation, also known as a Backlund transformation’. Iterated Darboux transformations (or Crum transformations) were considered in a work by Crum5 in 1955, where he also investigated their spectral properties. In 1975 Darboux transformations were used by Wadati‘ and collaborators to

65

generate multisoliton solutions of integrable equations associated to the ZS-AKNS spectral problem3. The role of the classical Darboux transformation in soliton theory was described by Calogero and Degasperis' in 1982. Zakharov and Shabat7 introduced a general matrix formalism for Darboux transformations (see also the textbook2) in connection with the dressing method. Further results on Darboux matrix transformations were obtained by Neugebauer and Meine18 and by Matveev and Salle". Links between dressing methods and classical Darboux transformations where investigated by Levi, Ragnisco and Symg~lo.An excellent recent account on the subject has been given by Rogers and Schief" in which great attention is paid to geometric aspects; we refer the interested reader to this monograph and to the references therein. In the next section we briefly recall basic notions of integrability theory and sketch the role of Darboux transformations in this context. Details are scattered in a vast literature, see for In Section 3 we briefly display recent results related to the construction, by means of Darboux transformations, of solutions of nonlinear multicomponent wave equations with nonvanishing boundary values; details are given el~ewhere'~ ,22.

2. Darboux matrix trasformation and integrability (1) Lax mapping. This is the starting point: it maps one-to-one the phase space of dynamical variables u into an appropriate set of linear operators L ,i.e. u L(u). Examples. Let cri, i = 1 , 2 , 3 , be Pauli matrices: 0 harmonic oscillator: u = ( q , p ) L ( u ) = qal + p a 2 ; 0 KdV equation : u = u(x) L(u) = -$ u(z); 0 NLS equation : u = (q(x),p(rc)) L(u) = -ia3&+q(.z)crl+

+

P(Xk2

'

In the following we assume that L(u) is a differential operator, as in the last two examples. (2) Eigenvalues problem. Given the linear operator L ( u )it is natural to consider the eigenvalue problem in an appropriate vector space:

L(u)v = kv. This leads to the spectra1 problem: given the spectrum of L , k E A c @ , one can define a spectral transform of u S[u] : A H @. This defines a mapping u e S[u],i.e.

u E phase space

r S[u]E spectral space.

66

( 3 Darboux t r a n s f o r m a t i o n (DT) and Backlund t r a n s f o r m a t i o n ( B T ) . If D is an appropriate linear operator (e.g. it is a differential operator if L ( u ) is also differential), the similarity transL(u2), L(u2) D = D L('LLI), correspondingly formation L(u1) acts on the eigenvectors v(u1, k ) of L(u1) as the Darboux transfor~ ( 2 ~k2) ,= Dv(u1,k ) . Darboux transformations mation v(u1, k) naturally induce a corresponding transformation on the phase space, called Backlund transformation12:

-

u1

-

u2.

Moreover, a Darboux transformation induces also a corresponding transformation on the spectral space S[UlI

Sb21.

In this respect, it is important t o keep in mind that a Darboux transformation may even change the spectrum of L(u1). (4) C o m m u t a t i v i t y of Darboux and Backlund t r a n s f o r m a t i o n s .

In the framework of soliton theory Darboux transformations can be used (a) to introduce dynamics and (b) to construct particular solutions of integrable equations. (5) Dynamics. We just sketch the way: let u(t)be a trajectory in the phase space

-

and let t E (tl,t2); one can consider an infinitesimal BT u ( t ) u(t E ) , as induced by the infinitesimal similarity transformation

+

L(t)

-

L(t + E )

= D(t)L(u(t))D-l(t),

D(t)=I

+& M ( t ) .

67

At the first order in

E

one obtains the Lax equation

{

&(t, k) = M ( t )v ( t ,k) L(t)w(t, k) = k v ( t ,k)

and the Lax pair

whose compatibility condition yields the integrable PDE zit = F(u). In fact, by choosing different DTs one obtains a hierarchy of (possibly) commuting flows (see, e.g., ref.14)

d --L(tn) dtn

= [Mn(tn),L ( t n ) ] .

We now turn our attention to the construction of solutions u(t) via Darboux transformations. This technique stems from the commutativity property of DTs and BTs, which implies that a BT leaves invariant the integrable PDE ut = F ( u ) .

3. Darboux transformations for coupled wave equations 3.1. Coupled wave equations

A natural approach to construct integrable multicomponent wave equations, which are likely to be of interest in physical applications, consists in starting from the Lax pair associated to the scalar NLS equation, and in replacing scalar dynamical variables with (generally rectangular) matrices15. In this setting the Lax mapping is

where now the field variable U is a N ( - ) x N ( + ) matrix, N(+) and N ( - ) being arbitrary, positive integers. With no loss of generality let us assume N(+)< - “-1. Here CJ is the constant diagonal matrix

and S(+)and S ( - ) are, respectively, N(+)x N ( + ) and N ( - ) x N ( - ) diagonal matrices whose diagonal elements sL*) are signs, namely

S(*) = diag (si*), . . . ,

2

) , &*I= 1 .

(2)

68

A convenient formulation of the Lax pair of matrix ODES is given here by

) x ( N ( + )+ N ( - ) ) square matrix. As for our notation: the constant coefficient y is the real dispersion parameter, y = y*,while C(O) and are constant blockdiagonal matrices,

where the blocks C ( j ) ( + respectively ), C ( j ) ( - )are , N ( + )x PI(+), respectively N ( - ) x N ( - ) , constant square matrices. The dynamical variables appear in Q = Q(z,t ) which is an off-diagonal ( N ( + ) N ( - ) ) x ( N ( + )+ N ( - ) ) block matrix of the form

+

=

(

S ( + ) u + ( tx),S ( - )

U(x,t)

ON(-)xN(-)

while the auxiliary dependent variable W

W=(

W(+) ON(-)xN(+)

),

= W(z, t ) has

O N ( + )x N ( - )

W(-)

>-

the form

The diagonal entries W ( + )W , ( - )are square matrices of dimension N ( + ) x N ( + ) and PI(-) x N ( - ) , respectively. Finally, the compatibility condition &-t = $tx provides the following nonlinear matrix partial differential equations

U, = C(O)(-)u - rJC(O)(+)-

u,- u,C(l)(+)]+ [c(1)(-)

+ [w(-) u+ uW ( + )+] iy [u,,- 2 us(+)u+s(-)u], W ( + )= C(W+),

[

(54

ut s(-) u], wJ-1 = c(l)(-), us(+) iyt s(-) .

[

1

(5b) Several systems of coupled wave equations can be extracted from this general equation (5) by a reduction technique. An instance of such reductions

69

of applicative interest reads U (l) t = buil) - s1 wu(2)

+ iy [u"

- 2 (sl

u p = -hi2)+ sq w*u(1)+ i y [U2.'

+ sq 1pq2u(1)] ) , (u(1) l2 + sq (u(2)1)' ,(2)]

IU(l)IZ

- 2 (sl

,

wz = 2 bs1 sq u(1)u(2)*, and it features a mixing of Schrodinger-type dispersion and quadratic nonlinearity as it occurs in the three wave resonant interaction. For other examples of reductions and integrable systems, as well as for details, see13i15,16. The integrability of (5), guaranteed by the associated Lax pair (3), entails the possibility to solve the initial value problem by the spectral transform (or inverse scattering) method. However, this task is certainly easier if one assumes that the solution U ( z ,t ) vanishes at the boundary, namely as 1x1 -+ 00, while in the case of nonvanishing boundary values, dictated by various physical contexts, the spectral transform method is rather involved because of the many branch points which occur in the complex plane of the spectral variable k.

3.2. Darboux transformations Let us consider the Darboux transformation k-dependent formalism)

41(z, t , k)

-

$2(&

(in the equivalent

t , k) = q z ,t , k) $1(z, t , k) ,

where

D ( z ,t , k) = I

R ( x ,t ) , +k-a!

+

+

I is the identity matrix, R ( z ,t ) is a ( N ( + ) N ( - ) ) x ( N ( + ) N ( - ) ) matrix and a! is a given parameter. As pointed out in the introduction, the corresponding Backlund transformation maps a known solution Ul of (5) into a new solution U2 of the same equation. A crucial requirement in this construction is the explicit knowledge of a matrix solution $1 of (3), associated to U1. In other words, the problem can be formulated as follows: given U l ( z ,t ) ,solution of (5), such that the corresponding solution $1(x, t, k) of (3) is known explicitly, construct the residue matrix R ( z ,t ) and therefore a new solution U2(z,t ) of (5). This problem can be easily solved in the class of solutions whose boundary value (say as 1x1 -+ cm) vanishes; in this case, the pole a! is required to

70

lie off the real axis, a* # a, and this condition simplifies the construction which is well known2. The way to solve this problem is more complicated if the solution U(x, t ) has nonvanishing boundary values. This is so because, in this case, the pole a may instead be on the real axis, a* = a. We report here on this latter case and, for the solution of the former, we refer the reader to13 for full details and to17 for an application of this technique to construct new explicit solutions of the celebrated three waves resonant interaction (SWRI) equation. Assume a = a*;this condition implies13

where p ( z ,t ) is a real scalar function, the matrix P(z,t) is the projector

and C is the diagonal matrix

It turns out that finding R ( z ,t) = ip(z,t)P(z, t) requires solving two nonlinear differential equations for the projector P, and also two additional differential equations for the scalar function p which are Riccati-like equations13. Once these equations are explicitly solved, the new solution U2 reads (see13 for details)

where the vectors z(+) and z ( - ) have dimension N(+) and N ( - ) , respectively, and obtain from the block form of the vector z ( z , t ) :

+

i.e. the known ( N ( + ) N(-))-dimensional vector

4 2 , t ) = 1cI1(z,t ,a )zo

+

7

(11)

( N ( + ) N(-))-dimensional vector. It is important to notice that these explicit formulas have been obtained merely by algebra and local integration of differential equations. Therefore, at the end of this computation, one has to deal with two features of solutions which are important in applications, namely their boundary values and their zo being an arbitrary, constant

71

boundedness. While the latter issue can be only considered a posteriori, the former, namely the boundary values, is naturally related t o the position of the pole a of the Darboux matrix D , see (7). Indeed, in the construction of soliton solutions the pole a has a spectral meaning since it belongs t o the discrete spectrum of the operator L(U2) and therefore it is constrained by the properties of this operator (i.e. by the functional class of Uz(z,t)). In general, if U ( z ,t) belongs to the class of functions which vanish as 1x1 -+ 00 (and W ( z ,t) to those (bounded) solutions which vanish as z -+ -00), then the discrete spectrum of the operator L ( U ) cannot be on the real axis and only Darboux matrices D with a # a* give rise to physically meaningful solutions. Bright optical solitons are in this class. On the other hand, if U ( x ,t ) belongs instead t o the class of those (bounded) solutions which do not vanish as 1x1 -+ 00, and are characterized by the condition”

, = I?(+) lim S(+)Ut(x,t ) S ( - ) U ( zt)

, where [dl)(+), I?(+)] = 0 ,

lxI+c=

r(+)being

a given t-independent constant N(+) x N(+) matrix (with N ( + ) 5 N ( - ) ) ,then the correspending operator admits gaps of the con-

tinuum spectrum on the real axis. Therefore, in this case, the Darboux matrix D , with a = a* in one of these gaps, provides the appropriate transformation. Dark optical solitons are in this class. This study has been motivated, among others, by the discovery of accelerated solitons known as b o o r n e r o n ~ and ~ ~ provides ~ ~ * ~ ~a ~tool for their construction and spectral description. The phenomenology of these objects is described in details in the context of nonlinear optics. Acknowledgments The present work of one of us (A.D.) has been done within the French CNRS project GDR 3073. References 1. F. Calogero and A. Degasperis, Spectral Trunsform and Solitons, Vol. 1, North Holland Publishing, Amsterdam, (1982). 2. V.E. Zakharov, S.V. Manakov, S.P. Novikov and L.P. Pitajevski, The The-

ory of Solitons: The Inverse Problem Method, Nauka, Moskow, [in Russian], (1980). 3. M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equation and Inverse Scattering Transform, London Mathematical Society Lecture Note Series, 149,Cambridge University Press, (1991). 4. G . Darboux, Sur une proposition relative auz equations line‘aires, C.R. Acad. Sci. Paris 94, 1456-1459, (1882).

72

5. M.M. Crum, Associated Strum-Liouville systems, Q. J. Math. Oxford 6,121127, (1955). 6. M. Wadati, H. Sanuki and K. Konno, Relationships among inverse method, Backlund transformation and an infinite number of conservation laws, Prog. Theor. Phys. 53,419-436, (1975). 7. V.E. Zakharov and A.B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering transf o r m , Funct. Anal. Appl., 8, 226-235, (1974). 8. G. Neugebauer and R. Meinel, General N-soliton solution of the AKNS class on arbitrary backgroud, Phys. Lett. A., 100, 467-470, (1984). 9. D. Levi, 0 . Ragnisco, A. Sym, Backlund transformation us dressing method, Lett. Nuovo Cimento, 33,401-406, (1982). 10. D. Levi, 0. Ragnisco, A. Sym, Dressing method us classical Darboux transformation, I1 Nuovo Cimento, B83, 34-42, (1984). 11. V. B. Matveev and M.A. Salle, Darboux Transformations and Solitons, Springer-Verlag, Berlin, (1991). 12. C. Rogers and W.K. Schief, Backlund and Darboux Transformations, Cambridge University Press, (2002). 13. A. Degasperis, S. Lombardo, Multicomponent integrable wave equations I. Darboux-Dressing Transformations J. Phys. A: Math. Theor., 40:961-977, (2007). 14. 0. Babelon, D. Bernard, M. Talon Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press, (2003). 15. F. Calogero and A. Degasperis, New integrable equations of nonlinear Schrodinger type, Studies Appl. Math. 113,91-137, (2004). 16. F. Calogero and A. Degasperis, New integrable PDEs of boomeronic type, J. Phys. A: Math. Gen. 39, 8349-8376, (2006). 17. A. Degasperis, S. Lombardo, Exact solutions of the %wave resonant interaction equation, Physica D: Nonlinear Phenomena, 214(2): 157-168, (2006). 18. F. Calogero and A . Degasperis, Coupled Nonlinear Evolution Equations Solvable via the Inverse Spectral Transform and Solitons that Come Back: the Boomeron , Lett. Nuovo Cimento 16,425-433, (1976). 19. F. Calogero and A. Degasperis, Novel solution of the system describing the resonant interaction of three waves, Physica D: Nonlinear Phenomena, 200(3): 242-256, (2005). 20. A. Degasperis, M. Conforti, F. Baronio and S. Wabnitz, Stable Control of Pulse Speed i n Parametric Three- Wave Solitons, Phys. Rev. Lett., 97,093901 1-4, (2006). 21. M. Conforti, F. Baronio, A. Degasperis and S. Wabnitz, Inelastic scattering and interactions of three-wave parametric solitons, Phys. Rev. E 74, 065602(R), (2006). 22. A . Degasperis, S. Lombardo, Multicomponent integrable wave equations II. Soliton solutions, in preparation.

73

TRANSITIVE DECOMPOSITION OF n-BODY SYMMETRY GROUPS D.L. FERRARIO Dipartimento d i Matematica e Applicazioni University of Milano-Bicocca; Via R. Cozza 53, 20125 Milano ( I ) E-mail: davide.ferrarioQunimib.it

1. Introduction Periodic orbits of the n-body problem can be found as minimizers of the Lagrangean action functional in a suitable subspace of the Sobolev space of all loops in the configuration space. Some periodic orbits are symmetric minimizers, that is, they are minimizers in the space of all loops symmetric with respect to a suitable group: they play an important role in the n-body problem, as they are the “simplest” periodic solutions within that class. In some sense, they are the natural generalization of equilibrium (or homographic) planar solutions, which can be seen as minimizers of the Lagrangean action functional on the space of all loops symmetric with respect to SO(2) (in the plane). The simplest example for three bodies is the Lagrange solution: an equilateral triangle uniformly rotating around its center of mass. A smaller (often discrete) group of symmetries gives non-homographic solutions, as far as the action fulfills some simple properties, as it was shown by A. Chenciner and R. Montgomery in’ (see also2). The remarkable “figure eight” (see figure 1) solution is a minimizer in the class of loops which are symmetric with respect to a suitable dihedral group. An even simpler

Figure 1.

“Figure-eight” solution

symmetry group (the so-called italian anti-~ymmetry~-~) yields non trivial

74

orbits, such as the 6-bodies orbit in figure 2. More complex groups, such

Figure 2.

6 bodies under the constraint z ( t

+ T/2)= - z ( t )

as the icosahedral group of order 60 (which turns out to be the smallest non-abelian simple group), yield accordingly more complex orbits (figure 3 - see Given the importance of finding symmetric minimizers, it

Figure 3.

60 bodies, with icosahedral symmetry

is natural to try to classify (better if constructively) all finite symmetry groups for the n-body problem. While building a unified framework for the construction and classification problem, at the same time it is useful to extend the application range of the averaging and blow-up techniques, which are at the foundation of the existence results in the variational settings. In order to do so, we will introduce an equivalence relation defined between groups, ruling out differences thought as non-substantial. Then, we will define a decomposition of a symmetry group into a sum of (something like) irreducible components, termed the transitive decomposition. This is the main result ofl5 which we will explain and apply to some simple examples. It is easy to deduce from the (algebraic and combinatorial) properties of the irreducible components some useful properties of action-minimizing periodic orbits, such as existence, being collisionless, being homographic or non-homographic, or angular momentum. We start by reviewing some aspects of group actions related to the n-body problem.

75

2. S y m m e t r i e s Let z ( t )be a trajectory in the configurations space X c (Rd),, of n particles (indexed by integers in n = {1,2,. . . , n } ) with masses ml, m2,. . . , m, 2 0. Assume d = 2 , 3 . The Lagrange action functional with homogeneous potential in a rotating frame reads as

on the time interval [O,T].It can be defined on the Sobolev space H1(R,X) c (Rd)Rxnby restrictions to compact intervals. Consider now the following groups. The affine isometry group A(R) of all affine isometries (translations and reflections) of the real time line R.The orthogonal group O ( d ) , acting on the euclidean space Rd (in the rotating frame, the group must preserve the rotation axis). The symmetric group C,, which consists of all permutations on the n indices in n. Actually we will consider only those subgroups of C, preserving the values of the masses. We consider symmetry groups of the type G c A(R) x O ( d ) x C,. The inclusion homomorpism G 4 A(R) x O ( d ) x C, induces a linear (orthogonal) action of G on If1(&!, X),and provided masses and rotation axis are preserved, the action functional A, is G-invariant. By the Palais criticality restricted to the space of all principle, it follows that critical points of equiwariant paths rc(gt) = g z ( t ) (that is, the paths in H1(R,X)G,paths fixed by G) are critical points for A. Of course, even for small groups it is not easy to prove that equivariant minimizers exist and are collisionless (seeg).

3. R e d u c t i o n s Let Iso(R) denote the group of (affine) isometries of the time line R,generated by translations and reflections. For every T > 0 there is a surjective projection Iso(R) O(T), where T = R/Tz. Let G be a symmetry group and its cower in Iso(R) x O ( d ) x C,, that is the pre-image of G via the projection -+

Iso(R) x O ( d ) x C,

-+

O(T) x O ( d ) x

c,.

The projections on the factors will be defined as T , p and (T (resp. T , p , 6 for the cover). It is possible to show that there is a natural isomorphism

H1(R, X)"

E H1(T,X ) G .

76

Consider the following diagram:

e F + I s o ( W ) x O(d)

.i

4

The image j?(E) = p(G) c O(d) is a finite space point group (hence classified). The group 6 = (7 x ,G)(G)c Iso(R) x O(d) is called the normalized cover of G. After a change in the time scale, we assume that its generator acts as t H t f 1 on the time line. The first reduction is the reduction by rotating frames: in a frame rotating around the vertical axis with fixed angular velocity 8, the equation s(t) = eietq(t)induces an isomorphism 8,

8,

:

P(R, x)-+ H ~ ( Wx), ,

defined by 8,q = x.Given G, its cover is transformed by 8, into the (unique) such that

6 and 8'are equivalent (also, we say that they are equivalent if they correpond simply up to a time scale). It turns out that for each n there is only a finite number of equivalence classes (seeloyll) and that there is a minimal model which acts without rotations on Rd.The next step is to split the action into transitive components. The main point is that possible transitive (reduced) groups are globally finite. We consider the disjoint sum of groups as follows (it is a kind of Burnside decomposition of a G-set in transitive orbits). If this happens, we say that

(1) Definition. Let GI and Gz be two symmetry groups with the same normalized cover GI = G2 c Iso(lw) x O ( d ) and permutation homomorphisms 61 : GI+ En,, 62 : G z 4 En,. The disjoint sum GI G2 is the group having as normalized cover the group G1 = G2 and as permutation homomorphism the direct product 6 1 x 62: GI= G2 4 En, x En, C

+

En1+ n z . Each group can be decomposed into its Burnside transitive components:

77

(2) Let G be a symmetry group. T h e n there are afinite number of symmetry groups GI, . . . , G1 with normalized cover equal to G such that

G = G I + Gz

+. . . + G I

and each Gi acts transitively o n its index set. Any symmetry group G can be constructed from its cover GI the pair (GI6)and equivalently from the (1 + 1)-uple (G;H I , .. . , fii) consisting of the normalized cover and the 1 isotropy subgroups of all the transitive components. A transitive symmetry group hence is given by a pair (G; H ) . Disregarding n (which is the index [G : fi]),a constructive classification is A

,

.

nothing but a list of G and all possible subgroups.

4. Minimizers Let G be the normalized cover of a symmetry group and z = x ( t ) E A = H1(R,X ) G an equivariant local minimizer. Assume that at time t = 0 E R the trajectory z ( t ) collides, and all bodies in a cluster k c n collide (which means that other bodies might collide, but not with bodies in k). Given the colliding cluster k c n at time t = 0, let G, c G be the following subgroup:

G,

= {g E

G :g ( k ) = k , g(0) = 0},

and analogously G, = { g E G : g ( k ) = k , g ( 0 ) = 0) G, its normalized cover. It was proved ing that the blow-up trajectory q(t) is a G,-equivariant local minimizer with respect to compactly supported G,equivariant variations of the Lagrangian action sl,, restricted to the path space H1(R,X k ) G * .Hence local variations are needed for all possible G,. By analysing all possible groups G,, in5 the following result is proved (all listed groups yield non-colliding minimizers).

(3) Theorem. Let G be a symmetry group (not bound t o collisions) with a colliding G-symmetric Lagrangian local minimizer. If G, c G is the Tisotropy group of the colliding time restricted to the index subset of colliding bodies, then G, cannot act trivially o n the index set; i f the permutation isotropy of a transitive component of G, is trivial, then the image of G, in 0 ( 3 ) cannot be one of the following: I , C, (for p 2 l), D p (for p 2 2), T , 0 , y , pip, C p h . The first, and easiest to prove, criterion is actually a kind of averaging on a 0-sphere.

78

(4) If G, acts trivially o n k via CTk, then (by averaging on a 0-sphere) the colliding x ( t ) is not a local G,-minimizer.

The triviality of permutation actions implies that all the point particles in k at time t = 0 belong to a linear subspace of Rd, and that all transitive components are singletons. The next property is needed for averaging on a 1-sphere. A circle S c Rd (with center in the origin 0) is called rotating under G, for an index i if where Hic G, is the it is G,-invariant (via rotations) and S c isotropy of i with respect to the the permutation action of G, on the index set, via 0 (see’). ( 5 ) I f there is an index i E k and a circle S c Rd which is rotating under G, for the index i , then (by averaging over a 1-sphere) the colliding x ( t ) is not a local G, -minimizer.

The last p r ~ p e r t y on , ~ the orientation of the space action, is necessary (often the first two fail in this case), and it is needed for averaging on a 2-sphere (see5).

(6) Let G, be as above the symmetry group of a blow up solution Q. If det p(G,) = 1 (i.e. G , acts orientation-preserving on the space R3) and f o r one of the indices i E k the permutation isotropy Hi (restricted to G,) is trivial, then (by averaging over a 2-sphere) the colliding x ( t ) is not a local G, -minimizer. In the next section we present some examples to illustrate the process. 5 . Examples

(1) Example Let A and B be the matrices defined as -1 0 0

Let G

c Iso(R) x O(3) be the group generated by the two elements T

=

(t Ht + l , - I ) ,

h = (t H -t,B).

fi

If = G is the permutation isotropy of all particles, the permutation action on body indexes is trivial. It is easy to see that minimizers are collisionfree and that they are not homographic, for any n 3. The symmetry

>

79

Figure 4. Example (1) with n = 6 bodies.

constraint can be read as: for each i = 1. . . n the trajectory must fulfill the equalities

i

Zi(t

+ 1) = -zi(t)

~ i ( t=) Bzi(-t)

==+ zi(0)= Bzi(0).

So at time t = 0 mod 1 all the bodies are on the vertical line fixed by B , while the constraint zi(t+ 1) = -zi(t) forces the body to leave the line and to return to it on the other half-line. (2) Example Consider the matrices A=

J,

-10 0 1 0

[ o0

B = [ A Y , l ,

and -I as in example (1).The elements ,4 and B generate a group of order 4 in O ( 3 ) .Let k be the subgroup of Iso(R) x O ( 3 ) generated by

(t and

I?

-

t

1

4 7

( t +it , B ) 7

the group generated by the element

(t H t + 1,- I ) . .

.

A

Then (G, H) gives a (transitive) symmetry group, with collisionless minimizers and zero angular momentum. A possible global minimizer is shown in figure 5 . Of course, it is possible to consider the direct sum of k copies of such 4-body group, to obtain 4k-bodies collisionless periodic orbits. 6. Conclusions

As it turns out, equivariant minimizers of the n-body Lagrangean action functional are in fact collisionless periodic orbits for most of the symmetry groups. It is of course of some interest to understand what happens for the (exceptional) remaining groups. The main method for proving “how to

80

Figure 5.

Four bodies, symmetric with the group of example (2)

avoid collisions”-results is probably still a careful use of asymptotic estimates (see also12) and local averaging variations (dim=O, 1 , 2 , . . .), or level estimates, or combination of both. Averaging methods at the moment do not make use of the fact that colliding trajectories (asymptotically) go to central configurations. As an enumeration of all symmetry group can be used to give a global result on collisions for equivariant minimizers, without restriction to the number of bodies. At the end, generating and classifying all such symmetry groups is also a method for a (computer) program for periodic orbit generation. In fact, all the steps involved can be implemented in suitable computing environments: first symbolic computations on groups and finite dimensional representations, then finite-dimensional approximation of the orbits’ numerical minimization, and finally visualization and interactive post-processing.

References 1. A. Chenciner and R. Montgomery, Ann. of Math. (2) 152,881 (2000). 2. A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: from homology t o symmetry, in Proceedings of the International Congress of Mathematicians, Vol. 111 (Beijing, 2002), (Higher Ed. Press, Beijing, 2002). 3. A. Ambrosetti and V. Coti Zelati, Topol. Methods Nonlinear Anal. 3, 197 (1994). 4. A. Chenciner and A. Venturelli, Celestial Mech. Dynam. Astronom. 77,139 (2000). 5. D. L. Ferrario, Adw. i n Math. (2007), (to appear). 6. M. Salomone and Z. Xia, J . Differential Equations 215, 1 (2005). 7. K.-C. Chen, Ergodic Theory Dynam. Systems 23,1691 (2003). 8. K.-C. Chen, Arch. Ration. Mech. Anal. 170,247 (2003). 9. D. L. Ferrario and S. Terracini, Invent. Math. 155,305 (2004). 10. D. L. Ferrario, Arch. Rational Mech. Anal. 179,389 (2006). 11. V. Barutello, D. L. Ferrario and S. Terracini, Arch. Rational Mech. Anal. (2007), to appear. 12. V. Barutello, D. L. Ferrario and S. Terracini, On the singularities of generalized solutions to n-body type problems (2007), preprint.

81

NORMAL FORMS FOR COMMUTING VECTOR FIELDS NEAR A COMMON FIXED POINT T. GRAMCHEV Dzpartzrnento di Matematica e Informatica, Universitd d i Caglaari, Cagliari, 09124, Italy E-mail: todorQunica.it M. YOSHINO

Graduate School OJ Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan E-mail: yoshinoQmath.sci.hiroshima-u. ac.jp We outline some recent results for convergent and divergent normal form transformations (NFT) for families of commuting analytic vector fields having a common fixed point. First we treat vector fields having linear parts with nontrivial Jordan blocks. We propose geometrically invariant conditions leading to the existence of convergent N F T . Divergent solutions of some overdetermined systems of linear homological equations are constructed and sharp Gevrey estimates for formal divergent solutions are derived. Next, we investigate some classes of resonant vector fields and show that they do not admit convergent N F T and we compute the Gevery index of their formal N F T and describe completely the centralizers in the framework of the smooth vector fields.

Keywords: Commuting analytic vector fields; normal form transformations; Jordan block; homological equations; Gevrey spaces.

1. Introduction and the set-up of the problem

The paper outlines some recent results for classical issues in the Poincar6 normal form theory. The first goal is to propose sufficient conditions guaranteeing the simultaneous reduction by means of convergent NFT to normal forms of commuting analytic vector fields near a common fixed point. Secondly, in case the conditions fail, we try to investigate whether the sufficient conditions are “optimal” namely, for concrete families of vector fields construct divergent NFT (or divergent solutions of the linearized homological equations) and derive precise estimates of the Gevrey type of the divergence of the formal transformations,

We will consider analytic vector fields X in a neighbourhood of the origin 0 E @" with a singular (fixed) point a t 0. We can write

where x = ( X I , . . . , x,), A stands for the linear part (nx n matrix). Denote by spec ( A ) = {XI,. . . , A}, the spectrum of A. The basic idea, going back to Poincark, is to find a near-identity (formal) change of the coordinates which transforms 2 into a new vector field which has a "simpler" form. We recall that X (or spec A) is said to be in the Poincark domain (respectively, Siegel domain) if the convex hull of XI,. . . , A, in the complex plane does not contain (respectively, contains) 0. The Poincark-Dulac theorem asserts that if specA is in the Poincark domain then there are at most finitely many resonances and there exists a convergent NFT which reduces X to a (finitely resonant) normal form. We refer to the monographs14 for more details. The convergence question in the Siegel domain is more difficult since small divisors appear. In a fundamental work Bruno5 succeeded in proving a deep result of the following type: a formal normal form is convergent under an (optimal) arithmetic condition on the small divisors I < X,a! > --Ajl-', called the (Bruno) w condition and a hypothesis on the formal normal form, called the A condition. Recently, there has been a lot of interest in the crucial role of the symmetry properties for the convergence problem (e.g., ~ f . ~ - ' )and for simultaneous convergent normal forms of commuting smooth and analytic vector fields cf.10p13 The main goal of the present article is to present a recent approach developed by the authors in14 for deriving simultaneous reduction by convergent NFT to normal forms of families of commuting vector fields of the type (1) and, if there exist no convergent NFT, to investigate the divergence type in the framework of the Gevrey classes. We also show that our approach recaptures as particular cases some results on normal forms of a single vector field in the presence of symmetries. In particular, we are able to construct divergent Gevrey NFT and to describe completely the smooth centralizers for the resonant vector fields uxfOx, x z d x 2 , a # 0, d N, d 2 2. We recall some basic notions for the Gevrey spaces. Let 2 1. We say that a formal power series f(x) = x f a x a is in the formal Gevrey space

+

G U ( C n )if there exist C > 0 and

a

R>

0 such that one of the following

83

equivalent conditions holds

Ifal

5

cEZIa'(a!)"-1,

z."+

aE

(2)

+ +

where a! = a l ! . . . a n ! ,la1 = a1 . . . an. We point out that the Gevrey index 6 used by J.-P. Ramis15 equals c - 1. Therefore, in our notations, the convergent series correspond to the Gevrey index c = 1. Concerning the space of Gevrey functions G"(R), R c R",we have f E G'(Rn) iff for every K cc R there exist C > 0, R > 0 such that

The paper is organized as follows. In Section 2 we state a sufficient condition for the convergent analytic normal forms for commuting analytic vector fields. Section 3 contains results on divergent Gevrey NFT for two commuting vector fields in the presence of simultaneous diophantine conditions and nontrivial Jordan blocks in the linear parts. In Section 4 we investigate the formal Gevrey character of divergent NFT for some resonant vector fields and describe their centralizers in the C" category. 2. Convergent normal form for commuting singular vector

fields Let 6" denote a &dimensional Abelian Lie algebra of analytic germs a t 0 E R" of analytic vector fields vanishing at 0, where R = C or K = R.Let p be a germ of singular infinitesimal analytic Kd-actions p : K d 6". We denote by Act(Rd : K") the set of germs of singular infinitesimal analytic Kd-actions a t 0 E Rn. By choosing a basis e l , . . . , ed E R", the infinitesimal action can be identified with a d-tuple of germs at 0 of commuting analytic vector fields X 3 = p(e,), j = 1 , .. . , d . We can define, in view of the commutativity relation, the action

-

p

: Rd x

K"

-

K",

P(s;z ) = xgl 0 . . . 0 X,",(Z) =

(3)

x;;, . 0

* . 0

x;fd( z ) ,

where s = ( ~ 1 ,... , s d ) E Rd, for all permutations o = ( 0 1 , . . . ,cd)of (1,.. . , d } , where X { stands for the flow of X 3 . We denote by plzn the linear action formed by the linear parts of the vector fields defining p. We are interested in finding sufficient conditions guaranteeing the existence of a local analytic diffeomorphism g preserving the origin 0 E Kd such that g conjugates p to its linear part pi&, i.e., P(s;9 b ) ) = 9 ( p X ( s , z ) ) ,

(s, 2) E

K d x K".

(4)

a4

Following the arguments in16 we can find m E (1,.. . , n } such that K" is decomposed into a direct sum of m linear subspaces invariant under all A' = VX'(0) (t = 1 , .. . , d ) :

K"

= 1'1

s1

+ . .. + Ism,

+ . . . + ,s

dim Isj

= sj,

(5)

= 72.

The matrices A', . . . ,Ad can be simultaneously brought in an upper triangular form, if K = C,and we write again Ae for the matrices,

Ae =

7

. . . A&

os,xs2

OS,XSl

(6)

where

A$ A!,l2 0 A$ .. ..

.

0

. . . A$,ls, . . . A$,2s,

... 0 . . . A$ .

with As, AS,"@E C.If the matrices A', . . . , Ad are semisimple they can be reduced simultaneously to diagonal matrices. Assuming K = R,the matrices A', . . . , Ad can be chosen as 2x 2 block matrices in upper triangular forms (see14 for more details). We define i j by ik= '(At,. . . , E Km, k = 1,.. , ,d. We assume that

Ak)

il. , , x'd

W".

(7)

& = t ( ~ f , . . . , ~ j d ) ~ ~ d , j = 1 , . . . ,m, := {G, . . . , A m } . We define the cone r[A,] by

(8)

'

'

are linearly independent in

One can easily see that (7) is invariantly defined. Set

and A,

Definition 2.1. We say that the Kd-action p is a Poincari! morphism if there exists a basis A, c K" such that r[A,] is a proper cone in Km, i.e. it does not contain a straight real line.

85

Remark 2.1. We note that the definition is invariant under the choice of the basis A,. For an alternative but equivalent definition of the Poincark morphism we refer to Definition 6.2.1,11 (see14 for more details). Next, we introduce the notion of simultaneous resonance. For or = (al,. . . ,a,) E K m , P = (PI,. . . ,P,) E K", we set (a,P) = ET=lavPu. For a positive integer k we define Zl;l(k) = {or E ZI;. lor( ; 2 k } . Put d

wj(or) =

C 1(Xv,or) - X Y ~ ,

j = 1,.. . , m ,

(10)

u=l

w(a)= min{wl(or), . . . ,wm(or)}.

(11)

The quantity w ( a ) "measures the simultaneous small divisors" (the BrunoStolovitch condition,

Definition 2.2. We say that Am is simultaneously nonresonant (or, in short p is simultaneously nonresonant), if w(or)

# 0,

vor E Z 3 2 ) .

(12)

If (12) does not hold for some or E Z1;(2), then we say that A, is simultaneously resonant and clearly the monomial xa is resonant for all vector fields X 1 , . . . , Xd. Clearly, the simultaneous nonresonance condition (12) is invariant under the change of the basis A,. We have the following analogue of the PoincarkDulac theorem for commuting vector fields (see14).

Theorem 2.1. Let ji be a Poincare' morphism. Then is conjugated to a polynomial action b y an holomorphic change of variables. If in addition p is simultaneously nonresonant, then ji is linearizable, i.e., ji is conjugated to blin. Remark 2.2. If the vector fields X', . . . , Xd have semisimple linear parts, the assertion above recaptures Theorem 2.1.4,'' Next, we show how to obtain results for symmetries in7)' as a corollary from Theorem 2.1.

Example 2.1. Let d = 2, n 2 3, m = 3. Then the pairs A' = (1,- 3 , 9 ) , X2 = (1,1,1)( ~ f for . ~n = 3), and X1 = (1,- 3 , 9 ) , X2 = (1,-2,4) (cf.,8 n = 3) are simultaneously nonresonant and define a Poincark morphism. Thus the linearization of X1 shown the aforementioned papers is a particular case of Theorem 2.1.

86

3. Gevrey estimates for divergent solutions of linear

homological equations Given a two-dimensional Lie algebra, choose a base XI,X2 with linear parts Aj E G L ( 4 ; C )satisfying Spec(A1) = {1,1,v,v} and Spec(A2) = {0,1,p , p } , respectively, where v < p < 0, (v,p ) $Z Q2, and

0000 0100

1000 0100 A 1 = (o o oo v vo ) > A 2 = (

o Oo O op Ep

)

'

(13)

where E # 0. We show a refinement of the divergence result in Gevrey classes inI4 for the solution 'u of the overdetermined systems of linear homological equations Ljv := V v ( x ) A j x - A j v = f j ( j = 1 , 2 ) , with the compatibility conditions for the RHS (see14 for more details).

Theorem 3.1. Let 1 / 2 5 ro < 00 be given. Then there exists

Eo C { ( v , p ) E (R\Q)';v < p < 0 , v does not satisfy the Bruno condition} with the density of continuum such that for every ( u , p ) E Eo, there exists analytic f = t ( f l ,f 2 ) E ( C $ { Z } ) ~satisfying , the compatibility condition for the overdetermined system and such that the unique formal solution v ( x ) is not in Ul 0 we have p(x) =pox:, q(x) = 40x2 while d

P(S) = x l v k (xzexp((a(d - 1 ) ) - ' ~ ; ~ + ' ) ) ,

provided d is odd and a < 0. Proof. It is well known cf.8 (see also the comments on p. 137,5) that the y2 = x2 qh(z1),where qh satisfies

first component of the (divergent) transformation y1 = X I while

..$#)'(a) + 4qx1)= x1

+

(39)

90

We write

4 as a

formal power series, plug it in (39), and we obtain by 03

l + k ( d - 1)

inductive arguments that 4(x1) =

4kx1

, where 4 0

= 1,

k=O k

H(I+ j ( d - I)),

4 k + l = (-l)k+lak+l

IC E

z+,

j=O

+

which leads to 4 E G1+l/(d-l)\ G", B < 1 l / ( d - 1). In order to describe the commuting vector fields p(x)d,, p , q E C", it is sufficient to solve the linear PDEs

axfd,,p

+ x2dx,p

axfd,,q

-

+ q(x)d,,,

adxf-lp = 0

+ x2d12q -

224 =

(40)

0

(41)

The method of the characteristics yields that

p(x1, x2) = x f p + (xz exp((a(d - 1 ) ) - ' ~ ; ~ + ' ) ) ) , q(x1,xz) = x2+*

(32

*XI

> 0,

,

> 0, 4 E Cm(R),T > 0, c > 0,

exp((a(d - l ) ) - l x l d f l ) )

where ipk,+* E Cm(Iw). We note that for given the function 4(x2 e x p ( c x l r ) ) is Co3([0, +oo[xIw) (up t o X I = 0) if and only if 4 is a constant. We conlcude by observing that the smoothness a t x1 = 0 0 implies that cp+(0) = p-(0), $+(0) = +-(0).

Acknowledgments The first author thanks G. Gaeta and S. Walcher for useful and stimulating discussions in the realm of the normal form theory.

References 1. V.I. Arnold and Yu. Ilyashenko, in: D.V. Anosov and V.I. Arnold (eds), Encyclopaedia of Math. Sci., vol. 1, Dynamical Systems I , Springer Verlag, New York, 1 (1988). 2. G. Belitskii, Normal forms, invariants, and local mappings, Naukova Dumka, Kiev, 1979 (in Russian). 3. Yu. Bibikov, Lect. Notes in Math. 702,Springer-Verlag, Berlin (1979). 4. G. Cicogna and G. Gaeta, Lecture Notes in Physics. New Series m: Monographs, 57,Springer-Verlag, Berlin, 1999. 5 . A.D. Bruno, Trans. Mosc. Math. SOC.25, 13 (1971) and 26, 199 (1972). 6. A.D. Bruno and S. Walcher, J . Math. Anal. Appl. 183,571 (1994). 7. D. Bambusi, G. Cicogna, G. Gaeta and G. Marmo, J . Phys. A 31, 5065 (1998).

91

8. G. Cicogna and S. Walcher, Acta Math. Appl. 70,95 (2002). 9. G. Gaeta and S. Walcher, J . Math. Anal. Appl. 269, 578 (2002). 10. M. Yoshino, in: A. Degasperis, G. Gaeta (eds) Symmetry and perturbation theory SPT 98 World Scientific, Singapore, 287 (1999). 11. L. Stolovitch, Publ. Math. I.H.E.S. 91, 133 (2000). 12. L. Stolovitch, Ann. Math., 161,589 (2005). 13. T.N. Zung, Math. Res. Lett. 9, no. 2-3, 217 (2002). 14. M. Yoshino and T. Gramchev, to appear in A n n . Inst. Fourier. 15. J.-P. Ramis, Mem. Amer. Math. Soc. 48, 1 (1984). 16. A. Katok and S. Katok, Ergodic Theory Dyn. Syst. 15,569 (1995). 17. M. Hibino, Publ. RIMS, Kyoto Univ. 35,893 (1999). 18. T. Gramchev, in S. Abenda, G. Gaeta and S. Walcher eds, Symmetry and perturbation theory, Cala Gonone, 16-22 May 2002, World Scientific, Singapore, 106 (2003). 19. G. Iooss and E. Lombardi, J. Differential Equations 212, 1 (2005).

92

A MONKEY SADDLE IN RIGID BODY DYNAMICS HEINZ HANSSMANN Mathematisch Instituut, Universiteit Utrecht Postbus 80.01 0, 3508 TA Utrecht, The Netherlands

A rigid body with three equal moments of inertia is moving in a nonlinear force field with potential z 3 . Next to the S1-symmetry about the vertical axis and a further S1-symmetry introduced by normalization, there is a discrete symmetry due to a special choice of the mass distribution. The continuous symmetries allow to reduce to a one-degree-of-freedom problem, which exhibits bifurcations related to the elliptic umbilic catastrophe. This bifurcation carries over from the integrable approximation to the original system and further to perturbations that break the S1-symmetry of the potential.

1. Introduction The rotational motion of a rigid body with three equal principal moments of inertia, fixed at one point and not subject to external torques or forces, is a three-degrees-of-freedom system where all motions are periodic. Such maximally superintegrable systems are easily analysed. Indeed, as every axis through the fixed point is a principal axis of inertia, any motion will consist in a rotation about such an axis, which is parallel to the (fixed) angular momentum. Similarly, all bounded motions of the spatial Kepler system are periodic. In fact, while the latter can be turned into the geodesic flow on S 3 ,the flow of the isotropic Euler top is the geodesic flow on SO(3) with respect to the bi-invariant metric. Placing the fast top in the vertical force field 6'= -3z2e', amounts to perturbing by the weak potential E . z 3 with E inversely proportional to the square of the velocity, cf. [3]. The more integrals an integrable Hamiltonian system possesses, the more difficult it becomes to study perturbations of that system. In the present problem, since the component p3 of the angular momentum along the vertical axis e', remains an integral of motion, one can reduce to two degrees of freedom. The unperturbed reduced system in two degrees of freedom is still superintegrable (this would not be true if the principal moments of inertia were

93

not all equal). The periodic orbits define an S1-symmetry, and normalization pushes this S1-symmetry through the Taylor series of the perturbation. The normal form of order one can be computed by simply averaging along the periodic orbits. In this way the total angular momentum 1pl becomes a further integral of motion. By construction the normal form can be reduced to one degree of freedom. Here it is important to consider external parameters like E or the various moments of the mass distribution as fixed constants, while the values of the internal or distinguished parameters p3 and Ipl are given by the initial conditions and thus allowed to vary. Since G is a positional force, the latter enter only as the quotient

3.

2. The mass distribution

Let us recapitulate the main facts about the Euler top with three equal principal moments of inertia. The reader may find a comprehensive introduction in [1,2,6]. We choose a set of axes Z‘,Zy, e‘, fixed in space, with e‘, pointing in the (vertical) direction of the force, and a body set of axes Z1,&, Z3. The configuration space is the group SO(3) of orientation preserving three-by-three matrices which specify how to transform Zz, ZY,e‘, into Z1, &, Z3. The phase space is the cotangent bundle T * S 0 ( 3 ) ,the space of positions and (angular) momenta. An element a E T * S 0 ( 3 )yields the components p l , p z , p 3 of the angular momentum with respect to the spatial frame Zz,Zy,Zz and the components e l , e 2 , f?3 with respect to the body set of axes. As the three principal moments of inertia are equal, 11 = 1 2 = 13 = 1, the kinetic energy is given by

In the same way that a constant force only acts on the centre of mass (the first moments of the mass distribution), a linear force would only act on the second moments of the mass distribution, i.e. not at all since these are equal. Similarly, the force field ? (. = -3&z2e‘, only “sees” the third moments

of the mass distribution drn. The freedom of orientation of the “principal axes of inertia” Z1, ZZ, Z3 may be used to reduce the number of external parameters Mijk from ten to seven, but here we restrict even further to the special case Mill = M222 = M333 = 1 and Mijk = 0 else. As shown in [9]

94

it is possible to construct corresponding (non-homogeneous) rigid bodies. The centre of mass, principal moments of inertia or higher moments of the mass distribution do not enter the potential energy

+

The Hamiltonian function H = T V not only admits the continuous symmetry of rotations about the vertical axis Zz, but is furthermore invariant under the discrete symmetry group of all permutations of {&?I,Z2, &}, isomorphic to the dihedral group D3. The S1-symmetry of rotations about e', is the same S1-symmetry that already appears in the heavy rigid body; its reduction is well-known and goes back to Poisson. In the body representation

T*S0(3) a'

-+

the S1-symmetry amounts to

SO(3) x R3 (S,C)

-

SO(3) x R3 (exp,og,l) where exp, E S O ( 3 ) stands for the rotation by the angle p about the third axis. Dividing S1out of SO(3) yields a sphere and the isomorphism

S1 x ( S O ( 3 )x R3) ( P , ( 9 ,el)

SO(3) IS1

g

(mods')

t-i

S2 g-'(!)

=:y

clarifies the geometrical meaning of this sphere. I t is the space of possible positions of the vertical axis e', measured in the body set of axes ZI, &, e'3. In particular the radius of this sphere is 1. The vector y is called the Poisson vector. The Hamiltonian function H = T V reads 1

H(y,l) =

+ ,(e?+e;+e;) + E ( y ; + y ; + y ; )

after this reduction to two degrees of freedom.

3. The normal form The normalization procedure consists in finding a change of co-ordinates that transforms H into its average

95

along the flow pt of the unperturbed system XT (which is periodic with 7, depending on the initial condition) plus higher order terms. period ~ ( e) Repeating this process yields higher order normal forms, but for our purposes the normal form of order one turns out to be sufficient. As shown in [9] the truncated normal form (2) is given by

By construction, H is invariant with respect to the 5“-action defined by the flow pt of XT ; the Hamiltonian function no longer depends on 7. The kinetic energy in H = T + becomes a “constant” without dynamic meaning and may be omitted after the reduction to one degree of freedom. enter only is a positional force, the angular momenta IpI, p3 and As as quotients and In particular (3) defines a family of one-degree-offreedom systems depending on the parameter v := E [-1, 11. We also and are left with the Hamiltonian function write {i :=

v

&.

fl

{ctl&}

on the sphere with radius 1. The Poisson bracket is given by = - f y k &, where the alternating Levi-Civita symbol f z j k denotes the sign of the permutation (,’,”%) and vanishes if two of the i , j ,k are equal. From the perturbed rigid body X, inherits a Ds-invariance. However, the Poisson bracket is not D3-invariant; for u E 0 3 we have {‘(Ez)iB(cj))

=

{&(z)icu(j)}

=

=

- s g n a f y k ~ ( l ‘ k ) = sgno*u({Jz,JJ)).

-Eu(z)u(j)u(k) & ( k )

While even permutations, a. e. rotations about the axis along with an an, k E 723, are therefore symmetries of the Hamiltonian system XX,, gle the transpositions only lead to time-reversing symmetries.

(:)

Remark 3.1. The Hamiltonian system defined by (4) has another reversing symmetry, the reflection

+:c

-4

(5)

about the origin. While the Poisson bracket is invariant under +, the Hamiltonian (4) satisfies X,o+ = -X,,. In particular, the composition $ 0 7 with a transposition 7 E 0 3 is a (non-reversing) symmetry of the Hamiltonian system that multiplies both X, and the Poisson bracket by -1.

96

Fig. 1. Phase portraits [12] for parameter values v = 0.8629 and v = 0.864

The symmetry (5) is induced by the time-reversing reflection about the origin ($11 $21 $3)

-z2,

(-$Il

-$s)

of the perturbing potential (1). Similarly, the reflection e', H -ZZ induces the time-reversing symmetry u +-+ -u of the family X x , . At v = 0 this latter symmetry enforces all points to be equilibria. For v = the equilibria fulfill = Jz = J3, i.e. = . These J;i( two equilibria are fully D3-symmetric, and they exist for all values of v. Further occurring equilibria break the &-symmetry and taking furthermore the reversing symmetry (5) into account these equilibria form sextuples. All equilibria in the family ( X ~ , ) v + l , l ~ retain a remaining reversing symmetry defined by one of the three transpositions.

*fi

c:

Proposition 3.2. For u

{E

2

E S

# 0 the

equilibria of

I I1 = ~2

or

G

= J3

Xxv all

i)

lie in the union

or 12 = J3 }

of three great circles.

Proof. When u il

=

#

&&the last factors in

(& %}

=

iZ

=

{[Z,xv}=

i3

=

{&ixv)

=

; E ~ ( [ z - &)

( (5v2- 3)62&3 -k

3

3Ev(J3 - [I) ((5v2- 3)(1&?-k qEv(E1 -($) = 0 while the additional variables q = ( q l , . . . , q m ) satisfy & ( q ) = 0, the quantity ($,q ) = Qi q i1due to Green's formula, is such that Dt($, q ) E im D , and consequently is the density of a conservation law on E. If the corresponding nonlocal variable presents in the solution of the defining equation &(@) = 0 then the operator will contain the term pD-' o $I, where 'p is a symmetry of E . In a similar way, for any symmetry 'p of the equation & the quantity ( p , ' p ) determines a conservation law on the 1*covering that provides terms 9'D-l o 'p to the solutions of &(a*) = 0 on C * ( E ) , 'p' being a symmetry as well.

xi

4. Example: the dispersionless Boussinesq equation, Ref. [6]

Consider the system, see Ref. [2],

The l- and 1*-coverings are obtained by adding to ( 1 ) the following systems:

106

and

c* ( E ) :

{

+ 2u;p2,

p i = u1p;

-

p: = p i

3dp3, - 2 u y ,

-

u2p;

P; = P:. The defining equations for recursion and Hamiltonian operators are

Dt(R1)= D,(R2), Dt(R2)= u1O,(R1)+ ukR1 + D,(R3),

(2)

D t ( R 3 ) = -u2Dz(R1)- 3u2R1 - 3u1D,(R2)- u;R2 and

D,(H1) = D z ( H 2 ) , D t ( H 2 )= ulD,(H1)

+ u;H1 + D z ( H 3 ) ,

(3)

D t ( H 3 ) = - u 2 D z ( H 1 ) - 3u2H1 - 3u1D,(H2) - u k H 2 , resp. Skipping coordinate representation of solutions for ( 2 ) and ( 3 ) obtained in Ref. [ 6 ] , we present them in a more structured way.

Theorem 4.1. T h e space of solutions t o system ( 2 ) is a n associative alRz] = gebra R with two generators R1 and R2 satisfying the relation [RI, R2 o R2. All the recursion operators pair-wise commute w.r.t. the Nijenhuis bracket. Theorem 4.2. T h e space of solutions t o system ( 3 ) is a free left R - m o d u l e with one generator Ho. T h e operator corresponding t o Ho is a Hamiltonian one. A pair (a1R1 +a2R2, H o ) f o r m s a Poisson-Nijenhuis structure iff a1 = 4 and a2 = 3. T h e operator R = 4 R 1 3Rz gives rise t o infinite series of Hamiltonian structures H , = R" o Ho. T h e y pair-wise commute w.r.t. the Schouten bracket. T h e first two of t h e m are local while all the rest are nonlocal.

+

Theorem 4.3. T h e s y m m e t r y Lie algebra for equation (1) splits into the direct s u m of rank 1 R-modules symE = Q a , 1 CB Q a , 2 CB Q a , 3 CB Q a , a CB & a , a .

T h e module

Qa

= Qa,l @

@ Qa,3

[am,.,Q a , a ] c Q * , a ,

is and Abelian ideal and, in addition,

[@a,.,

&*,a]

c @ * , a CB G a , a .

T h e ideal Q a consists of (x,t)-independent local symmetries, while @ a , a is formed by (x,t)-independent nonlocal ones. Symmetries that belong to G o , , depend o n x and t and are nonlocal.

107

Acknowledgments T h i s work w a s supported i n part by the NWO-RFBR grant 047.017.015 and RFBR-Consortium E.I.N.S.T.E.I.N. grant 06-01-92060.

References 1. V.A. Golovko, this volume. 2. H. Gumral and Y . Nutku, J. Phys., A27 (1994), 193-200. 3. P.H.M. Kersten, I.S. Krasil’shchik, and A.M. Verbovetsky, Acta Appl. Math. 83 (2004), 167-173, http://arxiv.org/pdf/math/0310451. 4. P.H.M. Kersten, I S . Krasil’shchik, and A.M. Verbovetsky, J. Geom. and Phys. 50 (2004), 273-302, http://arxiv.org/pdf/nlin/0305026. 5. P.H.M. Kersten, I.S. Krasil’shchik, and A.M. Verbovetsky, J. Phys. A37 (2004), 5003-5019, http://arxiv.org/pdf/nlin/O305026. 6. P.H.M. Kersten, I S . Krasil’shchik, and A.M. Verbovetsky, Acta Appl. Math. 90 (ZOOS), 143-178, http://arxiv.org/pdf/nlin/0511012. 7. Y. Kosmann-Schwarzbach and F. Magri, Ann. Inst. H. Poincark Phys. ThCor., 53 (1990), no. 1, 35-81. 8. A.Ya. Maltsev and S.P. Novikov, Phys. D156 (2001), 53-80. 9. P.J. Olver, J. Sanders, and J.P. Wang, J. Nonlinear Math. Phys. 9 (2002) Suppl. 1, 164-172. 10. A. Verbovetsky, V. Golovko, and I. Krasil’shchik, Scientific Bulletin of MSTUCA, 91 (2007), 13-21 (in Russian). 11. A.M. Vinogradov and I.S. Krasil’shchik, Acta Appl. Math. 15 (1989), 161209, http://diffiety.ac.ru/djvu/non-loc-trends.djvu.

108

ANOMALY OF LINEARIZATION AND AUXILIARY INTEGRALS BORIS KRUGLIKOV Institute of Mathematics and Statistics, University of Troms0, Troms0 90-37, Norway E-mail: [email protected] In this note we discuss some formal properties of universal linearization operator, relate this t o brackets of non-linear differential operators and discuss application t o the calculus of auxiliary integrals, used in compatibility reductions of PDEs. Keywords: Linearization; evolutionary differentiation; compatibility; differential constraint; symmetry; reduction; Jacobi bracket; multi-bracket.

Introduction Commutator [A,V] of linear differential operators A , V E Diff(.rr,.rr) in the context of non-linear operators F, G E diff ( T ,T ) is up-graded to the higher Jacobi bracket { F ,G } , which plays the same role in compatibility investigations and symmetry calculus. The linearization operator relates non-linear operators on a bundle 7r with linear operators on the same bundle, whose coefficients should be however smooth functions on the space of infinite jets. The latter space is the algebra of %-differential operators and we get the map

t : diff(T,.rr)

--f

%Diff(.rr,.rr)= C"(J".rr)

@.C=(M)

Diff(.rr,.rr),

defined by the formula3

t ~ ( s )=h & F ( s

+ th))t,o,

F E diff(n,.rr), s, h E Cm(.rr).

However it does not respect the commutator: [eF,tG]

# e{F,G}.

Example: Consider the scalar differential operators on and J"(.rr) = R"(x, u , p = p l , p 2 , . . . ) . Choose

F

= p 2 ,G = p

+c

, X;

{ F , G }= 2 c p

R,so that

* t{F,c)= 2cDz.

.rr = 1

109

[e~,e,]

If we commute e~ = 2pVx and eG = V x ,we get: = -2p2Vx, so that we observe an anomaly. There are two reasons for this. The first is that the operator of linearization disregards non-homogeneous linear terms, which are important for the Jacobi bracket. The second is the non-linearity itself. The goal of this note is to discuss reasons and consequences of this anomaly (this also plays a significant role in investigation of coverings and non-local calculus2). ACKNOWLEDGEMENT. The results were obtained and systematized during the research stay in Max Planck Institute €or Mathematics in the Sciences, Leipzig, in April-May 2007. 1. Anomaly via Hessian

The Jacobi bracket of non-linear operators F, G E diff ( T ,T ) is expressed via linearization as follows:

{ F ,G} = ~ F G eGF. We also consider the evolutionary operators defined by duality:

3pG = ~ G F . Since & is a derivation in G, 38 is a derivation (satisfies the Leibniz rule) and their union can be treated as the module of vector fields. These operators have no anomaly, i.e. the map 3 : Co3(Jo3.rr) -+ Vect(JmT) is an anti-homomorphism: [ 3 F 1 3 G ]= -3{F,G}.

This instantly implies Jacobi identity for the bracket { F ,G}, so that (diff ( T , T ) , {, }) is a Lie algebra.3 The operators of universal linearization and evolutionary differentiation do not commute and this leads to the following

Definition 1.1. The Hessian operator diff ( T ,T ) x diff ( T , T ) + 56 Diff ( T , T ) is defined by the formula HessF G = [ 3 G ,e F ] . We will also write HessF(G, H ) = Hessp G ( H ) for F , G , H E diff ( T , T ) and note that HessF = 0 for linear operators F , because in this case .!F = F , so the claim reduces to the commutation of left and right multiplications. Next we note that the Hessian HessF is symmetric:

Lemma 1.1. Hessp (G, H ) = Hessp (H , G) .

110

Indeed:

G~ HG HessF(G, H ) = ~ G ~ -F~HF ~ =

~ -H~ F F

~HG,

so that HessF(G, H) - HessF(H, G) equals [3G, ~

H ]-F eF{H, G}

=

- ~ { G , H-}~FF { HG} , = 0.

Now we can express the anomaly of linearization via the Hessian: Proposition 1.1. [ l ~ !G] ,- ! I F , G )

= HessG F - HessF G.

Indeed we have:

Finally let us express the Leibniz identity for non-linear operators and the Jacobi bracket. For linear operators it is well-known, but for non-linear ones there's an anomaly: Proposition 1.2. { F ,&H}

+

= ~{F,G}H &{F,

H } - HessF(G, H )

This is obtained as follows:

2. Coordinate expressions

A local coordinate system (zz,Q) on T induces the canonical coordinates ( z z , p 3 , )on the space J"T, where g = (21,. . . , in) is a multi-index of length 1 ~ =1 i l + . . . + in. The operator of total derivative of multi-order (and order )1.1 is Do = Vil . . . V : , where V,= aZt+ C~3,+~,a,:. The linearization !F of the operator F = (F1,... , F") is

where .OF1denotes the operator Do applied to the j - t h component of sections from C" ( T ) . The evolutionary differentiation 30 corresponding to G = (G1, . . . , G") equals

3Gi-- C ( D , G ' ) .

111

where dP;li] denotes the operator dP; applied to the i-th component of sections from Cm(n). Then i-th components of the Jacobi bracket is given by =

{F,G}Z

C(F;; . D , ( G ~ )- G;; . D , ( F ~ ) ) .

These formulas are known.3 I t is instructive to demonstrate the Jacobi identity in coordinates. For this we need the following assertion.

Lemma 2.1. In canonical coordinates on Jan:

C DT-,api

d p ; ~ T=

a-x

(the difference of multi-indices 0 - N is defined whenever summation is by N counted with multiplicity. This follows from iteration of the formula [dp;,Di]= d

c

N

.

P3,Li

g),

the

. Thus

F ~ ; D U - - X ( GD,+,(Hk)--Fi. ~:) P i D u - - x ( ~ ~DT+H(Gk) $) G i ; p t D T ( H k ) D u ( F j+) Hi;p5DT(Gk)Du(F j)

{Fl{ G ,H}}a = -

-

G$tD,-,(Hk~

P:LX

)D,(Fj)

+ H&D,-,(G~s

)DT(Fj), T-x

which yields C c y C l i c { {FG , ,H } } = 0. Now we write the Hessian: HessF(G, H ) i =

Piip5 D,Gj

’

DTHk,

and its symmetry in G , H and vanishing for linear F is obvious. The compensated Leibniz formula can be written as follows:

(iF,! G H )

C F:; D, -

- e{F,G}H

- l G { F ,H > ) i =

(

(G$ ) D, + ( H ) ~ G:;

( F ; ; ~ .D, (Gk ) +F;:

DT-

K(G;;-

x )

-

p 5 D~ (H’)

+G:: D~ ,(H‘.P”,x 1) D, (

G ; ; ~ :D~ ( F ~- G )::

- Gii (Do--w(F$)%+x(H k ) - Du-,(H&)DT+,(F

-

(F?. )) .D, pa-=

D T - ~

k )) = -HessF(G,

~) -j ( ~ j )

H)Z

112

This gives an alternative proof of Propositions 1.2 and 1.1. 3. Auxiliary integrals T ) is called an auxiliary integral Definition 3.1. An operator G E diff (n, for F E diff (n, n) if

{F,G} = txF

+ t,G

for some operators X E diff(T, n) and p @ VDiff (n, T ) . F \ ( 0 ) . The set of such G is denoted by Aux(F). We denote Aux,(F) the space of G satisfying the above formula with a fixed p E diff(7,n). This is a vector space. Then Aux(F) = U, Aux,(F). We can assume ord(p) < ord(F) for scalar operators, i.e. rankn = 1. With certain non-degeneracy condition for the symbols of F, G the following statement holds:

Theorem 3.1. A non-linear differential operator G is an auxiliary integral for another operator F a f f the system F = 0 , G = 0 is compatible (formally integrable). The generic position condition for the symbols of F, G is essential. If T

= 1 is the trivial one-dimensional bundle, this condition is just the

transversality of the characteristic varieties Char'(F) and Char@(G)in the bundle PcT'*M (after pull-back to the joint system F = G = 0 in jets); in this form it is a particular case of the statement proved in5 . For rankn > 1 the condition is more delicate and will be presented elsewhere. Notice that Auxo(F) = Sym(F) is the space of symmetries of F . This is a Lie algebra with respect to the Jacobi bracket. It can be represented as a union of spaces Syme(F) = { H : ~ F = H lo+~F}, 6' E diff(n,.rr), which are modules over Sym, ( F ) .More generally we have the graded group: Symef( F ) Sym,,, (F)c Symo,+,,, (F) Let us assume G E Aux,(F), H E Sym,(F), i.e.

+

{F,G}=txF+l,G,

{ F , H } =&IF.

Then denoting adH = { H , . } = t~ - 38 we get:

113

+ {G, adF H } = -{H,exF + epG} + {G,CeF}

adF{G, H } = {adF G, H }

= e { x , ~ } F + e x { F , Hf H ) e s s ~ ( X , F+ ) ~{,,H}G+~,{G,H}+H~~~H(~,G) -

e{e,G}F - eO{F, G}

+

= ( ~ { x , H } [ex, eel

-

e{e,G}

+

HessG(@iF ) HessH - HessG 0 ) F -

+ (&,HI

+ e,{G, H } - &+&+ Hesw p)G.

Thus {G, H } is an auxiliary integral for F if ! e l , = l ? ( , , ~fHessH ) p (the “iff’ condition means the difference annihilates G), which can be written as p E Ker[(!e

+ ladH

-

HessH) 0 el.

Such a pair 0 E sym*(F) = Sym(F)/Symo(F), H E Syme(F) determines the action of the second component adH : Aux,(F)

-+

Aux,(F).

Also since

we have:

Cp{G,H }

+ (!{,,HI

-

tee,

+ HessH p)G = -(adH +&)l,G.

Thus if H E SymO(F),i.e. (adH +!e)F = 0, and p E Ker[(adH + l o ) i.e. (adH +&)lP = 0, then adH : Aux,(F)

-+

o

el,

Sym(F).

4. Symmetries and compatibility

It has been a common belief that if G E Sym(F), then the system F = 0 , G = 0 is compatible, which forms the base of investigation for automodel solutions. This is however not always true.

Example: Let F , G be two linear diagonal operators with constant coefficients. Then {F,G} = 0 (in this case the Jacobi bracket is the standard commutator), so that G is a symmetry of F . However the system F = 0 , G = 0 is usually incompatible: for generic F , G of the considered type the only solution will be the trivial zero vector-function.

114

More complicated non-diagonal operators are possible, but it would be better to consider non-homogeneous linear operators. Then if the coefficients are constant and generic, the linear matrix part commute, but the system F = 0, G = 0 may have no solutions at all. For instance if we take

then {F,G} = 0, so that G E Sym(F), while the system F = 0, G not compatible, and moreover its solutions space is empty.

=

0 is

Thus the flow ut = G(u) on the equation F = 0 has no fixed points (no auto-model solutions). Here t is an additional variable (x is the base multi-variable for PDEs F = 0 and G = 0), so that G E Sym(F) can be expressed as compatibility of the system

F(u)= 0,

~t = G(u),

while symmetric solutions correspond to the stationary case ut = 0, i.e. compatibility of the system F ( u ) = 0, G(u) = 0. However if the non-degeneracy condition assumed in Theorem 3.1 is satisfied, then auto-model (or invariant) solutions exist in abundance, namely they have the required functional dimension and rank as Hilbert polynomial (or Cartan test) predicts, see.7

Remark 4.1. Symmetric solutions are the stationary points of the evolutionary fields and they are similar to the fixed points of smooth vector fields on R n , which must exist provided the vector field is Morse at infinity. The non-degeneracy condition plays a similar role. Many examples of auto-model solutions and their generalizations can be found inlis , non-local analogs use the same technique and similar t h e ~ r y . ~ > ~ Compatible systems correspond to reductions of PDEs and are sometimes called conditional symmetries by analogy with finite-dimensional integrable systems on one isoenergetic surface. But the rigorous result must rely on certain general position property for the symbol of differential operators, otherwise it can turn w r ~ n g .The ~ ? ~method based on this approach makes specification of the general idea of differential constraint and is described

115

5. Conclusion I n this note we described t h e higher-jets calculus corresponding t o symmetries a n d compatible constraints, basing on t h e Jacobi brackets. Another approach to integrability of vector systems is given by minimal overdetermination a n d it uses multi-brackets6 of differential operators

{. . . } : Am+' diff (m. 1,l)4 diff ( m. 1,l) Following this approach a minimal generalization of symmetry for F = E diff ( T ,T ) with T = m . 1 is such G E diff ( T , 1) t h a t

(F1,. . . , F,)

{ F I , . . . , F,, G } = l o , Fi

+ . . . + &,F,

W i t h certain non-degeneracy assumption6 this implies t h a t t h e overdetermined system F = 0, G = 0 is compatible (formally integrable). A more advanced algebraic technique would yield another higher-jets calculus producing anomaly t h a t manifests in non-vanishing of t h e expression { ~ F3 I'

' '

, [F,,+I

1

-

c { F ~ , . . .,Fm+l).

Implications for vector auxiliary integrals and generalized Lagrange-Charpit method follow t h e same scheme.

References 1. G. W. Bluman, S. Kumei, Symmetries and differential equations, Appl. Math.

Sci. 81, Springer, 1989. 2. P. Kersten, I. S. Krasilschik, A . Verbovetsky, Hamiltonian operators and e*coverings, J. Geom. and Phys., 50 (2004) 273-302. 3. I. S. Krasilschik, V. V. Lychagin, A.M. Vinogradov, Geometry of jet spaces and differential equations, Gordon and Breach (1986). 4. B. S. Kruglikov, V. V. Lychagin, A compatibility criterion for systems of P D E s and generalized Lagrange-Charpit method, A.I.P. Conference Proceedings, Global Analysis and Applied Mathematics: International Workshop on Global Analysis, 729, no. 1 (2004), 39-53. 5 . B. S. Kruglikov, V. V. Lychagin, Mayer brackets and solvability of PDEs - I I , Trans. Amer. Math. Soc. 358,no.3 (2005), 1077-1103. 6. B. S. Kruglikov, V. V. Lychagin, Compatibility, multi-brackets and integrability of systems of P D E s , prepr. Univ. Tromse, 2006-49; ArXive: math.DG/0610930. 7. B. S. Kruglikov, V. V. Lychagin, Geometry of Differential equations, in: D. Krupka, D. Saunders, Handbook of Global Analysis (2007); prepr. IHES/M/07/04. 8. P. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics, 107, Springer-Verlag, New York (1986).

116

NATURAL VARIATIONAL PRINCIPLES D. KRUPKA Department of Algebra and Geometry, Faculty of Science, Palacky University, Tomkova 40, 779 00 Olomouc, Czech Republic, and Department of Mathematics, La D o b e University, Melbourne, Victoria 3086, Australia, e-mail: krupkaOinf.upo1.cz We study global variational principles on jet prolongations of fibred manifolds, whose Lagrangians are differential invariants. In particular, a relationship between solutions of the Euler-Lagrange equations and conservation laws, related with vector fields on the basis of underlying fibred manifolds, is established. Keywords: Differential invariant; Lagrangian; Euler-Lagrange equations; Lepage form; conservation law. M S classification: 49805, 53A55, 58A32, 58330.

1. Introduction

This paper represents a written version of a plenary lecture, delivered at the conference Symmetry and Perturbation Theory, Otranto, June 2007. Its aim is to discuss properties of global, higher order variational functionals on fibred spaces, known as invariant with respect to diffeomorphisms. Many examples are known in Riemannian geometry, the theory of linear connections, and field theory (see e.g. Anderson, Horak, Horndeski, Krupka, Lovelock, P. Musilova and J. Musilova, Novotny, and Rund); a most wellknown one is the Halbert variational functional in the general relativity theory. In agreement with the theory of natural bundles, Lagrangians for these variational principles are called natural. A characteristic property of a natural Lagrangian is that it is a differential invariant. Its domain is usually a higher order jet prolongation J'Y of a natural fibre bundle Y ;the domain can equivalently be understood as the type fibre of J'Y. This implies, in particular, that a variational functional, defined by a natural Lagrangian, extends canonically to a category of underlying fibred manifolds Y . To understand this scheme more precisely, we give in Section 2 basic definitions

117

of the theory of differential invariants as applied in this lecture. Then in Section 3 we present two theorems on geometric structure of higher order variational functionals, the second one being the fundamental infinitesimal first variation formula. Finally, in Section 4,we introduce categories of manifolds that appear in the theory of natural Lagrangians. Then we establish a fundamental consequence of the first variational formula for these Lagrangians, an identity, involving the Euler-Lagrange expressions on one side, and conserved currents, related with diffeomorphisms of underlying manifolds, on the other. The exposition in this paper is based on the works [1,2], and [3], where many other references can be found. Throughout, standard concepts of analysis on manifolds are used: d is the exterior derivative operator, i z the contraction of a differential form by a vector field Z, and dz the Lie derivative operator by Z The r-jet prolongation of a vector field ( is denoted by Jr 1:

~ , ( z i , y " , ~ ~ , y " , a. j. l.. . , j ~ y u ) = o ,

i ~ a s m .

(6)

Variational equations. Let r 2 1. A horizontal n-form X on J'Y (where n = d i m X ) is called a Lagrangian of order r. Following Krupka (1973) we say that a form p is a Lepage equivalent of X if h p = A, and p l d p is a dynamical form. If d i m X = 1 then every Lagrangian has a unique Lepage equivalent, OX, called the Cartan form. If dimX > 1,Lepage equivalent of X is no longer unique: we have p = O+p where O = h p + p l p is related with the Lagrangian and p is an arbitrary a t least 2-contact n-form. However, only for r = 1 the form O is determined by the Lagrangian: it is denoted by O X and called the Poincark-Cartan form. For r > 1,the a t most l-contact part 8 of p is no longer determined by the Lagrangian: it splits to 0 = Ox + p z d v , where Ox comes from the Lagrangian and v is an auxiliary term; moreover, for r 2 3 the splitting is n o t invariant under transformations of fibred

127

coordinates (which means that we do not have global higher-order PoincardCartan forms for r > 2). The dynamical form Ex = p l d p = pldOx is called Euler-Lagrange form of A. It is global and unique (does not depend on a choice of a Lepage equivalent of A). Hence, components E, of E are determined by the Lagrangian; they are called Euler-Lagrange expressions. Paths of Ex are extremals of the Lagrangian A. Lagrangians giving rise to the same Euler-Lagrange form (i.e. having the same Euler-Lagrange expressions) are called equivalent. Variationality conditions. Variational equations arise as equations for paths of certain dynamical forms, however, not every dynamical form is an Euler-Lagrange form of a Lagrangian. A dynamical form E on J ' Y is called locally variational if to every point in J ' Y there exists a neighbourhood U , an integer r 5 s, and a Lagrangian X defined on T ' , ~ ( U such ) that Elr~= Ex. E is called globally variational if X is defined on J ' Y . As proved by Anderson and Duchamp (1980) and Krupka (1981), necessary and suficient conditions for a dynamical form E = Euwu A wg o n J ' Y be locally variational read

(7) for all CT,v , and 0 5 1 5 s. A local Lagrangian (of order s) then can be obtained using the Tonti- Vainberg fomnula 1

L

= yo

E , ( x i , u y w , u y y , . . . ,~ y ~ ~ , , , ~ , ) d u .

(8)

As proved by Vanderbauwhede (1979) every variational system of ordinary differential equations has local Lagrangians of order s / 2 (resp. (s l ) / Z ) if s is even (resp. odd). For partial differential equations a complete answer to the problem of local order-reducibility of Lagrangians is yet not known.

+

3. Exterior differential systems for variational equations

Lepage (n+1)-forms.Let E be a dynamical form on J ' Y . An (n+1)-form a is called Lepage equivalent of E if E = p l a and da = 0. The next theorem reflects a deep relation between locally variational forms and closed forms: Theorem 3.1. T h e 1-contact part E of a Lepage equivalent of E is a locally variational form. Conversely, every locally variational form E has a Lepage equivalent. If d i m X = 1, the Lepage equivalent of E is unique and projectable onto J ' - l Y .

128

If LY is a Lepage equivalent of a locally variational form E then locally dp where p is a Lepage a-form; X = hp is a (local) Lagrangian for E . Equations for extremals of a locally variational form E (Euler-Lagrange equations) take the intrinsic form (4). With help of a Lepage equivalent of E we obtain these equations in an equivalent, but geometrically more significant form as follows: LY =

Theorem 3.2. Let E be a locally variational f o r m on J " Y , (Y its Lepage X is an extremal of E equivalent defined on J k Y . A section y of r : Y if and only if J k y x i ~ c = u 0 for every rk-vertical vector field on J k Y . --f

0.

Since

d" ~,) (25) dzn the Bernoulli functions can be thought of as primitives of the periodic delta function. The following hyperfunctional equation holds. - ( B , (x - [z])) = n! (1 - 6

Proposition 4.1. Let q*

6, ( q + )

= e2si(z*io),z

+ (-l)n6,

((q-)-l)

E R. For all n E Z we have = (27ri)-n

K n(x).

(26)

Our aim is to provide a hyperfunctional version of the classical Lipschitz formula. This formula states that

The following result is based on the properties of periodic hyperfunctions described above, according to the commutative diagram (21).

Theorem 4.1. For all n E Z we have

C pB, + k) (T

kEZ

= 2i

6-n ( 4 ) (27~i)-~ (--I)~-' 6-n (q-l)

{

-

i f 141 < 1 (% > 0)

, i f 141 > 1, (ST < 0) (28)

whereas the usual Lipschitz formula corresponds t o n 5 -1. Here Bn is the [0,1]) which restriction to [0,1] of B, and (PB, is the function in O1 represents the hyperfunction Bn

(c\

139

Explicitly, formula (29) reads

where R,-1 is t h e polynomial of degree n-1 such that B, (T) log (1 - 1 / ~ ) + R, (T) E O1 [ O , l ] ) . T h e previous construction can be proposed in t h e case of t h e more general Appell polynomials considered in section 2 (sees). Here we limit ourselves to observe that, in this case, one introduces extended delta rational functions, obtained from t h e Fourier expansion of t h e periodic version of t h e considered Appell poynomials. Writing t h e analogue of the inversion equation ( 2 3 ) , one can get a generalization of Theorem 4.1 a n d construct families of hyperfunctions within t h e same framework.

(c\

References 1. A. Adelberg, Universal higher order Bernoulli numbers and Kummer and related congruences, J. Number Theory 84 (2000), 119-135. 2. V. M. Bukhstaber, A. S. Mishchenko and S. P. Novikov, Formal groups and their role in the apparatus of algebraic topology, Uspehi Mat. Nauk, 26(2(158)): 63-90, 1971. 3 . F. Clarke, The universal Von Staudt theorems, Trans. Amer. Math. Soc. 315 (1989), 591-603. 4. M. Hazewinkel, Formal Groups and Applications, Academic Press, New York, 1978. 5 . L. Hormander, The Analysis of Partial Differential Operators I , second edition, Springer-Verlag, 1990. 6. S. Marmi, P. Moussa, J.-C. Yoccoz, Complex Brjuno functions, J. Amer. Math. Soc. 14,n. 4 (2001), 783-841. 7. A. Kaneko, Introduction to Hyperfunctions, Kluwer Academic Publisher (1988). 8. S. Marmi and P. Tempesta, Polylogarithms, hyperfunctions and generalized Lipschitz summation formulae, Preprint Scuola Normale Superiore (2007). 9. N. Ray, Stirling and Bernoulli numbers for complex oriented homology theory, in Algebraic Topology, Lecture Notes in Math. 1370,362-373, G. Carlsson, R. L. Cohen, H. R. Miller and D. C. Ravenel (Eds.), Springer-Verlag, 1986. 10. G. C. Rota, Finite Operator Calculus, Academic Press, New York, 1975. 11. P. Tempesta, Formal Groups, Bernoulli-type polynomials and L-series, C. R. Math. Acad. Sci. Paris (2007), to appear. 12. P. Tempesta, New Appell sequences of polynomials of Bernoulli and Euler type, J. Math. Anal. Appl. (2007), to appear.

140

VARIATIONAL X-SYMMETRIES AND DEFORMED LIE DERIVATIVE P. MORANDO Dipartimento di Matematica, Politecnico d i Torino, Corso Duca Degli Abruzzi 84, 1-10129 Torino, Italy E-mail: [email protected] We introduce, in the spirit of Witten's gauging of exterior differential, a deformed Lie derivative that allows a geometrical interpretation of variational X-symmetries, in complete analogy with standard variational symmetries.

Keywords: X-symmetries; variational symmetries; conservation laws.

Introduction Symmetry analysis and reduction methods are well established tools to study nonlinear differential equations [1,4,8,17,18,20].I t was recently pointed out that the class of transformations which are useful for finding solutions of ODES is not limited to (Lie-point, generalized, non-local ...) proper Symmetries, but extends to a wider class of transformations, which were christened &symmetries, as they depend on a C" function A. We will not discuss here the relevance of this new class of symmetries, referring to [2,3,5,6,12-15,191for more details and applications. It should be stressed that X-symmetries are not symmetries in proper sense (i.e. they do not map solutions into solutions); nevertheless they can be used to perform symmetry reduction via exactly the same method used for standard symmetries. The aim of this paper is to give a geometrical interpretation of Xsymmetries in terms of a deformation of the usual Lie derivative. In particular, following Witten's idea [21] of gauging the exterior differential operator with a function f , we define the deformed Lie derivative C i f = e f C ( , f x ) . By using a generalization of this deformed Lie derivative we recognize that Xsymmetries can be characterized in complete analogy with standard symmetries just replacing CX with the deformed Lie derivative C$, where p = Ada: is a horizontal l-form on J 1 ( M )(see [ll]for a more detailed discussion and

141

extension to PDEs case). In particular we show that a vector field on Jk((M) is the A-prolongation of a vector field on M if and only if, for any contact form 19 in Jk((M),L$ (19) is a contact form (we recall that the standard prolonged vector fields on Jk((M)can be characterized by requiring that they preserve the contact ideal [8,18]). Finally we give a definition of X variational symmetries in terms of the deformed operator C$. Given a first order regular Lagrangian L , we denote by 0 the corresponding Poincarb-Cartan form (see, e.g. [9,10,18]) . By means of 0 we define a variational Asymmetry as a A-prolonged vector field X on J ' ( M ) such that L $ ( O ) is a contact form (again we recall that standard variational symmetries are defined as prolonged vector fields X on J 1 ( M ) such that L x ( 0 )is a contact form). In this geometrical framework we can easily associate to any A-symmetry X for the variational problem a A-conservation law (see below and [la] for the exact definition of this concept). Moreover we prove that our definition of X variational symmetries completely agrees with the corresponding ones given in local coordinates by Muriel Olver and Romero [la] (see [ll]for the generalization to the case of variational PDEs).

1. Preliminaries

In this section, in order to fix notations and for the convenience of the reader, we collect some definitions and preliminary results about jet bundles and geometry of differential equations [4,8,17,20]. Moreover we recall the definitions of standard symmetries and A-symmetries as given in [3,5,13]. Here and in the following we consider a fiber bundle ( M ,T O , B ) with 1dimensional base manifold B . In ( M ,TO, B ) we introduce local coordinates (z, ua) where a = 1,. . . , q. We will denote by r ( M ) the set of local sections of this bundle, by X ( M ) the set of vector fields on M and by h * ( M ) the graded algebra of differential forms on M . Finally h k ( M )will denote the set of Ic-forms on M . Similar notations will be used for any fiber bundle. The bundle ( M ,T O , B ) can be prolonged to the Ic-th j e t bundle ( J k ( M ) T, k , B ) with local coordinates (2, u a , u;,.. . ug).The total space of the jet bundle is also called the jet space, for short. The jet space J k ( M ) is naturally equipped with the canonical contact f o r m s that, in local coordinates, read

83" := du?3 - U?3+ldX.

142

We denote by C the exterior ideal generated by 29:, i.e. the set of all the forms in A* (J'"( M ) )that can be written as A where E A* ( J k ( M ) ) (here and in the following Einstein's summation convention on repeated indices is assumed). We will call C the contact ideal. We denote by V the distribution dual to the space of contact 1-forms. As well known, D is generated by a,, and

p i 197

D, :=

pi

(a/a~) + ~ ; + ~ ( d / d u,; )

where j = 1,.. . , k - 1. The vector fields D , define a first order differential operator called total derivative. A 1-form p E A' ( J k( M ) )is called horizontal if it annihilates all the vertical tangent directions in J k ( M ) .Thus, in local coordinates, a horizontal 1-form can be written as p = X(z, u a ,u,")dz. Given a section y E I ' ( M ) , we can consider its k - t h order jet-extension j'"(y) E I ' ( J ' " ( M ) ) ,requiring that j k ( y ) coincides with y on M and annihilates all the contact forms on J ' " ( M ) .Let A be a differential equation of order k

A

:=

F ( x , u a ,u:, . . . u;) = 0.

If A satisfies suitable regularity conditions we can see A as a submanifold of J ' " ( M ) .A section y E r ( M ) is a solution of A iff its k-th order jet-extension satisfy j'"(y) C: A. Given a vector field Xo E X ( M ) we define its k-th order prolongation as the unique vector field X E x ( J ' " ( M ) which ) reduces to Xo when restricted to M and which preserves the contact ideal C, i.e. L X (29) E C for any 8 E C. If Xo and X are given in local coordinates by

xo = E ( z , 4 & + F a ( z 1 4a m x = X o + Q3' au; "L

(1.1)

then the coefficients Q; of X satisfy the prolongation formula Q7+1

with

=

D,Q:

-

Ug+lDx
0.

(1) weakly Lie remarkable if E is the only maximal (with respect to the n) passing a t any 8 E I and inclusion) 1-dimensional equation in JT(E, admitting sym(E) as subalgebra of the algebra of its infinitesimal point symmetries; (2) strongly Lie remarkable if E is the only maximal (with respect to the inclusion) I-dimensional equation in J'(E, n) admitting sym(E) as subalgebra of the algebra of its infinitesimal point symmetries.

Of course, a strongly Lie remarkable equation is also weakly Lie remarkable. Some direct consequences of our definitions are in order. For each 0 E J T ( E , n ) ,let us denote by So(€) c TeJ'(E,n) the subspace generated by the values of infinitesimal point symmetries of E a t 0. Let us set S(E) dofUBEJ,.(E,n) So(€). In general, dim&(€) may change with Q E J T ( E , n ) .It is clear that dimsym(E) 2 So(€), for all 0 E J ' ( E , n ) . If the rank of S(E) at each Q E JT(E,n)equals dimsym(E), then S(E) is an involutive (smooth) distribution. The points of J T ( E n, ) of maximal rank of S(E) form an open set of J ' ( E , n ) (Ref. 12). It follows that E can not coincide with the set of points of maximal rank of S(E). The following statements (see Ref. 12) can be proved. (1) A necessary condition for the differential equation E to be strongly Lie remarkable is that dimsym(E) > dim€.

168

(2) A necessary condition for the differential equation E to be weakly Lie remarkable is that dim sym(E) 2 dim E. (3) If S(E)Ic is an Z-dimensional distribution on E C J T ( E , n ) ,then E is a weakly Lie remarkable equation. (4) Let S(E) be such that for any 0 6 E we have d i m s @ ( &> ) 1. Then E is a strongly Lie remarkable equation. Several examples of strongly and weakly Lie remarkable equations are provided in next sections. In the sequel, to make notation lighter, when n = 2 , we will use x and y instead of x1 and x 2 , respectively. 3. DEs characterized by their Lie point symmetries

Minimal surfaces in Rn+m. Let E = Rn+mendowed with the standard Euclidean metric. In the case n = 2 , m = 1, the mean and Gaussian curvatures are the real functions on J2(R3,2) defined by

+ + u;,uyy ,

1 (1 -t u;)uzx- 2uzuyuzy (1 H=2 (1 + u; + u 3 3 / 2

G=

uxxuyy - uE, (1 u:

+ +u y

'

The mean curvature can be generalized to surfaces in R2+m,with m 2 2 . The minimal surface equation E is then the equation { H = 0). A computation shows that the point symmetries of { H = 0} are the isometries and the homotheties of R2fm for 1 5 m 5 4. A more sophisticated computation proves that S(E) has maximal rank on an open subset of E. If d is the dimension of the minimal surface equation in R2+m and i is the dimension of the group made by isometries of R2+m and homotheties, then it turns out that, for m = 2 and m = 3 we have d > i while for m = 1 and m = 4 we have d = i. The above two arguments, together with the necessary and sufficient conditions of the previous section, allow us to prove the following theorem. Theorem 3.1. The equation of minimal surfaces in R4 and R5 is neither strongly nor weakly Lie remarkable, whereas it is weakly Lie remarkable in R3 and R6, provided that we remove a singular equation. Monge-AmpBre equations. We start with the following Monge-Ampbre equation

uxzuyy - dy= K.,

(1)

which, in the case K. = 0, is just the equation G = 0 for surfaces. If K. # 0, then Eq. (1) admits a 9-parameter group of point symmetries which span

169

a 7-dimensional distribution on the jet space except for a singular lowerdimensional submanifold. On the other hand, if K. = 0 then Eq. (1) admits a 15-parameter group of point symmetries whose associated distribution is 8-dimensional (provided that we remove a singular lower-dimensional submanifold). Hence the following result arises.

Theorem 3.2. Eq. (1) is weakly Lie remarkable if strongly Lie remarkable if K. = 0.

K.

# 0,

whereas it is

As shown by B ~ i l l a t Eq. , ~ (1) has the property of complete exceptionality. The use of such a property permitted to Boillat' to introduce higher order equations that are called generalized Monge-Ampbre equations. Among them one can find several Lie remarkable Just as an example, the third order Monge-Ampkre equation (uzzyuyyy

- UPy,)

+ X('(Lzzzuyyy - u z z y u z y y ) + X2(%zzUzyy

- UPzy)

+ p = 0,

where X and p are constants, is weakly Lie remarkable. 4. DEs which are uniquely determined by Lie algebras of vector fields on Rnfm

In what follows we shall consider only scalar partial differential equations in two independent variables, i.e., n = 2, m = 1, E = R3. We denote byZ(R3), d ( R 3 ) , P( R3)and C(R3), respectively, the isometric, affine, projective and conformal algebras of R3 with respect to the metric g = k l d s 8 d x kzdy 8 d y d u 8 du, where k l , k2 are non-vanishing real constants. The problem that we will recall here (see Ref. 13) consists in finding the strongly Lie remarkable equations associated with the previous Lie algebras of infinitesimal transformations.

+

+

The case 2(R3). The algebra Z(R3) has dimension 6. Then, in view of the necessary conditions, there can be strongly Lie remarkable equations only of order 1. In order to find them one prolongs the vector fields to the first jet space and computes the rank of the generated distribution. Such a rank decreases only on the singular submanifold u; u2 If-++=0, kl k2 which turns out to be a strongly Lie remarkable equation. Of course, the equation is nonempty if and only if the constants ki are not all positive.

170

The case A@’). The algebra d ( R 3 ) has dimension 12. By using the necessary conditions, we see that there can be strongly Lie remarkable equations of order 2 or 3. A computation13 shows that the strongly Lie remarkable second order equation is the homogeneous Monge-Ampkre equation G = 0, and that there exists also a strongly Lie remarkable equation of third order which has the following local expression: u:,utYy

+

+ 6ux,uxxxuxyuyyuyyy

- 6 ~ x x x ~ x , y u x ~ u t ~ (3)

- 6 u x x ~ x x x u x y y u t ~- 6 ~ : ~ u x y u x y y u y y y- 6 ~ : ~ u X x y ~ y y u y y y

+ 9uxxu:xyu~y+ ~ u ~ x u : y y ~ y y + 1 2 u ~ x x u : y u x y y u y y + 1 2 u x , u , x y ~ : ~ u y y y - 18u,,uX,yu,y~xyyuyy -8uxxxu:yuyyy

= 0.

To the authors’ knowledge, equation (3) has not been heretofore described in literature. The case P(R3). The algebra P ( R 3 )has dimension 15. Then, as in the previous case, equations G = 0 and ( 3 ) are strongly Lie remarkable. The case C(R3). The algebra C(R3) has dimension equal to 10. We have to look for second order strongly Lie remarkable equations. By analyzing the rank of the matrix of 2-prolongations of the vector fields we realize that the unique second order equation which is strongly Lie remarkable with respect to the conformal algebra is G = H 2 . By a direct computation, we realize that the unique second order scalar differential invariant I of the algebra formed by Z(R3) with the addition of homothetiesa is I = H2/G. Then I = k , with k constant, is a weakly Lie remarkable equation. Therefore we could look for strongly Lie remarkable equations among the equations I = k . From the above discussion, it follows that I = 1 is the strongly Lie remarkable equation we were looking for. Acknowledgments

This research has been supported by Departments of Mathematics of the Universities of Messina and Salento, PRIN 2005/2007 ( “Propagazione non lineare e stabilitb nei processi termodinamici del continuo” and “Leggi di conservazione e termodinamica in meccanica dei continui e in teorie di recall that a differential invariant is a function on a jet space which is invariant under the prolonged action of the vector fields of the given L:e algebra.

171

campo” ), G N F M , GNSAGA, joint EINSTEIN Consortium - RFBR project “Hamiltonian formalism for general P D E s and t h e BRST approach to quant u m field theories”.

References 1. K. Andriopoulos, P.G.L. Leach,l and G.P. Flessas, J. Math. Anal. Appl., 262 (2001), 256-273. 2. G. Baumann, Symmetry analysis of differential equations with Mathematica (Springer, Berlin, 2000). 3. G . W . Bluman and J. D. Cole: Similarity methods of differential equations (Springer, New York, 1974). 4. 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 f o r Differential Equations of Mathematical Physics, I. S . Krasil’shchik and A. M. Vinogradov eds. (Translations of Math. Monographs 182,AMS, 1999). 5. G. Boillat, Det Kgl. Norske Vid. Selsk. Forth., 41 (1968), 78-81. 6. G. Boillat, C. R. Acad. Sci. Paris, 313 (1991), 805-808; C. R. Acad. Sci. Paris SBr. I. Math., 315 (1992), 1211-1214. ~ 2000). 7. W.I. Fushchych, Collected W O T(Kyiv, 8. W.I. Fushchych and I. Yehorchenko, Acta App. Math., 28, (1992), 69-92. 9. N . H. Ibragimov, Transformation groups applied t o mathematical physics (D. Reidel Publishing Company, Dordrecht, 1985). 10. J. Krause, J. Math. Phys., 35 (1994), 5734-5748. 11. G. Manno, F. Oliveri and R. Vitolo, Proc. XI11 Int. Conf. on Waves and Stability in Continuous Media (R. Monaco, G. Mulone, S. Rionero, T. Ruggeri editors), World Scientific, Singapore, 2005, 420-431. 12. G . Manno, F. Oliveri and R. Vitolo, J. Math. Anal. Appl. 332 (2007), 767786. 13. G. Manno, F. Oliveri and R. Vitolo, Theor. Math. Phys., 151 (2007), 843850. 14. F. Oliveri, Note di Matematica, 23 (2004/2005), no. 2, 195-216. 15. P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. (Springer, 1991). 16. P. J. Olver, Equivalence, Invariants, and Symmetry (Cambridge University Press, New York, 1995). 17. V. Rosenhaus, preprint F. 18 Acad. Sci. Estonian SSR - Tartu (1982), Algebras, Groups and Geometries, 3 (1986), 148-166, and 5 (1988), 137-150.

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MOVING FRAMES AND DIFFERENTIAL INVARIANTS FOR LIE PSEUDO-GROUPS PETER OLVER School of Mathematics, UnaVeTSity of Minnesota, Minneapolis, M N 55455, USA E-mail: [email protected] http://www.math. umn. edu/- olver

JUHA POHJANPELTO Department of Mathematics, Oregon State University, Cornallis, OR 97331, USA E-mail: juhaamath. oregonstate. edu http://oregonstate. edu/Npohjanpp We survey a recent extension of the moving frames method for infinite-dimensional Lie pseudo-groups. Applications include a new, direct approach to the construction of Maurer-Cartan forms and their structure equations for pseudogroups, and new algorithms, based on constructive commutative algebra, for uncovering the structure of the algebra of differential invariants for pseudogroup actions. Keywords: Pseudo-group; moving frame; differential invariant; Grobner basis

1. Introduction Our goal in this contribution is to provide a brief survey of the moving frame theory for general Lie pseudo-groups recently put forth by the authors in Refs. 8-11, and in Refs. 3,4 in collaboration with J. Cheh. The moving frame construction is based on the interplay between two jet bundles: the infinite jets D ( ~ c) JM(MIM ) of local diffeomorphisms and the infinite jets J""(111, p ) of pdimensional submanifolds of M. Importantly, the invariant contact forms on D(Do)will play the role of Maurer-Cartan forms for the diffeomorphism pseudo-group and restricting these to the pseudo-group subbundle @") c D(O0)yields a complete system of Maurer-Cartan forms for the pseudo-group. Remarkably, the restricted Maurer-Cartan forms satisfy an "invariantized" version of the infinitesimal determining equations for

173

the pseudo-group, which can immediately be used to produce an explicit form of the pseudo-group structure equations. The construction of a moving frame is based on a choice of local crosssection to the pseudo-group orbits in J n ( M , p ) . The moving frame induces an invariantization process that projects general differential functions and differential forms on J" ( M ,p ) to their invariant counterparts yielding complete local systems of differential invariants and invariant coframes on J"( M , p ) . The corresponding invariant total derivative operators will map invariants to invariants of higher order. The structure of the algebra of differential invariants, including the specification of a finite generating set of differential invariants along with their syzygies or differential relations, will then follow from the recurrence formulae that relate the differentiated and normalized differential invariants. It is worth emphasizing that this final step requires only linear algebra and differentiation based on the infinitesimal determining equations of the pseudo-group action. Except possibly for some low order complications, the structure of the differential invariant algebra is then governed by two commutative algebraic modules: the symbol module of the infinitesimal determining system of the pseudo-group and a new module, named the "prolonged symbol module", containing the symbols of the prolonged infinitesimal generators. 2. The Diffeomorphism Pseudo-Group

Let V = V ( M ) denote the pseudo-group of all local diffeomorphisms of a smooth m-dimensional manifold M and, for each 0 5 n 5 00, let V(")c J"(M, M ) stand for the nth order diffeomorphism jet bundle. Local coordinates ( z , Z ) = (zl,. . . , zm, Z1,. . . , Zm) on M x M induce local coordinates ( z , Z(")) for g(") = j,'pl, E D ( n ) ,where the components 2; of Z(")represent the partial derivatives aB'pa/BzB of 'p at z . The cotangent bundle T*V(") splits into horizontal and vertical (or contact) components, cf. Refs. 1,7, with an induced splitting d = d, + d, of the exterior derivative. In local coordinates g(O0) = ( z , Z(")), the horizontal subbundle is spanned by the one-forms dza = d, z a , a = 1,.. . ,m, while the vertical subbundle is spanned by the basic contact forms m

Tg = d, 2;

= dZ; -

C Zz,c c=

dZC,

u = 1,.. . ,m, # B 2 0.

(1)

1

Composition of local diffeomorphisms induces an action of 1/, E V by right multiplication on diffeomorphism jets: R~(j,'pl,) = j,(p ~ q ! - ~ ) l + ( , ) .

174

A differential form p on V(")is right-invariant if RZ p = p, where defined, for every $ E V. The horizontal derivatives oa = d , 2" = 2: d z b , a = 1,.. . , m, of the right-invariant coordinate functions 2" : V(O)4 R form an invariant horizontal coframe, while their vertical derivatives pa = d,Z" = Ta are the zeroth order invariant contact forms. Let Dzl,. . . ,D,, be the total derivative operators dual to the horizontal forms CP. Then the higher-order invariant contact forms are obtained by successively Lie differentiating the invariant contact forms pa, pUag = D;pa = D$Ta,

where Dg p("3) = (

k = # B 2 0,

(2)

w e view the right-invariant contact forms . . . ) as the Muurer-Curtun forms for the diffeomorphism

= DZbl

... &

a = 1,.. . ,m,

. . . D Z b k .

pseudo-group. Next let p [ H ] denote the column vector whose components are the invariant contact form-valued formal power series

depending on the parameters H = ( H 1 , .. . , H"). Further, let d Z = p [ O ] CT denote the column vector of one-forms with entries d Z a = pa oa.

+

+

Theorem 2.1. T h e complete structure equations f o r the diffeomorphism pseudo-group are given by the power series identity

dpuHn = w u H n A muHn - dz),

where V H p [H ] =

(

aPa I[ H

1)

dg=VHp[O]Ac,

(3)

denotes the Jacobian matrix.

3. Lie Pseudo-Groups The literature contains several variants of the precise technical definition of a Lie pseudo-group, see e.g. Refs. 2,5,6,12,13. Ours is:

Definition 1. A sub-pseudo-group Q c V will be called a Lie pseudogroup if there exists no > 1 such that for all finite n 2 no: (a) the corresponding sub-groupoid 6'") c 2%") forms a smooth, embedded subbundle, (b) every smooth local solution 2 = p(z) to the determining system Q(") belongs to G,

175

(c)

@")

= pr("-"O)

is obtained by prolongation.

Let g c X denote the (local) Lie algebra of infinitesimal generators of the pseudo-group, i.e., the set of locally defined vector fields whose flows belong to 6. In local coordinates, we can view J"g c J"TM as defining a formally integrable linear system of partial differential equations

L(")(z,p )= 0

(4)

for the vector field coefficients v = C,"==, C a ( z ) a / a z a ,called the linearized or infinitesimal determining equations for the pseudo-group. A complete system of right-invariant contact forms on @") c D(")is obtained by restricting the Maurer-Cartan forms (2) to G("). Remarkably, constraints among the restricted forms can be explicitly characterized by an invariant version of the linearized determining equations (4).

Theorem 3.1. The linear system

L(")(2,p(")) = 0

(5)

provides a complete set of dependencies among the Maurer-Cartan forms p(") on ($"). The structure equations for the pseudo-group 4 are obtained b y imposing (5) o n the diffeomorphism structure equations (3).

4. Pseudo-Group Actions on Extended Jet Bundles For 0 5 n 5 00, let J" = J n ( M , p ) denote the bundle of nth order jets of p-dimensional submanifolds of M , cf. Ref. 7. We employ the standard local coordinates z(") = ( z , u ( " ) )= ( .. . 52 . . . uy . . . ) on J" induced by a splitting of the local coordinates z = ( 2 , ~=) (x', . . . ,zP,u',. .. ,uQ) on M into p independent and q = m - p dependent variables, Ref. 7. The pseudogroup 2, and their jets D(")act on J" in an obvious fashion. Let 'Id")denote the bundle obtained by pulling back the pseudo-group jet bundle g(") -+ M via the projection ?: J" -+ M . Local coordinates on X(")are given by (5, u("), g ( " ) ) , where the coordinates g(") parametrize the pseudo-group jets.

Definition 4.1. A moving frame p(") of order n is a G(") equivariant local section of the bundle 'H(") -+ J". Thus, in local coordinates, a moving frame defines a right equivariant map g(") = ~ ( ~ ) (u'")) x , to the pseudo-group jets. Moving frames are constructed through a normalization procedure based on a choice of a cross-section K(") to the pseudo-group orbits which we

176

typically choose by fixing the values of T, of the coordinates (2,u(,)). Then the group component g(") = y(")(z, u'")) of the moving frame is determined by the condition that g(,) . (2,d")) E K("). With a moving frame at hand, the invariantization of a function, differential form, etc., is obtained by replacing the pseudo-group parameters by their moving frame expressions in the transform of the object under the pseudo-group action. In particular, invariantizing the coordinate functions on J" leads to the normalized differential invariants

Hi =b(zi),

i = 1,.. . , p ,

IJ"= ~ ( u : ) ,

Q=

1,.. . , q ,

# J 2 0, ( 6 )

collectively denoted by (HII(,)) = ~ ( z~, ( ~ 1 Of ) . these, those corresponding to the constant coordinates determining K(") will be the constant phantom differential invariants, while the remaining basic differential invariants form a complete system of functionally independent differential invariants of order 5 n for the prolonged pseudo-group action on submanifolds. Secondly, invariantization of the basis horizontal and contact one-forms

mi= L(d.i),

d z = L(e;;),

(7)

i = 1,.. . , p , Q = 1,.. . , q , # K 2 0, under a complete moving frame pa leads to an invariant coframe on J". The associated total differential operators V l , . . . , Dp dual to wi commute with the pseudo-group action and, consequently, map differential invariants to new differential invariants. 5. Recurrence Formulae

The recurrence formulae connect differentiated invariants and forms with their normalized counterparts. Remarkably, they are established, through just linear algebra and differentiation, using only the formulas for the infinitesimal determining equation and the cross-section. Let v(O0)= ( P ( ~ ) ) *p(") denote the pulled-back Maurer-Cartan forms, which, in view of Theorem 3.1, are subject to the linear relations

L(")(H)I,V(")= ) L [ L ( n ) ( z , p ) ) ]= o , where we set

71.20 ,

(8)

$? = L ( F ^ " J ) = @?(I@), v(")),

(9)

~( (Pl2(ld*(4E TI, } c s^. (14)

It can be proved that, when the pseudo-group admits a moving frame,

the module J I z ( ' ) coincides with the symbol module associated with the prolonged infinitesimal generators. To relate this construction to the differential invariant algebra, we invariantize the modules using a moving frame. In general, the invariantization of a prolonged symbol polynomial

a=l # J > O

is given by

a=l # J > O

which we identify with the differential invariant

&=1 # J > O

Let 3 l(H,I(l)) = L(JI,(~)) denote the invan'antized prolonged symbol submodule. The recurrence formulae for the differential invariants I s take the form

V iIs

= Isi

+ Mb,i,

(16)

in which, unlike in (12), when degij >> 0, the leading t e r m ISi is strictly of higher order that the correction term. Now iteration of 16 leads to the Constructive Basis Theorem for differential invariants.

179

Theorem 6.1. Let D be a Lie pseudo-group admitting a moving frame o n a n open subset of the submanifold j e t bundle at order n*. T h e n a finite generating system f o r its algebra of local differential invariants i s given by: a ) the differential invariants Iv = I,, , where ul, . , . ,u1form a Grobner basis f o r the invariantized prolonged symbol submodule and, possibly, b) a finite number of additional differential invariants of order 5 n*.

3,

We are also able to exhibit a finite generating system of differential invariant syzygies. First, owing to the non-commutative nature of the the invariant differential operators Vilwe have the commutator syzygies

D J I5 - Dy15 = M5,j - M 5 , =~ NJ,y,,,

whenever

y= r ( J )

(17)

for some permutation r. Provided dega > n*, the right hand side NJ,y,z is of lower order than the terms on the left hand side. In addition, any algebraic syzygy satisfied by polynomials in 3 I(H,I(l)) provides an additional syzygy amongst the differentiated invariants. In detail, to each invariantly parametrized polynomial q(H,I ( 1 )s) ; = C J q J ( H ,I ( l ) ) s JE R[s] we associate an invariant differential operator q ( H ,1");D)=

q J ( H ,I ( ' ) ) D J ,

(18)

J

where the sum ranges over non-decreasing multi-indices. In view of (16), whenever Z ( H , I ( ' ) ;s, S ) E 3 l ( H , r ( i ) ) ,we can write

q ( H , 1");D) 1 5 ( H , r ( 1 ) ; s , s )= Iq(H,Io);s)z ( H , r ( l ) ; s , s + ) Rq,z,

(19)

+

where Rq,G has order < degq deg3. Thus, any algebraic syzygy - q v ( H ,I ( 1 )s), c v ( H ,I ( ' ) ;s, S ) = 0 of the Grobner basis polynomials of 3 I ( H , I ( l ) ) induces a syzygy among the generating differential invariants,

El=,

Cy=lqv ( H ,I ( ' ) ,D)IeV = R, where orderR < max {degqv + deg 5,). 1

Theorem 6.2. Every differential syzygy among the generating diflerential invariants is a combination of the following: ( a ) the syzygies among the differential invariants of order 5 n*, (b) the commutator syzygies, (c) syzygies coming f r o m a n algebraic syzygy among the Grobner basis polynomials. I n this manner, we deduce a finite system of generating differential syzygies f o r the differential invariant algebra of our pseudo-group.

180

Further details and applications of these results can be found i n o u r papers listed in the references.

Acknowledgments T h e first author is supported in p a r t b y NSF Grants DMS 05-05293. T h e second author is supported in p a r t by NSF Grants DMS 04-53304 and OCE 06-21134, and thanks PRIN “Leggi d i conservazione e termodinamica i n meccanica dei continui e in teorie di campo”, GNSAGA of Istituto Nazionale di Alta Matematica, and Dipartimento d i Matematica “E. De Giorgi” , Universith del Salento, Italy, for additional research support. References 1. I. M. Anderson, The Variational Bicomplex. (Utah State University, 1989). http://math.usu.edu/-fgdlp.

2. E. Cartan, Sur la structure des groupes infinis de transformations, Oeuvres ComplZtes, Part 11, Vol. 2 (Gauthier-Villars, Paris, 1953), pp. 571-714. 3. J. Cheh, P.J. Olver, J. Pohjanpelto, J . Math. Phys. 46, 023504 (2005). 4. J. Cheh, P.J. Olver, J. Pohjanpelto, Found. Cornput. Math., to appear. 5 . C. Ehresmann, Introduction B la theorie des structures infinitesimales et des pseudo-groupes de Lie, in Ge‘ometrie Diffe‘rentielle, Colloq. Inter. du Centre Nat. de la Rech. Sci. (Strasbourg, 1953), pp. 97-110. 6. A. Kumpera, J. Diff. Geom. 10, 289 (1975). 7. P. J. Olver, Equivalence, Invariants, and Symmetry, (Cambridge University Press, Cambridge, 1995). 8. P.J. Olver, J. Pohjanpelto, Regularity of pseudogroup orbits, in Symmetry and Perturbation Theory, eds. G . Gaeta, B. Prinari, S. Rauch-Wojciechowski, S. Terracini, (World Scientific, Singapore, 2005), pp. 244-254. 9. P. J. Olver, J. Pohjanpelto, Selecta Math. 11,99 (2005). 10. P.J. Olver, J. Pohjanpelto, Canadian J . Math., to appear. 11. P.J. Olver, J. Pohjanpelto, On the algebra of differential invariants of a Lie pseudo-group, (University of Minnesota, 2007). 12. J.F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, (Gordon and Breach, New York, 1978). 13. I.M. Singer, S. Sternberg, J . Analyse Math. 15, 1 (1965).

181

PERIODIC SOLUTIONS FOR A CLASS OF NON-LINEAR SCHRODINGER EQUATIONS IN D > 1 SPATIAL DIMENSIONS MICHELA PROCESI

Dipartimento di Matematica, Universitd d i Roma 3, 00100 Rome, Italy * E-mail: [email protected] In this report I will outline some recent results - o b t a i n e d in joint work with G. G e n t i l e on the existence of periodic solutions for a class of non-linear equations modelling waves, with Dirichlet boundary conditions on the D > 1 dimensional square. The main idea is t o combine a Lyapunov-Schmidt reduction and a renormalisation technique, in order to deal with the small divisors. In the massless case this implies finding a non-degenerate solution of the infinite dimensional bifurcation equation.

Keywords: Non Linear Schrodinger Equations; Periodic solutions; Small divisor problems.

1. Introduction The problem of finding small periodic solutions close to an elliptic equilibrium is a very classical one; to illustrate it let us consider the Hamiltonian: n

H h q )=C W j=1

h , ”

+ 4;) + H 3 h d

where w3 E R+ and H3 is some smooth function of degree higher than two. H ( p , q ) models a set of n harmonic oscillators coupled by a non-linearity. By the Lyapunov-Center Theorem, it is known that if -for i n s t a n c e wJ/wl @ Q for all j > 1, then periodic solutions persist. If the frequencies are all rational, then the situation is more complicated but the existence of the periodic solutions is still ensured by the Weinsteinl or the FadellRabinowitz2 theorem. In extending such results to infinite dimensional systems the level of difficulty of proving the persistence of periodic solutions is greatly increased by the fact than one in general has to solve a small

182

divisor problem (which in finite dimensional Hamiltonian systems occurs only when considering quasi-periodic solutions). Let us now introduce our models. Let S be the D dimensional square [0,.rrlD,and let 89 be its boundary. We consider the following classes of equations: (iat

+ P(-A) + p ) w = f(z,w ,v ) ,

U(5,t) = 0

i i

+

-

A

t)E

(Z,t)

(8tt (P(-A) v(z,t)= 0 (att

(2,

+ p ) 2 )21 = f(z,w,a),

+ p ) 21 = f(z, V ) , 21,

v(z,t)= 0,

E

s x R,

as x R,

t ) E s x R, ( z , t )E as x R,

(1)

(2,

(z, t ) E

s x R,

(z, t ) E

8s x R,

(2)

(3)

where A is the Laplacian operator, P ( z ) is a strictly increasing convex C1 function with P ( 0 ) = 0 and f ( z l v , V ) is analytic in all its arguments, super-linear in w and V and such that f ( z ,v,V ) = -f(-z, -w, - V ) :

+

with a,,,(%) odd for T s even and even otherwise. In (1) we require moreover that f is of the form

(4) where H ( z ,w,V ) is a real-valued function. For such classes of equations we prove existence of small periodic solutions with frequency w close to the linear frequency wg = P ( D ) , for all w in an appropriate “Cantor-like” set of positive measure (the restriction to a Cantor set is due to the small divisors). In general when looking for small periodic solutions for PDE’s one has to deal with two separate problems:

(A) one may need to solve a “small divisor problem” due to the fact that the eigenvalues of the linear term accumulate to zero, in the space of T-periodic solutions, for any T in a positive measure set. (B) If the linearised equation has infinitely many periodic solution with the same period- i. e. the equation is resonant- then the bifurcation equation is a (possibly very complicated) infinite dimensional non-linear integro-differential equation.

183

If the system is both resonant and has small divisors then in order to get non-trivial results one needs some non-degeneracy condition on the solution to the bifurcation equation. The case of non-resonant equations with Dirichlet boundary conditions in one space dimension was widely studied in the 90’s by using KAM theory by K ~ k s i n & P o s c h e and l~~~ Wayne;5 existence results were then extended to periodic boundary conditions by Craig&Wayne‘ and then to higher spatial dimensions by Bourgain.’-’ The main idea is to use a Lyapunov-Schmidt decomposition scheme to isolate the bifurcation equation and a Newton scheme to solve the small divisor problem. The two techniques are somehow complementary. The Lyapunov-Schmidt decomposition is more flexible: it can be successfully adapted to non-Hamiltonian equations and to “resonant” equations discussed above (see for instance Refs 10-12). On the other hand KAM theory provides more information, for instance on the stability of the solutions. The existence of periodic solutions for PDE’s in D > 1space dimensions was first proved by Bourgain7i8 by using a Lyapunov-Schmidt decomposition and the Newton algorithm. Subsequently Bourgain’ also proved the existence of quasi-periodic solutions for the nonlinear Schrodinger equation, with local nonlinearities, in any dimensions. Recently Eliasson and Kuksin13 proved the same result by using KAM techniques. Finally, Yuan14 uses a variant of the KAM approach to show the existence of quasi-periodic solutions for very large classes of equations. In his work the stability of the solutions is not obtained, but the proof greatly simplifies with respect to that given in Ref.s 9’13. The results I will describe are obtained by using first the classical Lyapunov-Schmidt decomposition and then the so-called “Lindstedt series method” (see for instance Ref. 11) to solve the small divisor problem. In Ref.s 15,16 we have used such techniques to find periodic solutions also in some non-Hamiltonian and in resonant cases, where the result was not known in the literature. In equations ( l ) , ( 2 ) and ( 3 ) we introduce a smallness parameter by rescaling

We now pass to the equation for the Fourier coefficients:

u(2,t) =

c

(YO ,m)s u €ZD+’

U~eiuotSm’x’

184

which gives &I/(&)

where -for while

UY(E)

example-in

= Efv({U},E),

Equation 1 one has

v E

zDfl,

&(E)

= -wn

+

(5) P(lm1) p

+

Since f ( z ,u,fi) is odd the Dirichlet boundary conditions imply that for all i = 1 , .. . , D uv = f u ~ where ~ ~&(Y) ) is the linear operator that changes the sign of the i-th component of v. Generally speaking the main feature which is used to solve the small divisor problem is the “separation of the resonant sites”. If D = 1 one can prove the following separation property: (1) if 16,+(~)I< a then Ikl > Ca-cO (this is generally obtained by restricting to a Cantor set). (2) if both IS,(E)I < a and IS,(E)I < a then either h = Ic or Ih - Icl 2 C(min{ IN7 Ikl

E

>Y.

Here t o and ( are model-dependent parameters, and C is some positive constant. In the case of periodic boundary conditions (2) should be suitably modified. It is immediately clear that (2) cannot be satisfied by our equations as the linear eigenvalues depend only on \mI2so that all the eigenvalues S,,, with n1 = n and ImlI = Iml are equal to &n,m.Ref.s 7,8 deal with this problem by weakening condition (2) in the following way: (2’) For each K sufficiently large, the sets of Ic E ZDfl such that lS,+l < 1 are separated in clusters, say C j ( K ) with j E N, such that each cluster has a diameter smaller than KE1 and dist(Ci, C j ) 2 K for some positive K independent [I. Now, in order to apply Spencer and Frolich’s method in the Newton algorithm, one has to control the eigenvalues of appropriate matrices of dimension comparable to ICjI. Such dimension goes to infinity with R and at the same time the linear eigenvalues go to zero, so that obtaining such estimates is a rather delicate question.

185

2. Main results We consider the quite general class of equations:

D(w) 4 2 , t ) = f(z,4 5 ,t ) ,u ( z ,t ) ,E ) , ( 2 , t ) E as x T, u(2,t)= 0,

D(w) u(5,t ) = E f ( Z ,

(z, t ) E 9 x ( 2 ,t ) E

4 2 , t ) ,u ( z ,t ) ,E ) ,

T,

T D x T,

(6)

(7)

where f(z,u,fi) is defined as in the examples, T := R/27rZ and D(w) is a linear (possibly integro-)differential wave-like operator with constant coefficients depending on a real parameter w = wo E . For u E ZD+l set u = (VO,m), with vo E Zand m = (vl,., . , V D ) E Z D and /vI = IvoI+lml = IvoI+lvll+. . .+Iv~l.For x = ( t , z )= ( t ,X I , . . . , Z D ) E RDtl set also u . x = vot m . x = vot vlzl . , . V D X D . Finally set = (0) u N.

+

+

+

z+

+ +

Assumption 1 (Conditions on the linear part). (1) D(w) is diagonal in the Fourier basis {eaVOt+m'x}yE~D+~ with real eigenvalues 6, ( E ) analytic in both u and E . (2) For all u E ZD+l one has either 6,(0) = 0 or I6,(0)/ 2 y ~ l v I - ~ Ofor , suitable constants yo,7-0 > 0 , and cl14co
0.

We now pass to the equation for the Fourier coefficients. We write

U€ZD+'

which gives an equation of the same type as 5: 6,(E)

U,(E)

= EfU({U},E),

v E ZD+l.

(8)

Definition 1. Following the standard Lyapunov-Schmidt decomposition scheme we split ZD+l in two subsets called and rl and treat the equations separately. By definition we call 11 the set of those u E ZD+l such that 6,(0) = 0; then we define = ZD+l \ rl. The equations 8 restricted to the and rl subset are called respectively the P and Q equations. Assumption 2 (Conditions on the Q equation). (1) For all v E 0 one has A,(&) := E - ~ ~ , ( E ) 2 c > 0 .

186

(2) T h e Q equation at

E =0

q(o) =

has a non-trivial non-degenerate solution

c

U;o)eivot+im.z

u p ) E R.

1

UEU

Definition 2. Let EO be a small positive constant. Given E E [0,EO] and y > 0, we set R ( E ) := {Y E ? !3 : lb,,(~)I < y/2} and RE, = U,,pEolR(&). Finally we call TI the subset \ Rco. Assumption 3 (Conditions on the set There exists a partition of RE, into disjoint sets { C j } j € ~such that, setting p j = minuEC, IvI, the following holds for positive constants C1 , Cz and appropriately small a , ,b' > 0.

+

(1) For all j E N one has ICj I 5 CI ( ~ o p j p,") . ( 2 ) For all E E [O,EO] there exist sets A,(&) c C j such that diam(Aj(E)) 5 ClPp,

diSt(Ai(E),Aj(E)) 2 Cz(min{P,lPj))Pl

R ( E )= U j E w A j ( & ) . (3) If by(&) versa.

=

S $ ( E ) ~ ; ( E ) , when

I~$(E)/ 5 y/2

then

I ~ ; ( E ) I 2 1, and

vice-

Theorem 1. Consider a n equation in the class (6), such that the A s s u m p tions 1, 2 and 3 hold. There exist a positive constant E O and a Cantor set E c [0,E O ] , such that for all E E E the equation admits a solution u(x,t ) , which is 2n-periodic in time and sub-analytic both in t i m e and in space, and such that

ju(x,t ) - q ( 0 ) ) 5 CE,

(9)

uniformly in ( x , t ) , where q ( O ) is defined in Assumption2. T h e set @ has positive Lebesgue measure and lim E-O+

meas(@n [0,E]) E

= 1,

where meas denotes the Lebesgue measure. Theorem 2. T h e model equations ( l ) , (2) and (3) satisfy the A s s u m p tions 1,2 and 3 provided that the linear m a s s t e r m p $ Q. Equation (1) with P ( x ) = 1 p = 0 satisfies the Assumptions 1,2 and 3 provided that f (x,u , V ) satisfies equation (4) with g(i) of degree higher than three in i and H ( x ,u , V ) = lvI4 + H ( x ,v,i) with H of degree higher than four in u and 'i.

187

2.1. A sketch of the proof

We follow closely the strategy and the notations of Ref. 16. Group the equations (8) for v E R by collecting together the v belonging to the same C j . Set d j = lCjl (number of elements in Cj); we assume that the sets C j = ( C j (l),. . . , Cj ( d j ) ) are ordered once and for all. Finally, define

Dj(j(w>= did6u(E)},,c3

Fj = { f ” ) U € C 3 ,

uj = { U ” } V E C 3 ,

parametrised by j E

(11)

N,so that the P equations spell

i 1 Dj(w) Uj 21” = Ed;

=E

F ~ ,j E N,

(&)f”, Y

(12)

E R.

We want to introduce a correction Gj to Dj( w ) acting for each fixed E only on the eigenspaces of those eigenvalues 6 , , ( ~ )which are small; moreover we want Gj to depend smoothly on E . We introduce a function E C”(R+, [0,1]) such that XI(.) = 1 if x < y/8, (x)= 0 if x > y/4, and all its derivatives are bounded uniformly in x. We also introduce the matrix

and define the renormalised P equations

h

where 7 is a real parameter, while Mj = Z l , j M j % l > jwhere , Mj and Lj are d j x dj symmetric matrices. The renormalised Q equations are: X u ( & ) 21”

= fv({~u’}u’EZD+’),

v E 0.

(15)

where the uvf such that v’ E are determined recursively by (14). By hypothesis (15) has a non-trivial non-degenerate solution at rl = 0 so that setting uu = do) U , for all v E Uwe obtain for U = { u , } , , ~a~linear equation:

+

+ C & v , f ~ ( { ~ $ ) }=)G(u(O), u v l u,{ I W ~ } ~ / / E ! ~ ) ,

L(&)Uv

(16)

U’EU

where G is either a t least linear in uU”or quadratic in U . By hypothesis the matrix in the left hand side of (16) is invertible. We have obtained two coupled recursive equations, which we solve by power series expansion provided that all the matrices D j ( w ) Gj are invertible with appropriate bounds.

+

188

T h e parameter q and the “counterterms” L, will have t o satisfy eventually the identities ( “compatibility conditions”) h

q=E,

M, = L,

(17)

for all J E N. We proceed as follows: we consider (E, M ) as parameters and we define a Cantor set D(y) such t h a t for all (E,M ) E D(y) the eigenvalues of D,(w) M,are bounded from below by yp,‘. For(E,M) E D(y) we construct as convergent power series the sequences U,(q, M , E ) E Rd3 and L,(q, E, M ) E Mat(d, x d,), which satisfy Equation (14). u , ( q , M , ~ )and L , ( q , & , M ) are analytic in q for lql 5 70, and such t h a t for all j E W

+

A

IU, (7,M , E ; 431 5 lqlKoe-“P~

(18)

I L, ( q ,E , M)(C,(a),C,(b))I 5 KoJrlle-“IC,(”)-C3(b)11’2

(19) To solve the compatibility equation we extend t h e matrix L3(&,E, M ) t o a C1 function -say L;(E, E , M ) - of (E, M ) and then apply t h e Implicit Function Theorem: we obtain M = M ( E ) .Finally we verify t h a t the Cantor set E := { e : ( E , M ( E ) )E D(2y) is of positive measure. References 1. A. Weinstein, Invent. Math. 20(1973), 47-57. 2. E.R. Fadell, P.H. Rabinowitz, Invent. Math. 45(1978) no. 2, 139-174. 3. S.B. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Mathematics 1556, Springer, Berlin, 1994. 4. S.B. Kuksin, J.Posche1, Ann. of Math. (2) 143 (1996), no. 1, 149-179. 5 . C.E. Wayne, Comm. Math. Phys. 127 (1990), no. 3, 479-528. 6. W. Craig, C.E. Wayne, Comm. Pure Appl. Math. 46 (1993), 1409-1498. 7. J. Bourgain, Internat. Math. Res. Notices 11(1994)(electronic). 8. J. Bourgain, A n n . of Math. (2) 148 (1998), no. 2, 363-439. 9. J. Bourgain, Green’s function estimates for lattice Schrodinger operators and applications, Annals of Mathematics Studies 158, Princeton University Press, Princeton, NJ, 2005. 10. B.V. Lidskii, E.I. Shul’man, Funct. Anal. Appl. 22 (1988), no. 4, 332-333. 11. G. Gentile, V. Mastropietro, M. Procesi, Comm. Math. Phys.256 (2005), no. 2, 437-490. 12. M. Berti, Ph. Bolle, Duke Math. J . 1 3 4 (2006), no. 2, 359-419. 13. L.H. Eliasson, S. Kuksin, K A M for non-linear Schrodinger Preprint, 2006. 14. X. Yuan,A K A M theorem with applications to partial differential equations of higher dimension Preprint, 2006. 15. G. Gentile, M. Procesi, Comm. Math. Phys. 262 (2006), no. 3, 533-553. 16. G. Gentile, M. Procesi, Periodic solutions for the Schrodinger equation with nonlocal smoothing nonlinearities in higher dimension, Preprint.

189

SINGULAR PERTURBATION FOR DISCONTINUOUS ORDINARY DIFFERENTIAL EQUATIONS M.A.TEIXEIRA IMECC- UNICAMP, CEP 13081-970, Campinas, Sdo Paulo, Brazil E-mail: teixeiraQime. unicamp. br P.R. DA SILVA

IBILCE-UNESP, Rua C. Colombo, 2265, CEP 15054-000 S. J. Rio Preto, Sdo Paulo, Brazil E-mail:[email protected] In this article some qualitative aspects of non-smooth systems on Rn are studied through methods of Geometric Singular Perturbation Theory (GSPTheory). We present some results that generalize some settings in low dimension, that bridge the space between such systems and singularly perturbed smooth systems. We analyze the local behavior around typical singularities and prove that the dynamics of the so called Sliding Vector Field is determined by the reduced problem on the center manifold.

Keywords: Regularization; vector fields; singular perturbation; discontinuous vector fields; sliding vector fields.

1. Introduction

One of the common frontiers of Mathematics, Physics and Engineering is the study of non-smooth dynamical systems. Certain phenomena in control systems, impact in mechanical systems and nonlinear oscillations are the main sources of motivation of our study concerning the dynamics of those systems that emerge from differential equations with discontinuous right-hand sides. We deal with non-smooth differential equations i = X O ( Z ) around p E Rn having a codimension-one submanifold M as its discontinuity set. More precisely let F : (JRn,p) (R, 0) be a C' function having 0 E R as a regular value with T big enough for our purposes. We denote F-' (0) by M . We write 0' = Slr (Rn,F ) the space of vector fields Xo such

-

190

that

XO(d

=

{ Xx,2(q)t ( q )

if 4 E M+ = { F ( q ) b 01, if q E M - = { F ( q ) < 0 } ,

(1)

where X i = ( f l , ...,fn),X ; = (gl, ...,gn) are C' functions. We write X O = ( X i ,X i ) , which we will accept to be multivalued a t the points of M . Filippov3 has considered differential systems with discontinuities in the right-hand sides. We are considering the case that such discontinuities occur on a smooth hypersurface M . There is a distinguished region C C M characterized by the following property: when a trajectory y(t) of X Omeets M a t p E C it slides on M for positive time. The study of this flow is our principal concern here. See figure (1). Sotomayor and Teixeira5 introduced a regularization process to study discontinuous vector fields. In this process for given small E O > 0 we get a one-parameter family of smooth vector fields X,,O< E 5 E O , such that: lim,+o X , = X O ;X,, is equal t o X i in all points of M+ whose distance to M is bigger than E O ; and X,, is equal to X i in all points of M - whose distance to M is bigger than € 0 . The dynamics of such families was studied by means of blow-up techniques.' To illustrate such a setting we briefly discuss an example:

Fig. 1.

Sliding vector field.

Example. Let X O = (X,',X,") be a non smooth system in R2 with X,'(x, y) = (y, 1) and Xi(x, y) = (1,l)with discontinuities occurring on M = {x = O}. Such a system induces on the sliding region, that is represented by C = { (0,y)ly < 0}, an invariant flow defined by X,"(x, y) = ( 0 , l ) . We consider now the following one-parameter family of smooth vector fields XE(x, y) = ((y 1)/2 cp(x/~)(y- 1 ) / 2 , 1 ) where p(s) is a smooth real

+

+

191

function which is equal to 1 if s 2 1 and equal to -1 if s 5 -1. Changing to polar coordinates z = T cos(0), E = T sin(0) we get

or equivalently, for

T

> 0, after

a time rescaling,

Our concern is to study its phase portrait for T -+ O + , near the set M = { ( 8 ,y)I (y 1) /2+cp(cot 0) (y - 1)/2 = 0}, which is, for T = 0, a continuum of singular points of (3) and a slow manifold of (2). It is the graph of a decreasing function which is 0 for 0 = 0 and tends to -cm when 0 -+ 7 r . For E = 0 the flow of (2) goes in the positive direction of the y-axis and the flow of (3) is horizontal. Observe that 0 > 0 for 0 < 0 < 0(y) and 8 < 0 for 8(y) < 8 < 7r with 0(y) given implicitly by y (cp(cot0) 1) - (cp(cot0) - 1) = 0. The phase portrait of the fast and slow dynamics of the singular problem for E = 0, and the phase portrait of the regularized vector field, for small c > 0, are illustrated in figure (2). In this paper we apply singular perturbation methods to studying general ordinary differential equations in Rn with discontinuous right hand side. Our main results state that the flow on the sliding region is determined by a reduced problem on the center manifold of a singular perturbation problem. In the next section we present all definitions with the necessary rigor and we state the main results.

+

+

Fig. 2. Fast and slow dynamics of the SP-Problem corresponding t o the fold case and its regularization.

192

2. Preliminaries and statement of the main results Let Xo E OT(IRn,F ) . In what follows we describe the regularization process introduced by Sotomayor and T e i ~ e i r a . ~ IR is a transition function if We say that a C' function cp : I% cp(z) = -1 for z -1, cp(z) = 1 if z 2 1 and cp'(z) > 0 if z E ( - 1 , l ) . The cp-regularization of Xo = (X,', X i ) is the 1-parameter family X , E C' given by

-

0. For the sake of simplicity we assume throughout the text that the transition map satisfies that p ( s ) = s for any -1/2 < s < 1/2. Denote V, = { g E IRn : F ( q ) E ( - ~ / 2 , ~ / 2 ) } .Our first result is:

Theorem 2.1. Consider X O E R T , r 2 2, X , its cp-regularization, and p E M . There exists an ordinary differential equation (*) 20' = Y(w,E), with Y a C' vector field and Y(w,O) = 0 for any w E M , such that the trajectories of X , in V, are in correspondence with the solutions of (*). Moreover, if (X,' - X i ) ( F ) ( p )# 0 then there is a Cr-' coordinate system ( 5 1 , ...,x,) such that (*) is written in the form = Ehi(z1,..., x,,E),

with hi

X; =

hn(z1,...,z,,E),

(5)

= 1,...,n.

E C'-',i

Systems like (5) are known as singular perturbation problems (SPproblems). AS usual in GSP-theory, we consider the time rescaling t = ET to get Xi

= hi(z1,..., x,,

E),

EX, =

h,(zl, ...,x,,E)

(6)

with xi = x i ( t ) , i = 1,...,n - 1. System (5) is called the fast system, and system ( 6 ) the slow system of SP-problem. The set S = { g E IRn : h(q,O)= 0) determines the singular points of the fast system, while the slow system characterizes the dynamics on it: h,(zl, ...,z, 0 ) = 0 , X i = hi(z1, ...,z,,O).It is called the reduced problem, and S is called the slow manifold. We refer to2 for an introduction to the general theory of singular perturbations. Consider X O E R' and p : IRn IR with p ( g ) being the distance between g and M . We denote by the vector field given by z(q) =

-

P(9)XO( 4 ) .

193

In what follows we identify Rn+' given by (z,(q),O).

zEwith the vector field on [(Rn\ M ) x R]c

Theorem 2.2. Consider X O E R' and p E M . There exist an open subset V of ((4, E ) : q E R",E 2 0 } , with ( p , 0 ) E V , a ( n 1)-dimensional manifold M , a smooth function @ : M R n f l and a SP-problem Y, on M such that @ sends orbits of YE)@'-l(V) to orbits of zEIV.

-

+

We distinguish the following objects in M : the sewing region is M I = { p E M : X,'F(p) . X i F ( p ) > O } ; the escaping region is M2 = { p E M : X A F ( p ) > 0, X i F ( p ) < 0 ) ; and the sliding region is M3 = { p E A4 : X,'F(p) < 0, X i F ( p ) > 0). The sliding vector field associated to X O is the vector field X: tangent to M and defined at q E M3 by X:(q) = m- q with m being the point where the segment joining q X,'(q) and q X,"(q) is tangent to M . It is clear that if q E M3 then q E Mz for - X O . Then we can define the escaping vector field on M associated to X O by X { = -(-XO)'. Here we use for both cases the notation X y . Moreover, for a discontinuous vector field X O = (X,', X i ) E R T ( U )with U C R",with discontinuity set M given by F , we denote by SR(X0) the union M2 U M3. According to the rules due to Gantmaher and Filipov3 if a point of the phase space which is moving in an orbit of X O = (X,', X i ) falls onto M I then it crosses M I over to another part of the space and the solutions of X O = (X,', X,") through points of Mz U M3 follow the orbit of Xf.

+

+

Theorem 2.3. Let M be represented by F ( x 1 , ...,xn) = x1 and X O E 0'. If for any q E M we have that X,'F(q) # 0 or X,"F(q) # 0 then the following statements hold: (a) There exists a singular perturbation problem (**) 0' = cy(r,O,p), p' = rp(T,B,p), with T 2 0, 6 E (O,n),p E M and a and 0 of class C', such that the sliding region S R ( X 0 ) is homeomorphic to the slow manifold c y ( O , O , p ) = 0 of (**). (b) The sliding vector field X f and the reduced problem of (**) are topologically equivalent. In Section 3 we prove Theorem 2.1 and we study vector fields with only one discontinuous coordinate. In Section 4 we prove Theorems 2.2 and 2.3.

3. Proof of Theorem 2.1 Proof of Theorem 2.1: Consider X O 6 R' with XA = ( f l , ...,fn),X i = (91,...,gn). The trajectories of X , on V, are the solutions of the differential system x i = (fi g i ) / 2 + F ( f i - gi) /2&, for i = I, ...,n. The time rescaling

+

194

+

+

gives the system xi = Y , = E (fi g i ) / 2 F (fi - g i ) / 2 . Thus we whereq = ( X I , ...,2,) andobservethat takeY(q,E) = (Yl(q,&),...,Y"(q,&)), Y(q,0 ) = 0 if q E M . If p E M then the eigenvalues of the linear part of Y for E = 0 are the solutions of the equation A" - ( X i - X;)(F)(p)X"-l = 0. Observe that the eigenvalue associated to the eigenvector transversal to the tangent space a t p E M is nonzero. So the Feniche12 theory (Lemma 5.3, pp. 67) can be applied and the desired coordinate system is obtained. I Now we consider the following class of discontinuous vector fields (0); = { X O= ( X , ' , X ; ) E RT : X i = ( h , , f z ,...,fn), i = 1 , 2 } . Moreover we suppose that F ( x 1 ,..., 2,) = x1 and that the transition function p satisfies p(s) = s for any -1/2 < s < 1/2. A direct computation proves that the trajectories of X, given by (4),on 1x11 < &/2 are the solutions of the singular system 7 =t/E

E i l

= h(X1,. . . y

Z,,E)

ii = f i ( 2 1 , .", x")l

(7)

+

with i = 2, ...,n and h = E ( h l hz) /2+x1 (hl - hz) / 2 . Moreover we have that: (a) the slow manifold is the set MU { q E Rn,O : hl(q) = h z ( q ) } ;(b) p E M I if and only if hl(p)hZ(p)> 0; ( c ) p E M Z if and only if h l ( p ) > 0 and hz(p) < 0; (d)p E M3 if and only if h l ( p ) < 0 and hz(p) > 0; (e) if p E M1 U Mz U M3 and hl ( p ) # h2 ( p ) then p is a normally hyperbolic point of (7). The last assertion follows of the linear part of system (7) a t ( x , ~=) (O,x~,...,xn,O) and that X = (1/2)(h1(0,22,..., x n , 0 ) - h 2 ( 0 , x z ,...r x n , O ) ) is the eigenvalue associated to eigenvector ( 1 , 0 ,...,0). The proofs of the following two propositions are similar to the ones in low dimension,* and they will be omitted. Proposition 3.1. Consider X O E (0); and p E Mz U M3. Then there exists a neighborhood V c (R", 0 ) with p E V such that the trajectories of the sliding vector field X f on V are solutions of reduced problem of the SP-problem ( 7 ) . Proposition 3.2. Consider X O E (52); and its p-regularization X, given b y ( 4 ) . If q E M2 U A43 is a hyperbolic singular point of Xf, then there exists E O > 0 such that for 0 < E < E O , X , has either a saddle point or a node point near q . 4. Proof of Theorems 2.2 and 2.3

Consider the discontinuous vector field X O E Rr and p E M . We take local coordinates such that F ( x 1 ,...,xn) = x1. The application of these theorems in the study of the local phase portrait around typical singularities

195

for

72

= 2 and n = 3 can be found in.ly4

Proof of Theorem 2.2. Define 1x1IXo(q), with q = (21, ...,z,).

% as the

vector field given by % ( q ) = The equations of its cp-regularization X , h

are

xi = (fi

+ S i ) /2 + p ( 5 1 / E )

% = 0, i

( f i - S i ) /2,

(8)

with i = 1,...,n. Consider 5 ': = { ( Z I , Z ) : Z ~ + ~ ~ = ~ ,Denote F _ > O T} .= 5': x Rn-1 x [0,m). Take @ : T EXn+' given by @(Tl,Z, ZZ, ...,z,,T) =

-

..-.

h

h

(rTl,zz, ...,z,,rS). There exists Y, on T such that @ * (YE)= X,. Denote M o = { (zl, ...,z, E ) : E 2 0} . We can join T with Mo \ ((0, zz, ...,z, 0)) where S: x EX"-' x ( 0 , ~is) identified with Mo \ ((0, z 2 , ...,z, 0)) by the map @. We get then

M

=

[TU(MO\((~,~Z,...,~C~,~)))I/(~-@(~)

For E = 0, on S i x R"-', we define Yo(@, 2 2 , ...,z,) taking 2 1 = r cos0, E = r s i n 0 and r = 0 in the system (8). On ((21 ,...,z,,0) : z1 # 0) we have YO= XO.For E > 0 we take the directional blow up 2 1 = T ~ . ? , E= ?. We observe that the directional blow up and the polar blow up are essentially the same. In fact, the map G(0,r ) = (cot 0, r sin 0) for 0 # O,T is such that p o G = p, with the directional component and cp the polar component. The trajectories of YOare solutions of the SP-problem h

j.2

+

= ( f i S i ) /2 = - sin0 [(fl

The trajectories of

+ cp (cote)

( f i- Si) / 2 ,

+ 91) /2 + cp (cote) (fl

-

91)/ 2 ] .

(9)

pEare the solutions of the SP-problem j.i

= (fi

C k 1 = (fl

+ S i ) /2 + p(T1) ( f i - S i ) / 2 , + Sl) /2 + Cp(E1) (fl - 91) /2.

So the theorem is proved,

(10)

I

Proof of Theorem 2.3. First we apply the regularization process and then we have that the trajectories of the regularized vector field X , are the solutions of the differential system & = (fi g i ) / 2 cp(z1/~)(fi - g2)/2, d = 0, with i = 1,...,n. Next we consider the polar blow up coordinates given by 21 = r cos 0 and E = r sin 0, with r 2 0 and 0 E [0, T I . Using these coordina.tes the parameter value E = 0 is represented by r = 0 and the blow

+

+

196

u p induces the vector field on [0, +m) x [ O , T ] x M given by T' = T cos ((fl+ g1)/2 + 'p (cote) (fl - g1)/2) , 8' = - s i n e ((fl + 91)/2 + cp (cote) ( f l - 91)/2) z; = T ((fi + g2)/2 + cp(c0t 0) (fi - $2)/2), 2 = 2, ..., 72. 1

Denote a =

(fl

+

+ g1)/2, b =

(fl

- g1)/2, cy = - s i n e [a

+

+

(11)

+ 'p (cot 0) b] p =

((f2 + 92)/2 cp (cot 0) ( f 2 - 92)/2,"', ( f n gn)/2 'p (cote) (fn - gn)/2). T h e equations of the system are described below: (i) On the region ( T , 8,p ) E (0) x (0, n) x M : 8' = - s i n e [u cp (cote) b] , p' = 0; and (ii) On the region ( T , 8,p) E (0) x (0, T } x M : 8' = 0, p' = 0. NOW the discontinuity set M is represented by 5': x M = {(cosO,sin8,p)le E [ O , T ] , ~E M } . T h e manifolds 0 = 0 and 6' = T are composed of singular points. T h e fast flow on S: x M is given by the solutions of system (i) and the slow flow is given by the solutions of the gi)/2 cp(cot8) (fi - gi)/2, i = 2, ..., n. reduced problem ii = (fi T h e slow manifold for 0 E ( 0 , ~ is ) implicitly determined by the equation u(O,B,p) cp(cot8) b(O,Q,p) = 0. We have that b(O,O,p) = 0 if and only if f l ( 0 , p ) = gl(0,p). Since X,'F(O,p) = fl(O,p),X;F(O,p) = g1(O1p), and X,'F(O,p) . X ~ F ( 0 , p )< 0, for any ( 0 , ~ )E SR(X0) we have that b(O,O,p) # 0 for any 0 E (0,7r),(O,p) E SR(X0). Moreover -1 5 - a ( o , e , p ) / b ( o , e , p ) 5 1, for all ( 0 , p ) E SR(X0). Since is increasing on (-l,l), the equation a(O,Q,p) cp(cotB)b(O,O,p) = 0 defines a continuous graph. Then the statement (a) holds. According to the = Xd X ( X i - Xd) with X E II% definition of Xf we have that such that X,'(xl,...,xn) X(Xi - X;)(sl, ..., s n ) = (O,y2,...,yn) for some y i l i = 2, ..., n. Thus it is easy to see that the Xf is given by Xf = (0, (fm- fm)/(fl- g l ) , ...,(fign - f n g l ) / ( f l - 91)). The reduced problem is j.i = (fi gi)/2 cp(cot S)(fi - gi)/2, for i = 2, ..n,constrained to g1)(fl - 91). Then we must have cp(cot 0) = -a(O, 8,p)/b(O, 8,p ) = - ( f l x i = (flgi - figl)/(fl - g l ) , for i = 2, ..,TI. It follows that the flows of Xf and of the reduced problem are equivalent. I

+

+

+

+

+

+

+

+

+

Xy

+

References 1. Buzzi, C., Silva, P.R., and Teixeira, M.A. (2006). A Singular approach to discontinuous vector fields on the plane, J . Diff. Equations 231,633-655. 2. Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations, J. Diff. Equations 31,53-98. 3. Filippov, A. F. (1988). Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications ( Soviet Series), Kluwer Academic Publishers, Dordrecht.

197

4. Llibre, J., Silva, P.R. and Teixeira, M.A. (2006). Regularization of Discontinuous Vector Fields via Singular Perturbation, J. Dynam. Differential Equation.19, n2, 309-331. 5 . Sotomayor, J., and Teixeira, M.A. (1996). Regularization of Discontinuous Vector Fields, International Conference on Differential Equations, Lisboa , 207-223.

198

C O N T I N U A T I O N OF P E R I O D I C SOLUTIONS I N CONSERVATIVE A N D R E V E R S I B L E SYSTEMS ANDRE VANDERBAUWHEDE

Department of Pure Mathematics, Ghent University Gent, Belgium E-mail: [email protected] http://cage. ugent. be/-avdb In this note we show how very general continuation results can be used to describe (1) families of periodic orbits in conservative systems, and (2) families of symmetric and doubly-symmetric orbits in reversible conservative systems. We describe a general approach which repairs the lack of submersivity for such problems due to the presence of first integrals. Our results can in particular be applied in the Hamiltonian context; we very briefly describe how they can be used for the continuation of choreographies in the N-body problem.

Keywords: Continuation; Conservative Systems; Reversible Systems; MultiSymmetric Solutions.

1. Introduction

Periodic orbits in systems having a first integral typically appear in oneparameter families, and the same is true for symmetric periodic orbits in reversible systems. When the number of (independent) first integrals increases, also the families of periodic orbits will become higher dimensional. This leads to the problem on how to parametrize the periodic orbits along these families, and, from a numerical point of view, how to calculate these families. A typical approach would be to fix the values of some appropriate first integrals and to work on level sets. However, theoretical results show that this is not always possible; moreover, it leads to implicit equations which one may want to avoid in numerical calculations. In this note we describe an approach which avoids this type of problems. Instead of imposing internal constraints to reduce the problem we actually extend it by adding appropriate gradient terms to the equations; in the end these terms will appear to vanish along the solutions we calculate, such that we get indeed solutions of the original problem. But moreover,

199

these additional terms regularize the problem, in such a way that standard techniques can be used for the augmented problem. Although most of the applications will be found in the Hamiltonian and reversible Hamiltonian context we do not specialize to thzt particular situation here. However, we very briefly indicate how our results can be used for the continuation of choreographies in the N-body problem, for example when one changes some of the masses. More details, both on the theory and the applications, can be found in a series of papers by Muiioz-Almaraz et al.i-3 2. Continuation of zeros of constrained mappings Let xo E R" be a zero of some smooth mapping f : R" R";if f is submersive at xo, i.e. if ImDf(z0) = R",then by the Implicit Function Theorem the zero set of f forms locally near zo a smooth ( m - n)-dimensional submanifold of Rm (the submersivity condition requires m 2 n). There are situations where the structure of the mapping f prevents the submersivity condition to be satisfied. Consider in particular the case where f = g - h for some smooth g, h : Rm -i R",and such that the space ---f

3 := { F : R" + R I F

og =F

0

h)

contains some non-constant functions. Differentiating the identity F ( g ( x ) )= F ( h ( x ) )at a zero xo of f (i.e. g(z0) = h(zo) =: yo) we find that

DF(y0) . Dg(x0) . 1 = DF(yo) . Dh(x0) .5, V5 E R", from which we deduce that ImDf(z0) = Im(Dg(x0) - Dh(x0)) c

nFEFker DF(y0).

Or stated differently, ImDf(x0)

c W',

W := {VF(yo) I F E 3).

(1)

We call a mapping f such as described here a constrained mapping; we are interested in the continuation of a given zero xo of such constrained mapping. The structural lack of submersivity due to the inclusion (1) can be easily repaired, as follows.

Lemma 2.1. Let xo E R" be a zero of a constrained mapping f = g Then all solutions ( x ,w)E EX" x W of the equation

+

g(x) = h ( x ) w

-

h.

(2)

suficiently close to (x0,O) are of the f o m ( x , O ) ,with I% E Rm a zero o f f .

200

Proof. Let P be the orthogonal projection in R" onto W', and let Fi E T (1 5 i 5 k := dim W ) be such that (VFi(y0) 1 1 5 i 5 k } forms a basis of W . Then (2) and the definition of T imply

P ( g ( z ) )= P ( h ( z ) ) and Fi(g(z))= Fi(h(z)), (1 L i I k). Since the mapping y I+ ( P ( y ) , F l ( y ) , Fz(y), . . . ,Fb(y)) is locally near yo a diffeomorphism from R" onto W' x Rb it follows that g ( z ) = h ( z ) and w = 0. 0 It follows from the lemma that finding the zeros z E R" near zo of f is equivalent to finding the zeros (z,w) E R" x W near (z0,O) of the "augmented mapping" f : R" x W --+ R" defined by f(z, W ) := g ( z )- h ( z )- W . I t is a trivial observation that f is submersive at (zo, 0) if and only if the inclusion in (1)is actually an equality. This leads to the following definition. Definition 2.1. We say that zo E R" is a normal zero of the constrained mapping f = g - h : R" -+ R" if g(z0) = h(zo) =: yo and if ImDf(z0) = WL, where W := {VF(yo) I F E F}.

In combination with lemma 2.1 and the I F T this leads to our main abstract result. Theorem 2.1. Near each of its normal zeros the zero set of a constrained mapping f : R" + Rn forms a smooth ( m - n k)-dimensional manifold, with k := dim W .

+

3. Periodic orbits in conservative systems

A direct application of Theorem 2.1 is given by periodic orbits in conservative systems. Consider a smooth n-dimensional system i=X(z),

(3)

denote its flow by Z ( t ; z), and assume that the space

F := { F : R"

4

R I D F ( z ) .X ( z ) = 0 )

(4)

contains some non-constant functions. Typical examples of such systems are Hamiltonian systems having some (continuous) symmetries: in this case n is even (say, n = 2 N ) , X ( z ) = J V H ( z ) for some smooth H : R2N -+ R and with J E C(R2N) the standard symplectic matrix, and T = { F : RZN+ R I { H , F } 5z 0 ) .

201

Periodic orbits of ( 3 ) with minimal period T of the mapping

> 0 are given by the zeros

f : R x Rn --+ R”,( T , x )H f ( T , z ) := Z(T;Z) - 5. This mapping f is a constrained mapping, since F ( Z ( T ;z)) = F ( x ) for all F E F.Given a zero (TO, 2 0 ) of f one can easily verify that ImDf(T0,zo)

= RX(z0)

+ Im(M - Id),

where M is the monodromy matrix of the To-periodic solution Z ( t ; 2 0 ) .We say that 5 ( t ;2 0 ) is a normal periodic solution if (TO, 2 0 ) is a normal zero o f f , i.e. if

RX(z0)

+ Im(M - I d )

W := ( V F ( z 0 )I F E 3}.

= W’,

By Theorem 2.1 such normal periodic orbits belong to a k-parameter family of periodic orbits of ( 3 ) , where k := dim W ;indeed, m = n+ 1, so we have a (Ic f 1)-dimensional manifold of zeros of f , but this manifold is of course foliated by 1-dimensional periodic orbits. To calculate this family of periodic orbits we choose Fj E 3 (1 5 j 5 k) such that ( V F j ( z 0 ) I 1 5 j 5 k } forms a basis of W , and we solve the “regularized equation” k

~ ( T , z , c x ) : = Z ( T , Z ) -EXN- ~ V F J ( Z O ) = O

(5)

j=1

for ( T , z , a ) E IR x R” x W k near (To,zo,O). All solutions will be of the form ( T ,z, 0), with ( T ,x) a zero of f. There is however a different (and for numerical purposes better) way of regularizing the periodicity condition. It is based on the following lemma.

Lemma 3.1. Let F E 3 and let k ( t ) be a periodic solution of

i = X ( Z )+ V F ( 2 ) . T h e n V F ( 2 ( t ) )= 0 , and ?(t) is actually a periodic solution of ( 3 ) . Therefore, instead of solving (5) one can use the modified system k

j. = X ( Z )

+ C ajVFj(z),

(6)

j=1

and obtain the desired family of periodic orbits of (1) by solving the new periodicity condition

Zmo&

2,a ) = 2 ;

(7)

202

here 2 m o d ( t ; x,a ) denotes the flow of (6). In practice one needs to add to (7) some phase conditions' in order to prevent the calculation of some "trivial continuations" of the starting solution. 4. Reversible systems

In (time)-reversible systems there are different ways for obtaining some periodic orbits, but again, in the presence of first integrals similar regularity (submersivity) problems as explained in Section 3 do arise. The system (1) is reversible if there exists a compact group G of linear operators on W" and a non-trivial character (group homomorphism) x : G + { 1,-1) such that

X ( g . X) = x ( g ) g * X ( Z ) , V g E G, V X E R". Then the flow Z ( t ; x ) will be such that Z ( X ( g ) t ; g . x) = g . S ( t ; z ) . An operator R E I? such that x ( R ) = -1 is called a reversor of (1); then ?(--t,Rz) = RZ(t,x). A (maximal) orbit y of (1) is called R-symmetric if R ( y ) = y; it is easily shown that this will be the case if and only if y n Fix(R) # 0, where Fix(R) := {x E R" I R x = x}. Setting t = 0 at the intersection point the corresponding solution x ( t ) will be such that x ( - t ) = Rx(t) and z ( t ) = R2x(t), i.e. the orbit y is contained in Fix(R2). Therefore, when considering R-symmetric solutions one can w.1.o.g. assume that R 2 = Id. Roughly speaking, doubly-symmetric solutions are solutions (orbits) which are symmetric with respect to two reversors. More precisely, we use the following definition. Definition 4.1. Let Ro and R1 be two reversors of the system (1) (the case R1 = Ro is allowed and, as explained before, we will assume that R; = R: = Id). Then we say that a solution z ( t )of (1) is (Ro, Rl)-symmetric if there exist t o < tl such that s ( t 0 ) E Fix(R0) and x(t1) E Fix(R1); we call [ t o ,t l ] a basic domain of such (Ro,Rl)-symmetric solution. Usually we will set t o = 0 and tl = T > 0.

As shown in Muiioz Almaraz et aL3 both the Figure Eight choreography for three equal bodies and Gerver's Supereight choreography for four equal bodies are examples of such doubly-symmetric solutions. Assuming that the basic domain is [0,T] the (Ro, Rl)-symmetric solution x ( t ) will be such that z(-t) = Rox(t) and z ( T + t ) = R l s ( T - t ) ;from these it is easily shown that the solution s ( t ) can be extended for all t E R. Moreover, we have for each m E Z that

203

+

x(2mT t ) = (R1Ro)"~(t); ~ ( 2 m T= ) (R~RO)~"-'R~Z(~~T); ~ ( ( 2 ~ 7 21)T)= ( R ~ R O ) ~ R ~ 1)T). Z ( ( ~ ~

+

+

In case there exists some A4 such that ( R ~ R o=) Id ~ (something which appears in many applications) then z(t) is automatically 2MT-periodic1 and also (Ro, R0)-symmetric with basic domain [0, M T ] ,in other words, we have a Ro-symmetric periodic orbit. This holds in particular with M = 1 in the case R1 = Ro (due to our assumption that Ri = Id). Next we consider the continuation of such (Ro,Rl)-symmetric solutions. Since Rg = R: = Id we can split the phase space Rn as Rn

= Fix(R0) @ Fix(-&)

= Fix(R1) @ Fix(-Rl);

we denote the corresponding projections as .rr$ and T:, respectively: 1 1 := -(Id f R1). := Z(1d f Ro), 2 We obtain (Ro, Rl)-symmetric solutions with basic domain [0,T] by looking for those z E Fix(R0) which are such that . r r y Z ( T ; ~= ) 0. If the mapping f o , l : R x Fix(R0) + Fix(-R1) given by fo,l(T,z) := n,Z(T;z) is submersive a t some of its zeros (T0,zo) then this zero belongs to a ddimensional manifold of zeros, where d = 1+ dim Fix(Ro) - dim Fix( -R1). In many applications the phase space is even-dimensional ( n = 2 N ) and dimFix(+Ro) = dimFix(fR1) = N ; in this case, and when the submersivity condition is satisfied, (Ro,Rl)-symmetric solutions belong to oneparameter families of such solutions. From now on we will assume these conditions on the fixed point subspaces of Ro and R1. Similarly as for periodic solutions in conservative systems also here the existence of some first integrals for the reversible system may cause the submersivity condition to fail; however, this time not all first integrals come into play. More in particular, we will consider the subspace F o ,of~F given by

TO+

~1"

F o ,:= ~ { F E F I F is constant on Fix(R0) U Fix(R1)).

~ some non-constant functions then the mapping f 0 , l is When F o ,contains a constrained mapping, as follows. We consider f0,l as a mapping from R x Fix(R0) into the full phase space R Z N and write it in the form fo,l(T,x) = 5(T; Z ) - rT?(T;z). Then we have N linear constraints, given by x;?(T;x) = (.rr;)'?(T;x), and moreover, we have for each F E F o , ~ that F(5(T;z)) = F(z)= F(r;Z(T; z));

204

for the last equality we use the facts that 2 E Fix(Ro), ntZ(T;x) E Fix(R1), and that F is constant on Fix(R0) U Fix(R1). I t follows that at each zero (To,zo)E R x Fix(R0) of fo,l we have ImDfo,l(To,zo) c Wk1 n Fix(-RI), where

} W O ,= ~ (VF(z1) I F E F o , ~and Observe that

51

z1 := 1(To,zo).

E Fix(R1) and that V F ( q ) E Fix(-R1) for F E F o , ~ .

Definition 4.2. We say that a zero (TO, 2 0 ) E R x Fix(R0) of f 0 , l generates a n o r m a l (Ro,Rl)-symrnetric s o l u t i o n of the reversible conservative system (1) if ImD fo,l(To, 2 0 ) = W&

n Fix(-R1).

(8)

Theorem 4.1. A normal (Ro, Rl)-symmetric solution belongs to a (1 f lco,l)-parumeter family of such (Ro, Rl)-symmetric solutions, where k0,l := dim W O , ~ .

+

This (1 ko,~)-parameterfamily can be obtained by considering the augmented system k0,l

X =X

+C ajVFj(~),

(X)

(9)

j=1

where Fj E F o ,(1 ~ 5 j 5 ko,l) are chosen such that (VFj(x1) 1 1 5 j 5 I c o , ~ }forms a basis of WO,J.Denoting by 20,1(t;x , a ) the flow of (9) one has to solve then the equation n;&IJ(T;

2,a ) =

0

(10)

for (T,x,cr) E R x Fix(&) x RIco,l near (To,zo,O). This problem is now regular (i.e. the submersivity condition is satisfied) and all solutions will have the form ( T ,z, 0), with (T,2 ) a zero of f 0 , l . The proofs can be found in Muiioz Almaraz et aL3 5 . Application to N - b o d y p r o b l e m s

As we have already indicated above a large area of application of the foregoing ideas and results can be found in Hamiltonian systems, and in particular in the N-body problem from celestial mechanics. For example, they can be used to calculate continuations of the by now well known Figure Eight

205

and Supereight choreographies for respectively three and four equal bodies. These choreographies can be considered either as periodic orbits, or as R-symmetric periodic orbits for an appropriate reversor R, or as (Ro,R1)symmetric solutions for several choices of reversors Ro and R1. The first integrals are the total energy and the components of the total linear momentum and the total angular momentum; also, since these choreographies are planar, one can work either in R2 or in R3. There is a particular aspect of this application which we should mention here. The general N-body problem has in addition to the translational and rotational symmetries generated by the first integrals also a rescaling symmetry. When applying our continuation schemes for the continuation of the Figure Eight or the Supereight one finds families of choreographies which can be obtained from the starting choreography by symmetries and rescalings. So, this way we get nothing particularly new. The way out is to fix the period (this prevents rescalings) and to use some external parameter as one of the variables of the problem. For example, one can change one or more of the masses. Using this approach one in fact allows the system to change, which implies that one can only use those symmetries which are compatible with these changes. Again we refer to the papers of Muiioz Almaraz et al.lP3 and to the paper by Doedel et al.4 for the results of such continuation studies; on the website h t t p : //www.maia.ub.es/"malmaraz/investigacion/Jaca/jaca.~l one can find some numerical data and graphics related to this problem

References 1. F.J. Muiioz-Almaraz et al. Physica D 181,1 (2003). 2. F.J. Muiioz-Almaraz et al. Monografias de la Real Academia de Zaragoza 2 5 , 229 (2004). 3 . F.J. Muiioz-Almaraz et al. Celestial Mechanics & Dynamical Astronomy 97, 17 (2007). 4. E.J. Doedel et al. Int. J. Bifurcation and Chaos 13, 1353 (2003).

206

EMERGENCE OF SLOW MANIFOLDS IN NONLINEAR WAVE EQUATIONS FERDINAND VERHULST

Mathematisch Instituut, University of Utrecht PO Box 80.010, 3508 TA Utrecht, T h e Netherlands verhulst @math.uu.nl Averaging-normalization, applied to weakly nonlinear wave equations provides a tool for identification of slow manifolds in these infinite-dimensional systems. After discussing the general procedure we demonstrate its effectiveness for a Rayleigh wave equation t o find low-dimensional invariant manifolds.

Keywords: Slow manifold; wave equation; Rayleigh; normalization.

1. Singular perturbations and slow manifolds

Consider the equation j: =

f(.)

+ Eg(Z)

with E a small, positive parameter. Suppose that the equation y = f ( y ) that arises if E = 0, contains an invariant manifold M o . One of the basic questions is, does this invariant manifold persist if E > 0 in (slightly deformed) shape ME? For ODEs, there are many results, see13 for a survey, but for PDEs the literature is still restricted. The existence and approximation of invariant manifolds of differential equations is strongly related to hyperbolicity properties of Mo. In the case of singular perturbations of ODEs with initial values, these hyperbolicity properties are directly related to the attraction properties of the regular (outer) expansion; this also plays an essential part in the actual asymptotic approximations. Under additional assumptions such a regular expansion is associated with the existence of a so-called slow manifold. Theorems by Tikhonov, O’Malley-Vasil’eva provide the foundations for asymptotic approximations. The theoretical basis for the existence of slow

207

manifolds in ODES was given by Fenichel; see5i7i14 for references, details and applications. To be explicit, consider the autonomous system i=Ef(z,y)+Ez...

jr = g(z, y)

+

E..

.,

,

Z E D C P ,

y E

G c Rm,

where the dot denotes differentiation with respect to t , E is a small, positive parameter. Putting E = 0 we find from the second equation a family of equilibria given by g(z,y) = 0 with x a parameter. The basic assumptions are that all real parts of the eigenvalues of the linearization of the zero set with respect to y are nonzero and that the zero set corresponds with a compact manifold in Rn+m.In this case the zero set y = @(z)of g(z,y) corresponds with a first-order approximation MO of the n-dimensional (slow) manifold M E .

Usually, y is called the fast variable and z the slow variable. In this case of a slow-fast system, because of the presence of the parameter E , the slow manifold Mo is normally hyperbolic. If MO is a compact manifold that is normally hyperbolic, it persists for E > 0, i.e., there exists for sufficiently small, positive E a smooth manifold M E close to Mo. Corresponding with the signs of the real parts of the eigenvalues, there exist stable and unstable manifolds of M E ,smooth continuations of the corresponding manifolds of M o , on which the flow is fast. This idea has been very fruitful for finite dimensional systems; in this paper we will discuss extension to infinite dimensional problems. It turns out that the dimension of the invariant manifolds obtained in this way, is usually much lower than the dimension estimates obtained for inertial manifolds. 2. Extension to PDEs Extension to infinite-dimensional problems is possible but raises special difficulties, depending on the choice of operator and the type of problem formulation. A paper discussing parabolic and hyperbolic problems is' , see also213,15.In6 the emphasis is on the persistence of invariant manifolds in dissipative equations. A prominent technique is contraction which takes often the form of Gronwall's lemma. We will briefly discuss the results of2l3 where parabolic PDEs have been considered. The equations and their solutions are associated with a Banach space X and a C1 semiflow defined on X. First one has to identify

208

a compact, connected invariant manifold MO of the flow that is normally hyperbolic. One has to prove then that persists under perturbations of

the semiflow where persistence again means 'small quantitative deformation of MO without qualitative (topological) changes'. In the case of parabolic equations we have the additional problem that backwards solutions may not exist, so certain maps associated with these equations are not invertible. Also the finite dimensional geometric tools of dynamical systems theory like the smooth continuation of tangent bundles, have to be developed for infinite dynamical systems. Another problem is the lack of compactness that is typical for these systems. Applications often refer to systems of the form Ut

= AU

+ f ( ~ +) E

* * *

,

where u is an n-vector, f ( u ) represents the nonlinear terms. The manifold Mo is identified for E = 0 and conditions are provided in2v3so that it persists for E > 0. 3. Formulation for wave equations

For wave equations, a suitable linear operator usually generates a group instead of a semigroup. Variation of constants enables us to formulate a slowly varying system that permits normalization. This facilitates the identification of the invariant manifold MO and the persistence properties. This idea was first explored in" with an application to a parametrically excited wave equation: utt - C 2 u,, +&put

+ (w," + E Y C O S ~ ) U=

E Q U ~ t,

2

o,o < x < T ,

(1)

with Neumann boundary conditions u,(O,t) = u,(T,~) = 0 and ,l? >0 (damping), In" the experimental motivation for this model is discussed, for instance a line of coupled pendula with vertical (parametric) forcing or the behavior of water waves in a vertically forced channel. The analysis of slow manifolds of this equation can be found in" together with the bifurcational behavior of the invariant manifolds. More in general, consider semilinear initial value problems of hyperbolic type, Utt

+ AU = E ~ ( u ,t , ~ ) ~, ( 0=) Ut,

U O ,U t ( 0 ) = V O ,

(2)

where A is a positive, self-adjoint linear operator on a separable Hilbert space and f will be specified. Here we will be concerned with the case that

209

we have one space dimension and that for E = 0 we have a linear, dispersive wave equation by choosing:

Au

-u,,

=

+ 21.

To produce a system in vector form, one writes

and uses the operator (with eigenvalues and eigenfunctions), associated with this system. To focus ideas, consider the case of boundary conditions u(0,t) = U ( T , t ) = 0. In this case, a suitable domain for the eigenfunctions is { u E W12'(O, T ) : u(0) = U ( T ) = 0). Here W1i2(0,n) is the Sobolev space consisting of functions u E Lz(0,T ) that have first-order generalized derivatives in Lz(0,T ) . The eigenvalues are A, = w: = d m ,n = 1 , 2 , ' . . the corresponding eigenfunctions w,(z) = sin(nz) and the spectrum is nonresonant. Substitution into Eq. (2) the expansion M

n=l and taking inner products with v,(z), m = 1 , 2 , .. . , produces the infinite set of coupled second-order equations

ii,

+ wiu,

= &F(U),

(3)

with u representing the vector with elements u,, n = 1 , 2 , 3 , .. . . The next step is to transform system (3) into a slowly varying system by the (variation of constants) phase-amplitude transformation

u,(t) = ~ , ( t )cos(wnt

+ & ( t ) ) , & ( t ) = -w,m(t)

sin(w,t

+ $,(t)).

(4)

The resulting system is of the form f n = &Fl(T,,$,,t),

4, = &FZ(T,,$,,t),

n =1,2,...

(5)

with (Fl(r,, $, t ) ,F~(T,,$,,t ) ) an almost-periodic function in a Banach space, satisfying Bochner's criterion, see for instance14. Its average (F:, F i ) is defined by:

210

We can apply normalization by the averaging transformation, seel2)l4.An explicit example will be given in the next section. The normalized system will be of the form

+

i = EG(z) O(c2)

(7)

with 2 representing the infinite dimensional system of transformed phases and amplitudes. After introducing the normalizing transformation, we can still in principle obtain the exact solution by solving the resulting Eq. (7) including the O(E2)-terms to find r,(t) and qn(t).‘In principle’, because in nearly all cases the solution of the full system can not be given explicitly. Omitting however the O(E2)-terms and solving the resulting equations produces an approximation of the solutions in the following sense: Assume that the righthandside vector fields of system ( 5 ) are continuously differentiable and uniformly bounded on b x [ O , o o ) x [0,E O ] , where D is an open, bounded set in a suitable Banach space X. Solving system (7) to O ( E )produces and o(1)-approximation of rn(t),I,/,Jn(t), valid on the timescale I/&. For a more precise formulation see4iI4. Suppose that not all the linear normal modes, i.e. the decoupled one degree of freedom solutions in the case E = 0 (filling two-dimensional manifolds), are solutions of system (3) for E > 0, but applying averagingnormalization, the normal modes are solutions of the averaged system to O ( E ) The . Lyapunov manifolds, smooth continuations for E > 0 of the twodimensional normal mode manifolds, persist for the original system (2) or (3) if we have normal hyperbolicity of the normal mode solutions of the averaged system. In an amplitude-phase representation, this happens for instance if we have attraction in the amplitude equations and parallel flow from the phase equation to O ( E ) . We also have to check that in the spectrum the eigenvalues are sufficiently separated, preferably by a gap size, independent of the mode number. 4. A nonlinear Rayleigh wave equation

A benchmark example of a nonlinear wave equation was studied by Keller and Kogelman in8, who consider a Rayleigh type of excitation described by the equation (8)

21 1

with boundary conditions u(0,t ) = u(n, t ) = 0 and initial values u ( x ,0) = $(z),ut(x,O) = $(x) that are supposed to be sufficiently smooth. Apart from being a classical example, the equation plays an essential part in modeling self-excited vibrations of waves produced by an external wind field or other types of fluid flow perturbations. The authors of8 use multiple timing to first order, which yields the same results as averaging. We have for the eigenfunctions and eigenvalues

v n ( x ) = sin(nz), A, = w: = n2 + 1, n = 1,2,..., and to perform our averaging-normalization scheme,we propose to expand the solution of the initial boundary value problem in a Fourier series with respect to these eigenfunctions. Substituting the expansion C u,(t)w,(~) into the differential equation, we have M ..

w

M

w

C ii, sin n x + C(n2+ l ) u nsin n x = C tin sinnx - 5(Ctin sin n ~ ) ~ . ~~

&

E

n=l

" n=l

n=l

n=l

When taking L2-inner products with sinmx, m that we find the system

=

1 , 2 , . . ' , it is shown in14

where the dots stand for nonresonant terms. This infinite system of ordinary differential equations is equivalent to the original problem. For the variation of constants transformation we have to avoid amplitude-phase variables as the transformation is singular for normal modes. We have checked however, that amplitude-phase variables produce the same results. To start with, we use the transformation

un(t)= a,(t) C O S W , ~ + bn(t) sinw,t, tin(t) = -wnan(t) s h u n t wnbn(t)COSWnt.

+

(9) (10)

After averaging, more insight is obtained by using amplitude-phase variables rn,& from (4). Putting

a,2 we find

+ b,2 = r i = En, n = 1 , 2 , . . . ,

212

We have kept the same notation for the variables after normalization. To obtain asymptotic approximations we omit the O ( E ~ terms, ) replacing En, by their approximations En,4,. We have immediately a nontrivial result: starting in a mode with zero energy, this mode will not be excited on a timescale l / & .Another observation is that if we have initially only one nonzero mode (a normal mode), say for n = m, the equation for Em becomes

+,

We conclude that in the one-mode case we have stable equilibrium at the value 16 E* = O(E). 3(m2+1)

+

The results until this point can be found in the literature, usually without error estimates. We note, that the averaging theorem formulated above, yields that the approximate solutions have precision O ( E ) on the timescale 1 / ~It. can be shown that if we start with initial conditions in a finite number of modes, the error is O ( E )(see14). Slow manifold theory enables us to formulate stronger results. It follows from the normal hyperbolicity of the normal modes (E& has eigenvalue - E , the infinite number of other modes grow with eigenvalue + E ) , that for the original wave equation (8), for E > 0 but small, an infinite number of two-dimensional, unstable Lyapunov manifolds exist &-closeto the normal mode coordinate planes, containing the normal mode solutions in the case & = 0. It is not difficult to extend this to cases with more dimensions. Consider for instance the case of excitation of two modes, k and m. Putting En = 0, n # k , m , the system becomes

Em

=

&Em( 1

3

-

E(m2+ l ) E m

1

-

+

1+

i ( k 2 1)Eh

O(E').

Considering the system for Ek, Em, we find four critical points and three heteroclinic connections: (0,O) unstable node; ( 3($+1), 0) stable node; (0, 3 ( m +1) ) stable node;

',"

213

( E i ,E;) with EZE; > 0, saddle corresponding with an unstable 2-torus in the original system. The saddle point has heteroclinic connections with the other three critical points. The two degrees of freedom system in the full system is normally hyperbolic. Again we conclude from slow manifold theory, that for the original wave equation an infinite number of corresponding higher-dimensional manifolds exist. 5 . Discussion

In the example of the Rayleigh wave equation we have found an infinite number of finite-dimensional slow manifolds. They are unstable and difficult to observe. In a number of models, an asymmetric force field plays a part, for instance in the case of galloping motion of hanging cables in a wind field. For Eq. (8) this means the addition of a term ~ f ( uwith ) f(u)an even function that can be expanded in a Taylor series in u.In particular we have f ( 0 ) = O,df/du(O) = 0 with examples like f(u) = u 2 , u2 - u4 or usinu. The normal form analysis to first order does not change and we draw the same conclusions as in section 4. To deal with other wave equations, the difficulty is often the number of possible resonances. For instance, making the Rayleigh wave equation nondispersive (omitting u on the lefthand side in Eq. (8)) produces a very complicated normal form. Valid asymptotic approximations can be obtained by truncation methods, but one loses then the results on the existence of slow manifolds. The parametrically excited wave equation mentioned in section 3 is even more interesting as it shows bifurcations for rather small values of E involving the interaction of a finite number of modes as found experimentally in". This analysis can be found in".

References 1. P.W. Bates and C.K.R.T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Reported 2, (1989) pp. 1-38. 2. P.W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manzfolds for semiflows i n Banach space, Mem. Amer. Math. SOC.135, (1998) no. 645. 3. P.W. Bates, K. Lu and C. Zeng, Persistence of overflowing manifolds f o r semiflow, Comm. Pure Appl. Math. 12, (1999) pp. 983-1046. 4. R.P. Buitelaar, The method of averaging i n Banach spaces, Thesis (1993), University of Utrecht.

214

5. C.K.R.T. Jones, Geometric Singular Perturbation Theory, in R. Johnson, ed., Dynamical Systems, Montecatini Terme , Lecture Notes in Mathematics 1609, Springer-Verlag, Berlin (1994) pp. 44-118. 6. D.A. Jones and E.S. Titi, C1 Approximations of inertial manifolds f o r dissipative nonlinear equations, J. Diff. Equations 127, (1996) pp. 54-86. 7. T.J. Kaper, A n introduction to geometric methods and dynamical systems theory f o r singular perturbation problems, in Jane Cronin and Robert E. O’Malley, Jr., eds., Analyzing multiscale phenomena using singular perturbation methods, Proc. Symposia Appl. Math., AMS 56 (1999) pp. 85-131. 8. J.B. Keller and S. Kogelman, Asymptotic solutions of initial value problems f o r nonlinear partial diflerential equations, SIAM J. Appl. Math. 18, (1970) pp. 748-758. 9. Yu.A. Kuznetsov, Elements of applied bifurcation theory, 3d ed., Springer (2004). 10. Hi1 G.E. Meijer and Ferdinand Verhulst, Emergence and bifurcations of slow manifolds in nonlinear wave equations, to be published. 11. R.H. Rand, W.I. Newman, B.C. Denardo and A.L. Newman, Dynamics of a nonlinear parametrically-excited partial differential equation, Proc. 1995 Design Eng. Techn. Conferences 3 (1995) pp. 57-68, ASME, DE-84-1. (See also Newman, Rand and Newman, Chaos 9, pp. 242-253, 1999.) 12. J.A. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems, revised ed., Springer (2007). 13. F. Verhulst, Invariant manifolds in dissipative dynamical systems, Acta Appl. Math. 87, (2005) pp. 229-244. 14. F. Verhulst, Methods and applications of singular perturbations, boundary layers and multiple timescale dynamics, Springer (2005) 340 pp. For comments and corrections, see the website www.math.uu.nl/people/verhulst 15. C. Zeng, Persistence of invariant manifolds of semiflows with symmetry, Electronic J. Diff. Eqs vol. 1998, (1998) pp. 1-13.

215

SUPERINTEGRABLE SYSTEMS WITH SPIN IN TWOAND THREE-DIMENSIONAL EUCLIDEAN SPACES P. WINTERNITZ* and I.YURDU$EN** Centre de Recherches Mathe'matiques, Universite' de Montre'al, C P 6128, Succ. Centre- Ville, Montre'al, Quebec H3C 357, Canada *E-mail: [email protected]. ca **E-mail: yurdusenOcrm.umontrea1. ca The concept of superintegrability in quantum mechanics is extended t o the case of a particle with spin s = 1/2 interacting with one of spin s = 0. Nontrivial superintegrable systems with 8- and 9-dimensional Lie algebras of firstorder integrals of motion are constructed in two- and three-dimensional spaces, respectively.

Keywords: Integrability; superintegrability; quantum mechanics; spin.

1. Introduction

A superintegrable system in classical and quantum mechanics is a system with more integrals of motion than degrees of freedom. A large body of literature on such systems exists and is mainly devoted to quadratic superintegrability. This is the case of a scalar particle in a potential V(F)in an n-dimensional space with k integrals of motion, n 1 5 k 5 2n - 1, all of them first- or second-order polynomials in the momenta (see e.g. Ref. 1-6 and references therein). Maximally superintegrable systems have 2n - 1 integrals of motion and are of special interest. In classical mechanics all bounded trajectories in such systems are closed. In quantum mechanics these systems have degenerate energy levels and it has been conjectured that they are all exactly solvable.6 Quadratic integrability for a Hamiltonian of the form 1 H = -3' 2 V(F),

+

+

i.e. the existence of n second-order integrals of motion in involution, is related to the separation of variables in the Hamilton-Jacobi, or the Schrodinger equation, respectively. Quadratically superintegrable systems

216

are multiseparable. The non-abelian algebra of integrals of motion usually has several non-equivalent n-dimensional Abelian subalgebras, each of them leading to the separation of variables in a different coordinate system. The situation changes when one goes beyond Hamiltonians of the type of Eq. ( l ) ,or considers higher-order integrals of motion. If a vector potential is added in Eq. (1),corresponding to velocity dependent forces, e.g. a magnetic field, then second-order integrability no longer implies the separation of variables7-10 and the same is true in the case of third-order integrals of motion for Eq. (l).11-13 The purpose of this contribution is to report on a research program which investigates the concepts of integrability and superintegrability for systems involving particles with spin. Here we restrict ourselves to the simplest case of the interaction of two particles with spin 0 and spin 1/2, respectively. We write the SchrodingerPauli equation including a spin-orbit term as 1 HQ = K(r', z{vi(q, Q,

[--a :: +

+

z.i}]

where {, } denotes an anticommutator, 01, u2, u3 are the usual Pauli matrices, Q is a two-component spinor and L is the angular momentum operator. The Hamiltonian given in Eq. (2) would describe, for instance a low energy (nonrelativistic) pion-nucleon interaction. In this paper we restrict ourselves to first-order integrability. Thus we require that the integrals of motion should be first-order matrix differential operators .

3

3

3

with cro = I2. For particles with spin zero only components with p = 0 in Eq. (3) would survive and the condition [ H , X ]= 0 (with V1 = 0), would imply a simple geometric symmetry. 2. The Two-Dimensional Case Let us first consider the case when motion is constrained to a Euclidean plane. We assume Q ( q = Q(x,y), set p3 = 0, z = 0 and write the Schrodinger-Pauli equation given in Eq. (2) as

with

217

The operator Eq. (3) reduces to

X

=

+

+ +

+

+

(AOPI BOP^ 40)1+ (Alp1 B 1 ~ 2 41)a3 1 [ ( ( p lAo) (p2Bo))I ((plA1) ( P ~ B I ) ) Q ].

+z

+

+

(5)

The commutativity condition [ H ,XI = 0 implies 12 determining equations forthe8functionsA,(x,y),B,(x,y),~,(x,y) andV,(x,y) ( p = 0,1).l?rom these we obtain A, = wPy

+ a,,

B, = -wPx

+ b,

,

where w,, a, and b, are real constants and ( p , v)=(O,1).The above equations can be simplified by rotations in the xy-plane and by gauge transformations of the form

u = ( e;

= U-'HU,

e!q ,

a =a([),

Y [ = -X.

(7)

The gauge transformations leave the kinetic energy invariant but modify the potentials

d! Vl =v1+-,

1J2 1 2 vo=vo+(1+-)(--++vl)

X2

22

2

22

The results obtained by analyzing Eq. (6) can be summed up as follows: (1) Exactly one superintegrable system with V1

#

0 exists up to gauge

transformation, namely

H

=

- -1 A 2

1 2 2 + -y (x + y2) + ya3L3, 2

y = const.

(9)

I t allows an 8-dimensional Lie algebra C of first-order integrals of motion with a basis given by L*

=

qya, - x a y ) ( I f a3),

x* = (a F Y Y ) ( 1 f a3) ,

*

Y, = (ia, y x ) ( 1 & a3), I* = I f a 3 .

218

The algebra L is isomorphic to the direct sum of two central extensions of the Euclidean Lie algebra e(2)

L

E+(2) @Z-(2),

G ( 2 ) = {L*,X*,Y*,I*}.

(11)

The two Casimir operators of C and the Hamiltonian Eq. (9) are 1

c* = xi + Y i f 4yL*I*,

H = - (C++ c-). 8

Conjugacy classes of elements of the algebra

(12)

C can be represented by

X l = L + + X L - , X z = L + + X X - , X3=X++XX-, XER.(13) (2) Integrable systems (with one integral of motion in addition to H ) exist. They are given by (a)

vo = V o ( P ) , X

= (wg

+

K

p = ~ 1 ~ 3 ) L 3 , w p = const, = &(PI,

dFTi7, p =0,l.

(14)

(b)

Vl

= VI(2) ,

x = -28,

vo = Y2 -v12(2) + F ( z ) , - 03

i

Vl ( 2 ) d z .

(15)

Thus, the superintegrable system Eq. (9) involves one arbitrary constant y.The integrable systems Eq. (14) and Eq. (15) each involve two arbitrary functions of one variable. The integrals of motion can be used to solve the Schrodinger-Pauli equation for the superintegrable system (in several different manners). In the two integrable cases Eq. (14) and Eq. (15) they can be used to reduce the problem to solving ordinary differential equations. For all details see the original article Ref. 14. Before going over to the case n = 3 let us mention that two important features that simplify the case n = 2. The first one is that the Hamiltonian given in Eq. (4) is a diagonal matrix operator (since 02 and 0 3 do not figure). Hence we could restrict our search to integrals X that are also diagonal. The second is that there exists a zeroth-order integral X = 0 3 (for any VO and Vl), in addition to the trivial commuting operator X = I . Hence any integral of motion can be multiplied by 0 3 and there is a “doubling” of the number of integrals of a given order.

219

We have set the Planck constant ii = 1 in all calculations. Keeping h in the Hamiltonian and integrals of motion does not change any of the conclusions. In particular VOand V 1 do not depend on ii. 3. The Three-Dimensional Case

Let US now consider Eq. ( 2 ) and search for an integral of the form Eq. (3) which we rewrite as

X

=

(A0

+

+

+

+

+

+ +

a')pl (Bo B'*a')p2 (C0 C . a ' ) p 3 $0 6 .a - -a{ ( A o + A . Z ) x + ( B o + S . C ) , + ( C o + 6 . C ) , ) , (16)

2 where Ao, Bo, Co, 4 0 and A i , Bi, Ci, 4i (i = 1,2,3) are all functions of 7, to be determined from the commutativity condition [ H ,XI = 0. This commutator will have second-, first- and zeroth-order terms in the momenta. From the second-order terms we obtain

AO= b l - U

+

BO= b2 + U

~ Y U ~ Z ,

~ X U ~ Z C ,o

= b3 - U

+

~ Z ~ l y , (17)

where ai and bi are constants. We also obtain the following overdetermined system of 18 first-order quasilinear partial differential equations (PDE) for A i , B i , Ci and V1

For any V1 Eq. (18) has the following solution A1 =

0,

B1 = - Z W , c 1=

yw,

A2 = Z W , B2

== 0 ,

cz = -xw,

A3

= -YW,

B3

=XW,

c 3

= 0,

220

where w is an integration constant. The first-order terms provide a system of 9 first-order quasilinear PDE for v1 and & and 3 first-order quasilinear PDE for $0 and Ai,Bi, Ci. Tbey also provide 9 second-order PDE for Ai, Bi, Ci and V1, however, these are differential consequences of Eq. (18). The 12 first-order quasilinear PDE can be written as

+

+ z(AoKz + BoK, + Coviz) + = 0 , + + z(AoKz + BoKy + CoKz) - 43y = 0 , K ( b - alz + 2 ~ 4 1 + ) y(AoV1z + BoK, + CoVi,) + 432 = 0 , vl(b2 + - 2x43) + y(AoV~z+ BoK, + COVlz) - 41, = 0 , vl(b3 - a22 + 2x42) + z(AoKz + BoKy + COVlz) + 41, = 0 , vl(b3 + a11/ 21/41) + z(AoKz + BoKy + COVlz) =0, Vl(aZI/ + a3z 2I/$h - 2243) + 412. = 0 , K(Ulz + - 2x41 2643) + $2, = 0 , Vl(a1z + %I/ - 2241 - 2I/42)+ =0, 422

-

$21

V l ( h - QI/ 2I/43) K ( h a22 - 2242) a32

-

U3Z

-

($32

(20)

where Ao, Bo and COare given in Eq. (17) and

40z ~ O

+ (zAzz - zAzz) + (2-41, - yAi,) + (Cz - B3)) + V l z ( z A ~- I/&) + Vly(ZB2 - yB3) + Kz(zC2 - I / C ~ ) , = Y K ( ( Y B ~ ~ zB3,) + (zBzz - zBzz) + (zBI, - yB1,) + (A3 Cl)) =

K ( ( ~ A 3 z- 5A3,) -

-

+ Vly(ZB.3 - zB1) + Vlz(zc3 ZCI) , = K ( ( I / C ~-~Z C ~+~(zCzz ) - 2Cz2) + (ZCi, yC1,) + (Bi - Az)) +Kz(yAl - ZAZ)+ Vl,(I/B1 - z&) + VL(I/C1 - ~ C Z ) . (21) +Vlz(XA3 - zA1)

$02

-

-

The system of 9 PDE given in Eq. (20) has a solution if the following conditions are satisfied: 1. (i) If bi # 0, i = 1 , 2 , 3 , then V1 = p (ii) If bi = 0, Vi,then V1 = Vl(r).

Finally, the zeroth-order terms in the commutator provide 8 more PDE that also involve VO and are in general of second-order. In fact some of them are third-order differential equations, however, by using Eq. (18) they can be reduced the second-order ones. These equations are too long to be presented here. The complete discusion of the above determining equations is long and we cannot reproduce the details here so we just present some results.

221

(a) A superintegrable system. The entire overdetermined system of equations can be solved for VO= $ V1 = $. We obtain the Hamiltonian

1 H = --A+ 2

-r21+

1 r2

-

---(Z,J~)~

with a 9-dimensional Lie algebra C of integrals of motion:

Jz = La

+ -2102

ni = p i

1

- --Ei,lZ,cq r2

We see that frepresents total angular momentum, fi a “modified linear momentum” and s’ a “modified spin”. The algebra is isomorphic to a direct sum of the Euclidean Lie algebra e(3) with the algebra o(3)

c

N

e(3) CB o(3) = {Y-

g1 fi}

is’}.

(24)

These generators satisfy the following commutation relations

I t is interesting to note that the potentials in Eq. (22) are a purely quantum mechanical effect. Indeed if we reintroduce fi into the Hamiltonian Eq. (2) and integral Eq. (16) it will figure significantly in the determining equations Eq. (18), Eq. (20) and Eq. (21). The potentials in Eq. (22) are then modified to

In the classical limit h + 0 both VO and V1 vanish. Integrable and superintegrable quantum systems that have free motion as their classical limits also exist in the case of scalar particles11J2i’6 but they are related to third- and higher-order integrals of motion. (b) Spherical symmetry. For V1 = Vl(r), VO = Vo(r) we obtain- the well-known result that H + commutes with total angular momentum J = L + 43. A full discussion will be presented elsewhere. l5

222

4. Conclusions

We have shown that first-order integrability and superintegrability in the presence of spin-orbital interactions exist and are nontrivial. For n = 2 the superintegrable potentials do not depend on fi whereas for n = 3 they vanish in the classical limit h + 0. Work is in progress on the search for superintegrable systems invariant under rotations and allowing secondorder integrals of motion.

Acknowledgments We thank F. Tremblay who participated in the early stages of this project for helpful discussions. The work of P.W. was partly supported by a research grant from NSERC. I.Y. acknowledges a postdoctoral fellowship awarded by the Laboratory of Mathematical Physics of the CRM, Universitk de Montrkal.

References 1. P. Winternitz, Ya. A. Smorodinsky, M. Uhli?, and I. F’riB, Sou. J. Nucl. Phys. 4, 444 (1967). 2. A. Makarov, J. Smorodinsky, Kh. Valiev, and P. Winternitz, Nuovo Cim. A 5 2 , 1061 (1967). 3. M. A. Rodriguez and P. Winternitz, J . Math. Phys. 43, 1309 (2002). 4. E. G. Kalnins, J. M. Kress, and W. Miller Jr., J . Math. Phys. 47, 043514 (2006). 5. E. G. Kalnins, W. Miller Jr., and G. S. Pogosyan, J . Math. Phys. 48, 023503 (2007). 6. P. Tempesta, A . V. Turbiner, and P. Winternitz, J . Math. Phys. 42, 4248 (2001).

7. B. Dorizzi, B. Grammaticos, A . Ramani, and P. Winternitz, J. Math. Phys. 8. 9. 10. 11. 12. 13. 14. 15. 16.

26, 3070 (1985). F. Charest, C. Hudon, and P. Winternitz, J . Math. Phys. 48, 012105 (2007). G. Pucacco and K. Rosquist, J . Math. Phys. 46, 012701 (2005). S. Benenti, C. Chanu, and G. Rastelli, J. Math. Phys. 42, 2065 (2001). S. Gravel and P. Winternitz, J . Math. Phys. 43, 5902 (2002). S. Gravel, J . Math. Phys. 45, 1003 (2004). I. Marquette and P. Winternitz, J . Math. Phys. 48, 012902 (2007). P. Winternitz and I. YurduSen, J . Math. Phys. 47, 103509 (2006). P. Winternitz and I. YurduSen, in preparation. J. Hietarinta, Phys. Lett. A 246, 97 (1998).

223

GENERALIZATION OF HAMILTONIAN MONODROMY. QUANTUM MANIFESTATIONS BORIS ZHILINSKII Universite' du Littoral, U M R 8101 du CNRS, Dunkerque, France E-mail: [email protected] Within the qualitative approach to the study of finite particle quantum systems different possible ways of the generalization of Hamiltonian monodromy are discussed. It is demonstrated how several simple integrable models like nonlinearly coupled resonant oscillators, or coupled rotators, lead to physically natural generalizations of the monodromy concept. Fractional monodromy, bidromy, and the monodromy in the case of multi-valued energy-momentum maps are briefly reviewed. Keywords: Monodromy; energy-momentum map; bidromy.

1. Introduction

The main goal of the qualitative theory of excited quantum finite particle systems is to describe and to classify generic qualitative phenomena which can be presented in families of Hamiltonian dynamical systems depending on a number of control parameters. The basic tools of such analysis are quantum-classical correspondence, symmetry group actions, topological asand references therein). Initially the accent of the pects (for a review qualitative analysis was put on the "quantum bifurcations" 5-7 and the redistribution of energy levels between bandss-" in the energy spectrum under the variation of some control parameters. At the end of the nineties Richard Cushman brings to the attention of physicists the Hamiltonian monodromy phenomenon. The monodromy was known to exist in several simple classical mechanical problems like spherical pendulum12J3 and was tentatively suggested to be of certain importance for quantum pr0b1ems.l~Soon after, the presence of monodromy was demonstrated for a number of different integrable approximations for concrete physical systems like coupled angular m ~ m e n t a , ' ~hydrogen ~'~ atom in external fields,17-19 Hf molecular ion," CO2 molecule21~22 and many other simple atomic and molecular sys-

224

tems. As soon as all these real physical systems are quantum, the notion of “quantum monodromy” is needed. It was introduced in23124and the interpretation of quantum monodromy as a certain “defect” of the lattice formed by joint spectrum of several commuting observables for quantum problem was s ~ g g e s t e d . ~ ~ ~ ~ 2. Singularities of energy-momentum maps and monodromy

In its simplest form the Hamiltonian m o n o d r ~ m y l appears ~ ? ~ ~ for classical completely integrable problems with two degrees of freedom. Such dynamical systems can be considered as integrable toric fibrations defined by two integrals of motion in inv~lution.~’ Typical simplest images of energy-momentum (EM) maps shown in Fig.1 consist of regular values and singular values. The inverse images of regular values are two-dimensional tori (one or ~ e v e r a l ) . ~Singular ’ values on the boundary of the EM map image correspond to lower dimensional tori. Typical isolated singular values (codimension two singularities) which appear generically for Hamiltonian dynamical systems with two-degrees of freedom are associated with the so called pinched torus (one of the generating circles of the torus is shrinked to a point, Fig.2a).12,27The presence of an isolated singular fiber makes toric fibration non-trivial. The monodromy describes the global twisting of the family of tori parameterized by a closed path going through regular values of the EM map of the integrable system. It can be considered as an automorphism of the first homology group of regular fibers associated with the homotopy equivalent class of closed loops on the regular part of the image of EM map. From the dynamical system point of view the monodromy is the first obstruction to the existence of global action-angle variables for completely integrable problems. 13*26 Pinched torus singularity is structurally stable under small perturbations which preserve the integrability of the problem. That is why the presence of monodromy is important from the point of view of physical applications. Moreover, having a topological origin, the monodromy should in some sense persist even under small non-integrable perturbation. This fact was recently proven for nearly integrable system in the style of KAM theorem.31 For a family of integrable systems depending on parameters, the position of a singular value on the image of EM map can change and, in particular, this singular value can touch the boundary of the EM map (compare sub-figures (a) and (b) in Fig.1). The corresponding qualitative modifica12v27128

225

b Fig. 1. Typical images of the energy momentum map for completely integrable Hamiltonian systems with two degree of freedom in the case of (a) - all internal points are regular, no monodromy; (b) - isolated singular value with integer monodromy, (c) - “island” formed by second component with nonlocal monodromy, and (d) - “island” formed by second component with trivial monodromy. [Dashed line - “bitorus” singularity.]

tion of the image of EM map (often named as bifurcation diagram2g) is the Hamiltonian Hopf bifurcation. 18,32 Another possible qualitative modification which can happen with isolated singular value is another kind of bifurcation which leads to formation of second connected component (or a second fiber). Two components fuse together a t a singular line (see Fig.lc,d). Each regular point a t that line has a singular “bitorus” as inverse image (see Fig.2b). Note, that the Hamiltonian Hopf bifurcation leads to the formation of an island (second leaf) on the image of the EM map which can be surrounded by a closed loop possessing the same monodromy as the initial singular (pinched) torus (compare sub-figures (b) and (c) in Fig. 1).The corresponding quantum problem possesses two different lattices in the two-component region which fuse together along the singular line. Such structure appears in quadratic spherical pendulum,’8 LiCN, HCN molecule^,^^,^^ hydrogen atom in external fields. At the same time, it should be noted that the “island” formed by the second component can be formed without an initial singular pinched torus. In such a case there are only two exceptional singular values on the singular boundary of “island” associated with the two ends of the “bitorus” line. The corresponding fibers are singular tori shown in F i g . 2 ~The . monodromy associated with the closed loop around such an island is trivial (identity). The standard definition of Hamiltonian monodromy requires the existence of a closed loop in the plane of values of integrals which goes only through regular values of the EM map. An important generalization of this notion was recently proposed which allows the existence of some onedimensional singular strata which can be crossed by a loop. The corresponding singular fibers are curled tori (Fig.2d). The restriction imposed by the existence of such strata leads to the possibility to define monodromy only for certain subgroups of the first homology group of regular fibers because 19135

226

a

b

d

C

Fig. 2 . Two dimensional singular fibers in the case of integrable Hamiltonian systems with two degrees of freedom (left t o right): a - pinched torus, b -bitorus, c - singular torus, and d - curled torus.

a

b

C

Fig. 3. Typical images of the energy momentum map for completely integrable Hamiltonian systems with two degree of freedom in the case of fractional monodromy (black point correspond to essential singularity): (a) - singular line is formed by curled tori; (b) - “island” formed by second connected component with one of its boundary being a circle with nontrivial stabilizer (fuzzy fractional m o n ~ d r o m y ~(c) ~ ) -; “island” formed by second component (non-local fractional m ~ n o d r o m y ~ ~ ) .

only a subgroup of cycles can go through the singular stratum. The resulting fractional monodromy was first introduced in36>37for a problem of nonlinearly coupled resonant oscillators and was illustrated immediately on quantum example by the evolution of a multiple (double) cell along a closed path crossing once the singular stratum. Much more detailed analysis of the fractional monodromy is given in several recent publication^.^^-^' In its simplest version, the fractional monodromy is defined for closed paths which do not cross singular strata of the bitorus type, i.e. singular fibers associated with transformation of one regular torus into two regular tori. At the same time, similar to the case of integer monodromy, the pinched curled torus, i.e. the fiber corresponding to singular value a t the end of “curled torus line”, can be deformed into an island formed by a second connected component. In such a case, the nonlocal fractional monodromy which is illustrated in F i g . 3 ~arises. In this case the essential singularity, namely the pinched curled torus, gives the second component attached to

227

the main leaf through the bitorus line (see an example of the formation of such structure in the case of hydrogen atom in external fields in.35) Another possibility of formation of second component is shown in Fig.3b. Here the essential singularity remains on the main leaf, and the second component is attached to the main leaf through the bitorus line. This situation was the subject of Nekhoroshev’s study of arbitrary fractional m o n ~ d r o m y . ~He ’ has shown that locally, close to essential singularity, the notion of fractional monodromy can be conserved even if the closed path crosses the bitorus line where the discontinuity of actions takes place. The presence of these discontinuities results in the appearance of “fuzzy” fractional monodromy. The fuzziness becomes less pronounced when the crossing point between closed loop and the bitorus line approaches an essential singularity.

3. Bidromy Another possibility to cross the bitorus line and to go from the region of EM map with one connected component as inverse image into the region with two connected components was suggested by Sadovskii and Z h i l i n ~ k i i ~ ~ ? ~ ~ on the basis of their study of a realistic physical model describing threedimensional nonlinear oscillators in the presence of axial symmetry with 1:1:2 resonance and small detuning between a doubly degenerate mode and a non-degenerate one, which corresponds e.g. to the Fermi resonance in the GO2 molecule. In this problem the two-components and the bitorus line are again present in some region of the EM map image, but their arrangement now is completely different (see Fig.4). In fact, the two components are formed due to self-overlapping of one regular leaf of the EM map. As Fig.4 schematically shows, the two fibers (b’, b”) associated with the same values of the integrals are two different regular tori, which can be deformed one into another along a path going entirely through regular fibers. Such path starts and ends a t different components and, consequently, it is not closed even if its initial and final points have the same values of the integrals. In such situation the interesting possibility of defining a “bipath” appears.22 Let the “bipath” start a t point a (Fig.4) and go through regular tori till point c a t the bitorus line. At that point the path ac bifurcates into two component path. Each sub-path [(cb’) and (cb”)] evolves through regular tori on different components in the region of self-overlapping. The two subpaths of the bipath go independently through regular tori till the initial point where they fuse. The only singularity on the bipath is the bitorus c. Point a is special, but the corresponding fiber is regular. The following transformation of quantum cells of the joint spectrum

228

Fig. 4. Schematic representation of a bipath associated with bidromy.

Fig. 5 . Energy momentum diagram for the Manakov top together with joint spectrum of mutually commuting operators for the corresponding quantum problem (taken from44). Inverse images of regular points in four regions consist of two, two, two and four regular tori. Evolution of a quadruple cell along a closed path surrounding the central singularity consists in splitting of quadruple cell into two double cells when entering into region with four components. Th e transformation between the initial and the final cell is trivial.

lattice is suggested to be associated with such a bipath.22 We start a t a with the double cell which splits into two cells belonging to the different components when crossing the bitorus line. Further evolution of each cell is regular till the final point where two cells should fuse together. The resulting cell can be compared with the initial one. The transformation between initial and final cells is named the bidromy transformation. It is conjectured that this transformation does not depend on the place where the bipath crosses the bitorus line. It is quite important to note that in the case of the “island” monodromy similar crossing of the bitorus line leads to a result which apparently depends on the position of the crossing point. The Manakov top problem4244 gives an another interesting example of the EM map with an even more complicated system of connected components (see Fig.5). The recent paper44 uses a construction similar to “bipath”

229

and bidromy, which allows to define the evolution of a multiple quantum cell along a “multicomponent” path. Curiously, the generalized monodromy defined in this case becomes trivial, in spite of using rather complicated multi-paths. 4. Conclusions

We briefly reviewed the possible generalizations of the monodromy concept which lead t o fractional monodromy and bidromy. While presently the mathematically rigorous definition of fractional monodromy seems to be properly formulated, the description of “bidromy” is just at its initial stage. Further mathematical constructions are necessary, and quite demanding, in order t o propose mathematically satisfactory tools which allow t o treat the new qualitative features in dynamics which were heuristically described on several examples of quantum molecular models.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12.

13. 14. 15. 16. 17. 18. 19. 20.

L. Michel and B. Zhilinskii, Phys. Rep. 341, 11 (2001). L. Michel and B. Zhilinskii, Phys. Rep. 341, 173 (2001). B.I. Zhilinskii, Phys. Rep., 341, 85 (2001). B. I. Zhilinskii, in: Topology in Condensed matter, Ed. M. Monastyrsky (Springer Series in Solid State Sciences, vol. 150, 2006), p. 165 (2006). K. Efstathiou, D.A. Sadovskii, and B.I. Zhilinskii, SIAM J. Dynamical Systems 3, 261 (2004). I. Pavlichenkov, Phys. Rep. 226, 173 (1993). I. Pavlichenkov and B. Zhilinskii, Ann. Phys. (N. Y.), 184, 1 (1988). F. Faure and B. I. Zhilinskii, Phys. Rev. Lett. 85, 960 (2000). F. Faure and B.I. Zhilinskii, Lett. Math. Phys., 55, 219 (2001). F. Faure and B. I. Zhilinskii, Phys. Lett. A 302, 242 (2002). V.B. Pavlov-Verevkin, D.A. Sadovskii, and B.I. Zhilinskii, Europhys. Lett., 6, 573 (1988). R.H. Cushman and L. Bates, Global aspects of classical integrable systems. Basel, Germany: Birkhauser, 1997. J.J. Duistermaat, Comm. Pure Appl. Math. 33, 687 (1980). R. H. Cushman and J. J . Duistermaat, Bull. A m . Math. Soc., 19, 475 (1988). L. Grondin, D. A. Sadovskii, and B. I. Zhilinskii, Phys. Rev. A 142, 012105 (2002). D. A. Sadovskii and B. I. Zhilinskii, Phys. Lett., A 256, 235 (1999). R. H. Cushman and D. A. Sadovskii, Physica D 142, 166 (2000). K. Efstathiou, Metamorphoses of Hamiltonian systems with symmetry. Lecture Notes in Mathematics, vol. 1864, Heidelberg: Springer-Verlag, 2004. K. Efstathiou, R.H. Cushman, and D.A. Sadovskii, Physica D 194, 250 (2004). H. Waalkens, H.R. Dullin, & P.H. Richter, Physica D 196, 265 (2004).

230

21. A. Giacobbe, R.H. Cushman, D.A. Sadovskii, & B.I. Zhilinskii, J. Math. Phys. 45, 5076 (2004). 22. D. Sadovskii and B. Zhilinskii, Ann. Phys. ( N . Y.), 322, 164 (2007). 23. S. Vii Ngoc, Comm. Math. Phys., 203, 465 (1999). 24. S. Vii Ngoc, Adv. Math., 208, 909 (2007). 25. B.I. Zhilinskii, Acta Appl. Math., 87, 281 (2005). 26. N.N. Nechoroshev, Trans. Moscow Math. SOC.,26, 180 (1972). 27. V.S. Matveev, Sb. Math, 187, 495 (1996). 28. M.Symington, in Topology and Geometry of Manifolds, Athens, GA, 2001, Proc. Symp. Pure Math., V. 71, AMS, Providence, RI, 153 (2003). 29. A.V. Bolsinov and A.T. Fomenko, Integrable Hamiltonian systems. Chapman & Hall/CRC 2004. 30. V.I. Arnold, Mathematical Methods of Classical Mechanics. Springer, New York, 1989. 31. H.W. Broer, R.H. Cushman, F. Fassb, & F. Takens, Ergod. Th. & Dynam. Sys., t o appear (2007). 32. J.J. Duistermaat, Z. Angew. Math. Phys., 49, 156 (1998). 33. K. Efstathiou, M. Joyeux, and D.A. Sadovskii, Phys. Rev. A 69, 032504 (2003). 34. M. Joyaux, D.A. Sadovskii, and J. Tennyson, Chem. Phys. Lett., 382, 439 (2003). 35. K. Efstathiou, D.A. Sadovskii, and B.I. Zhilinskii, Proc. Roy. SOC.A 463, 1771 (2007). 36. N. N. Nelhoroshev, D. A. Sadovskii, and B. I. Zhilinskii, C.R. Acad. Sci. Paris, Ser 1335, 985 (2002). 37. N. N. Nek?loroshev, D. A. Sadovskii, and B. I. Zhilinskii, Ann. Henri Poincare' 7, 1099 (2006). 38. K. Efstathiou, R.H. Cushman, and D.A. Sadovskii, Adv. Math. 209, 241 (2007). 39. A. Giacobbe, Diff. Geom. Appl. in press (2007). 40. N.N. Nekhoroshev, Sbornic Mathematics, 198, 383 (2007). 41. D. A. Sadovskii and B. I. Zhilinskii, Mol. Phys., 104, 2595 (2006). 42. I.V. Komarov and V.B. Kuznetsov, J.Phys. A : Math. Gen., 24, L737 (1991). 43. S.V. Manakov, Funct. Anal. Appl., 11, 328 (1976). 44. E. Sinitsyn and B. Zhilinskii, SIGMA, 3, 046 (2007).

EXTENDED ABSTRACTS


233

Internal flow through a conducting thin duct via symmetry analysis Mina B. Abd-el-Malek

The American University i n Cairo, Dept. of Math., Cairo 11511, Egypt'

H. S. Hassan The Arab Academy for Science and Technology, Alexandria, Egypt

1. Mathematical formulation of the problem Flow subjected to an axial variation of the external heat transfer coefficient and nonlinear boundary conditions due to radiation, through a conducting thin duct subjected to variable heat transfer coefficient. The energy equation for flow in dimensionless form can be written as

together with the following boundary conditions = o a t T = 0, (ii) K ( T ) g + h(z)T = -uE [T4- q 4 ( z ) ] (iii) T ( Tz, ) = 0 at z = 0. (2)

at

T

= 1,

(2)

The thermal conductivity and the heat capacity, respectively, are defined by

K ( T ) = KoTa and

C ( T )= CoTb.

(3)

2. Solution of the Problem

At first, we derive the similarity solutions using Lie-group method under which (1) and (2) are invariant, and then we use this group to determine similarity variables. * O n leave from Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt

234

Lie point symmetries. Consider the one-parameter (&)Liegroup of infinitesimal transformations in (r,z ; T ,S, v,h, q ) given by

+ E$(T,z,T, S , 4, h, + O(E'), Z = z -tE ( ( T , z , T S,g, , h, V) + O(E'), T =T+ Z ,T , S , 4 , h, V ) + O(E'), S = S + t,T , S , 4 , h, V ) + O(E'), F =T

V)

E ~ ( T ,

= v + E V ( T , z , T , S , q , h , v )+ 0 ( E 2 ) ,

ET(T,

FL

= h +EH(T,Z,T,S,q,h,u)

+0(&2),

where "2' is a small parameter. Equation (l),under the transformations (4) will be reduced to:

Therefore, the general solution of the invariant surface condition is Q!

=r,

T ( r ,z ) = z P F ( a ) , S(r,z ) = ~ ( " + l ) ~ O ( aV)( ,T )

= *(a).

(6)

Hence the reduced problem is:

The boundary conditions (2) will be

where X =

is a constant, and ( b - 3)p = 1.

3. Results

(1) Temperature of the fluid increases with the increase of fluid velocity, and decreases with the increase of the heat transfer coefficient. (2) Combination of 2 scaling transformations reduced the nonlinear partial differential equation to nonlinear ordinary differential equation.

235

Contact Geometry of parabolic Monge-AmpBre equations R. Alonso Blanco Departamento de Matemdticas, Universidad de Salamanca, Plaza de la Merced, 37008 Salamanca, Spain E-mail: ricardoQgugu.usal.es

G. Manno Dipartimento d i Matematica “E. De Giorgi”, Universitd del Salento via per Arnesano, 73100 Lecce, Italy E-mail: [email protected] F. Pugliese

Dipartimento di Matematica ed Informatica, Universitd d i Salerno via ponte don Melillo, 84084 Fisciano (Salerno), Italy E-mail: [email protected]

Let (x,y, z ) be coordinates on the trivial bundle T = R2 x R. A parabolic Monge-Ampicre equation (PMAE) is a second order P D E on T of the form

~ 2 + ~Az,,) + Bzzy + Cz,, + D = 0 (1) B 2 = 4(AC + N D ) . Such an equation can be iden-

N(z,,zyy -

satisfying condition tified [l] with a Lagrangian subdistribution D c C, where C ={U = d z - zzdx - zydy = 0} is the contact distribution on J ~ ( TTherefore, ). classifying PMAE’s amounts t o studying the contact geometry of Lagrangian distributions in J1(7).This, in turn, can be “almost always” be reduced to the contact classification of generic vector fields X contained in C (here, “generic” means that 1-forms U , X ( U ) ,X 2 ( U ) X , 3 ( U ) are independent): in fact, in this case V can be be uniquely reconstructed from the orthogonal complement X of the derived ditribution D’ = D+ [V, D]. However, even in the non generic case the study of X is extremely useful for classifying the corresponding PMAE. By following this idea, we find [2] a list of normal forms for PMAE’s.

Definition 0.1. Let V

cC

be a Lagrangian distribution. Then

V is said

236

t o be of class ( i , j , k ) if d i m V ' = i , d i m V " = j , dimV"' = k. T h e only possible cases are ( 2 , 2 , 2 ) , ( 3 , 4 , 4 ) , ( 3 , 4 , 5 ) , ( 3 , 5 , 5 ) , t h e last one being generic.

Theorem 0.1. Let Eq. (1) be a n o n generic P M A E and V c C be the corresponding Lagrangian distribution. T h e n Eq. (1) can be reduced t o one of the following forms:

v

(1) zyy = 0 , if is of class ( 2 , 2 , 2 ) ; # 0 , if V is of class ( 3 , 4 , 4 ) ; (2) zyy = b ( x , y , z , z x , z y ) , (3) zyy 22 zZy z 2 z x x = b ( z , 9 , z , z,, zy),if V is of class ( 3 , 4 , 5 ) .

+

+

I n fact, t h e first [3]a n d t h e second [4]normal form were already known, while t h e third one was known only in t h e real analytical case [5]. As to t h e generic case, we extend to t h e smooth category a normal form which was previously known only in t h e real analytic case [5].

Theorem 0.2. A P M A E is locally contact equivalent t o a n equation of the fom

+

zyy- 2azZy a2.zxx = b, with a , b depending o n x , y , z , z x , z y , if and only if it admits a complete integral, i.e. a 3-parametric family of solutions. References 1. A. Kushner, V. Lychagin, V. Rubtsov: Contact Geometry and Non-Linear

Differential Equations, CUP 2007. 2. R. Alonso Blanco, G. Manno, F. Pugliese: Normal forms of parabolic MongeAmpkre equations, preprint 2007. 3. E. Goursat: LeCons sur l'integration des equations aux derive'es partielles du second ordre, vol. I, Gauthier-Villars, Paris, 1890. 4. D.V. Tunitskii: On the contact linearization of Monge-AmpBre equations, Izv. Math. 60 (1996), no. 2, 425-451. 5. R.L. Bryant, PA. Griffiths: Characteristic cohomology of differential systems. 11. Conservation laws for a class of parabolic equations. Duke Math. J. 78 (1995), no. 3, 531-676.

237

Complete Symmetry Groups and Lie remarkability K. Andriopoulos* Department of Mathematics, University of Patras, Rio, GR-26500, Greece * Email: kandOaegean.gr

Krause’s concept of a complete symmetry group (csg) of a differential equation (de) is the group underlying the algebra associated with the set of symmetries, be they point, contact, generalised or nonlocal, required t o specify the equation or system completely [4]. Subsequent work has added a third requirement for a set of symmetries to represent the csg of a de [l]:the group must be of minimal dimension. Des possess in general an infinite number of symmetries. Symmetries can be viewed as constraints on the structure of a de. A symmetry group is the set of all Lie point symmetries (Lps). The likelihood of admission of Lps is slim. The happy happenstance of the complete characterisation of an equation by Lps only is even slimmer. This is the main reason why Oliveri [6] noted that those equations should be termed Lie remarkable. The terminology, Lie remarkable equations, is indeed fitting with a limited occurence in des since the only admissible symmetries are Lp. Lie remarkability is straightforwardly embodied into csgs since it is just an expression that describes a point csg. Conjecture: Every (system of) de(s) is characterised completely by a set of symmetries, be they point, generalised, contact or even highly nonlocal. Consider the partial de [3] u;u,,uyy - cu;u:,

+ 2(c - l)u,uyu,yuyy + (1 - c)u,uyy = 0, 2

2

(1)

which, as far as we know, has never before been reported. Equation (1) possesses the 8-dimensional algebra of Lps

A, = a,, A2

=

A,

= u&,

= yay,

A,

a,, A3 = a,, A4 = xd,, A7

= udY,A, =

.ay

(2)

with the algebra {A2@A2}@s{Al@s3A1}.The csg of (1)comprises the Lps of (2) and in that sense equation (1) is said to be (strongly) Lie remarkable.

238

For c = 1 equation (1) reduces to the second-order Monge-Ampkre equation in two independent variables, uxxuyy

- u:,

= 0,

(3)

which, apart from the symmetries in (2), admits the additional Lps Ag = ~ 2 1 8 ,

+ yu8y + U ~ C L ,

+ +

+ +

= y&, A11 = x2& ~ y d y XU&, (4) xdu, A14 = u&, A 1 5 = XY& y2ay YU&, thereby constituting the algebra d ( 4 , R). Equation (3) (and a whole family of Monge-Ampkre equations) was examined in [5,6] and was proven t o be (strongly) Lie remarkable. For more details see: [3]. For c = 0 eq. (1) gives rise t o the two-dimensional Bateman equation A12 = Y&,

a 1 3=

uxxu;

+

UvYU:

- 2uxuyuxy= 0

(5)

which possesses the infinite-dimensional algebra P1

=41(UPy,

P2

P4

= 44(U)&,

P 5 = Y45(U)dy, P 6

= .42(u)ay,

P7

= 47(u)8,, Pa = x 4 a ( u ) 8 x ,

P3

+ Y243(U)dy,

= XY43(U)8x = x2$6(u)ax f

xy$S(u)ay,

(6)

0 9 = Y49(U)&

with the algebra (A1 BSd ( 3 , R)}M. When we select all pi symmetries with &(u) = 1, i = 1- 5 , 7 - 9, equation (5) can be proven t o be Lie remarkable. Very few equations are abundant in Lps and therefore cannot be characterised completely by an algebra comprising Lps only. Many equations have the misfortune to be devoid of Lps, even contact, and thence complete characterisation results only by the consideration of nonlocal Symmetries. In some cases [2] these nonlocal symmetries are endowed with such an uncommonly extraordinary structure that one would wish that it always be the case. Although symmetries play such a conspicuous role in the theory and applications of des, what further information can be obtained by the symmetries which comprise the csgs of those equations has yet to be reported. However, there must be some sort of significance, but this is yet t o be found. Note that one may follow the inverse procedure and determine those des that possess a particular algebraic structure. For more details: [ 3 ] .

References 1. 2. 3. 4. 5. 6.

K. Andriopoulos & P.G.L Leach, J . Nonlin. Math. Phys. 9-2, 10 (2002). K. Andriopoulos, P.G.L. Leach & A. Maharaj, (submitted). K. Andriopoulos, S. Dimas, P.G.L Leach & D. Tsoubelis, (in preparation). J. Krause, J. Math. Phys. 35, 5734 (1994). G.Manno, F. Oliveri & R. Vitolo, J. Math. Anal. A p p l . 332, 767 (2007). F. Oliveri, Note di Matematica 23, 195 (2004/2005).

239

Quantum wave equations in curved space-time from wave mechanics Mayeul Arminjon

Laboratoire 3S-R, CNRS €9 Universitt?s de Grenoble, France. E-mail: [email protected]

The usual way to write the wave equations of relativistic quantum mechanics in a curved spacetime is by covariantization: the searched equation in curved spacetime should coincide with the flat-spacetime version in coordinates where the connection cancels at the event X considered. This is connected with the equivalence principle. For the Dirac equation with standard (spinor) transformation, this procedure leads to the Dirac-Fock-Weyl (DFW) eqn, which does not obey the equivalence principle. Alternatively] in this work we want to apply directly the classical-quantum correspondence. The latter results1 from two mathematical facts.2 i) There is a one-to-one correspondence between a (2”d-order, say) linear differential operator:

P$

=

ao(X)

+ u:(x)a,$ + a y ( X ) d , d , $ ,

(1)

and its dispersion equation, a polynomial equation for the covector K:

IIx(K)

ao(X)

+ i a : ( X ) K , + i 2 ~ ~ ” ( X ) K , K=y0 ,

(2)

the latter arising when one looks for “locally plane-wave” solutions: $(X)= A exp[iQ(X)],with d,K,(Xo) = 0, where K , = ape. The correspondence from (2) to (1) is K , + -id,. ii) The propagation of the spatial wave covector k = ( K j ) ( j = 1 , 2 , 3 ) obeys a Hamiltonian system:

where W ( k ; X ) is the dispersion relation, got by solving IIx(K) = 0 for the frequency w -KO. Wave mechanics (classical trajectories=skeleton of a wave pattern) means that the classical Hamiltonian is H = hW. The classical-quantum correspondence follows1 by substituting K , -+ -id,.

=

240

This analysis shows3 that the classical-quantum correspondence needs using preferred classes of coordinate systems: the dispersion polynomial IIx(K) and the condition a,K,(X) = 0 stay invariant only inside any class of “infinitesimally-linear” coordinate systems, connected by changes satisfying, at point X considered, (d2x’P/dxfidxw) = 0. One such class is that of locally-geodesic coordinate systems at X for metric g : g M u , p ( X = ) 0 , p , v , p E { 0 , . . . , 3 } .Another class occurs if there is a (physically) preferred reference frame: that made of changes which are internal to this frame. Assuming one class or the other gives distinct wave equations. We may now apply this correspondence to a relativistic particle, also in a curved ~ p a c e - t i m e In . ~ each coordinate system, the energy component po of the 4-momentum defines a classical Hamiltonian H = -PO satisfying

g~”p,p,

-

m2 = 0

( c = 1).

(4)

The dispersion equation associated with this by wave mechanics is

gC””K,K,

-

m2 = 0

( h = c = 1).

(5)

Applying directly the correspondence K , -+ -ia, to it, leads to the KleinGordon equation. Instead, one may try a factorization:

IIx(K) = (gfiu(X)K,K,-m2)1

=

[ a ( X ) + i y C L ( X ) K , ] [ P+iC”(X)K,]. (X)

(6) Identifying coefficients in (6) (with noncommutative algebra), and then substituting K , -+ -ia,, leads to the Dirac equation:

(iyp

a, - m)$ = 0,

with ypy’

+ yV-f = 29,”

1.

(7)

Assume the first class (locally-geodesic systems). Then Eq. (7), derived in any system of that class, rewrites in a general coordinate system as:

( i y v D , - m) $ = 0,

(D,$),

= q, = a,+, + rg,qU.

(8)

(The r!,,’s are the Christoffel symbols of 9.)With the second class (preferred-frame systems), a different (preferred-frame) equation is got. These two equations are also distinct from the standard, DFW equation. References 1. M. Arminjon, Nuovo Czmento 114B,71-86 (1999). 2. G. B. Whitham, Linear and Non-linear Waves (Wiley, New York, 1974). 3. M. Arminjon, arXiv:gr-qc/0702048 (2007).

24 1

Nonlocal interpretation of A-symmetries Diego Catalan0 Ferraioli Dipartimento di Matematica, Universiti di Milano, via Saldini 50, 1-20133 Milano, Italy

A reduction method of ODEs not possessing Lie point symmetries makes use of the so called A-symmetries . These are not standard symmetries, nevertheless for any given A-symmetry of an ODE y one can always reconstruct nonlocal symmetries of y . As a consequence, using these nonlocal symmetries, the A-symmetry reduction method reduce t o a standard method of symmetryreduction.

1. Introduction

Local symmetries provide a unified approach to the reduction of ordinary differential equations (ODEs) by the method of differential invariants (MDI). Unfortunately, finding symmetries for ODEs is not always easy and often one encounters equations with a lack of local symmetries. In fact, for a Ic-th order ODE y in the unknown Y, local symmetries are defined by the solutions to a linear PDEs depending on the derivatives of Y up to order k - 1. Now, since the general solution to this PDE cannot be found unless one knows the general solution to y , one usually can only search for particular solutions depending on derivatives of Y up to order lc - 2. Therefore, y is not reducible by MDI whenever it has no symmetries of order less or equal to k - 2 (in particular, it has no Lie point symmetries). On the other hand, there are also examples of equations which can be solved by quadrature but with a lack of local symmetries. These examples, in particular, seem to prove that local symmetries are sometimes inadequate to handle equations which have not enough local symmetries and raise the question of whether an extension of the notion of symmetry would lead to a more effective method of reduction. Hence, in the last few years, various attempts in this direction have been done and some new classes of symmetries have been proposed. Among these, special attention has been devoted to Xsymmetries which were first introduced by Muriel and Romero [2]. Despite

242

their name, A-symmetries of an ODE y are not a t all standard symmetries of y , nevertheless their invariants can be effectively used to reduce the order of y . At first sight, A-symmetries may appear an outstanding new concept which deserve an increasing attention due to their applications. However, as discussed in Ref. 1, the A-symmetry reduction method is just a particular case of the MDI applied to nonlocal symmetries. This fact follows from the reconstruction theorem 2.1 below, which states that from any A-symmetry of an ODE y one can always reconstruct nonlocal symmetries of y , with respect to a covering of y determined by A. The reader is referred to Ref. 1, and references therein, for further details. 2. Reconstruction of nonlocal symmetries from

A-symmet ries Before stating theorem 2.1 below, we recall the definition of A-symmetries. Here by TO we denote the trivial one-dimensional bundle over R with standard coordinates ( t ,v) and by y a k-order ODE on 7r0 in normal form. The reader is referred to Ref. 3 for generalities on the geometry of differential equations.

Definition 2.1 (Muriel-Romero). A vector field X = pat + $13~o n Jo(7ro) is a A-symmetry ofy iff the vectorfield XIAikl = pa,+Cf=, $[X%ila,,, with $,[X,OI = $ and $ [ X , i I = Do($[X,i-lI)- D o ( p ) v i + A ($[X7%-11 - P V i ) (Do being the truncated total derivative operator o n J k ( 7 r 0 ) ) , is tangent t o y . A conceptual framework for nonlocal symmetries is provided by the notion of covering (see Ref. 3 for details). The following theorem gives an interpretation of A-symmetries in terms of nonlocal symmetries.

adTheorem 2.1 (Reconstruction Theorem [l]). y mats a A-symmetry X iff there exists a function x = X(t,v,v1,..., v k - l ) such that, with respect t o the covering determined by the system of ODEs {vk = f ( t ,v , ...,V k - 1 ) , w1= A}, the (infinite) prolongation of e” ( X is a nonlocal symmetry of y .

+ xa,)

References 1. D Catalan0 Ferraioli 2007 Nonlocal aspects of A-symmetries and ODEs reduction J . Phys. A : Math. Theor. 40, 5479-5489. 2. C. Muriel and J. L. Romero 2001 New methods of reduction for ordinary differential equations IMA J . Appl. Math. 66,111-125. 3. A. M. Vinogradov et al. 1999 Symmetmes and conservation laws for dzfferential equatzons of mathematical physics (American Mathematical Society).

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R-SEPARATION FOR THE CONFORMAL LAPLACIAN C . CHANU Universitb d i Torino, via Carlo Alberto 10, 10123 Torino, Italy E-mail: [email protected] M . CHANACHOWICZ and R. G . McLENAGHAN University of Waterloo, Waterloo, Ontario, Cunada, Inc, N2L 3G1 E-mail: [email protected], [email protected]

1. The conformally invariant (CI) Laplace equation

ci

The classical Laplace equation in Cartesian coordinates on EXn, dz",$ = 0, is usually extended to a general n-dimensional Riemannian manifold ( M ,g ) as = 0, where a = gij(didj - r$&)is the Laplace-Beltrami operator. A different generalisation of it, introduced for dealing with conformally invariant objects, such as spin zero fields, is

a$

where R, is the Ricci scalar; it is called the conformally invariant Laplace equation. Contrary to A$ = 0, Eq. (1)is conformally invariant - that is the solutions of the equation written on a conformally related manifold with metric = e2ffgijare of the form $ = e v ' $ , with $ satisfying (1). We recall that multiplicative R-separation of a single 2nd order PDE is the search for a complete solution of the form $ = R ( q l , .. . , qn) +i(qi,c,) with (c,) E iR2n-1. The study of R-separation of (l),instead of A$ = 0, seems more natural. [l]Indeed, R-separation is a conformally invariant property occurring on all conformally related manifolds. Criteria for R-separation of a single PDE [2] applied to (1)give:

n,

Theorem 1.1. R-separation for Eq. (1) occurs in the coordinates ( q i ) 2 8 : (i) ( 4 % )are orthogonal conformally separable coordinates i. e., they allow additive separation of variables for the null geodesic Hamilton- Jacobi Eq.; (ii) R is (up to separated factors) a solution of di In R = $ghkI'hr:i;

244

+y ,

(iii) the modified potential U = S R , is of the form U = ghh f h ( q h ) , f o r some suitable functions f h (compatibility condition). Conditions ( i )and (iii)are conformally invariant: ( i )means that g is conformal to a separable metric and implies that the PDE system determining R is integrable. The modified potential associated with ijhh = eP2*g hh is U = eP2"U and U satisfies condition (iii)iff U does. The three-dimensional case. We restrict ourselves to general conformally separable coordinates without conformally ignorable coordinates q k such that d k ( g i i / g j j ) = 0. In these coordinates the form of the metric is: [3] gii =

Qhi(qi)(qif2- q i + l )

..

z. = 1,. , 3 ( m o d 3),

where Q is the conformal factor and hi are functions of a single variable. Computing U and imposing (iii),we get the geometric condition: Theorem 1.2. O n a 3-manifold, R-separation of Eq. (1) occurs in general conformally separable coordinates iff the manifold is conformally flat. If there are conformally ignorable coordinates, then conformal flatness is a sufficient, but not necessary, condition. Applications to A$ = 0 include: Theorem 1.3. ( a ) If a 3-manifold satisfies R, = 0 , then R-separation of A$ = 0 occurs in general coordinates i f f the manifold is conformally flat. ( b ) O n a conformally flat 3-manifold R-separation of A$ = 0 occurs in general coordinates iff the Ricci scalar R, satisfies the compatibility condition (iii). Example 1.1. The 3-sphere S3 is a conformally flat 3-manifold with constant Ricci scalar R, > 0. Thus, R-separation of A$ = 0 occurs only in the conformally separable coordinates that are indeed separable. Indeed A$ = 0 admits R-separation in the same general coordinates allowing Rseparation of the Helmholtz Eq. A$ = E$ ( E E EX). On S3 Eq. (1)coincides with the Helmholtz equation for fixed value of the energy E = - R,. Hence there exist additional conformally separable coordinates allowing fixed energy R-separation of the Helmholtz equation for the value E = -R,/2. Example 1.1shows that the concept of fixed energy R-separation is useful also for non-zero energy cases and justifies the study of the CI-Laplacian. References 1. N. Kamran, R.G. McLenaghan, Lett. Math. Phys. 9, 65 (1985). 2. C. Chanu, G. Rastelli, Int. J . Geom. Methods Mod. Phys. 3,489 (2006). 3. C.P. Boyer, E.G. Kalnins, W. Miller, Trans. Amer. Math. SOC.242,355 (1978).

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Asymptotic behavior of a Bouncing Ball Anna Maria Cherubini, Giorgio Metafune, F’rancesco Paparella Dip. di Matematica E. De Giorgi, Universith di Lecce Lecce, Italy E-mail: { annu. cherubini,giorgio.metufune,francesco.paparella} @unile.it

The simplest and most widely used model of a bouncing ball (or grains of a granular fluid) assumes that the ball is a rigid body, and that an impact with the floor is an instantaneous event, which reverses the vertical component of the speed of the ball: to model energy dissipation caused by an impact, it is customary to introduce a positive coefficient of restitution T < 1, which is the ratio between the absolute values of the vertical speeds immediately after and before an impact. This model performs well when the ball does not experience too many impacts in the time unit, otherwise granular systems described in this way are generically subject to the phenomenon of inelastic collapse, where clusters of particles undergo an infinite number of collisions in a finite time [2] : after this time these models are meaningless. Popular models taking into account rotational degrees of freedom and exchanges of angular momentum at the impacts [3] or defining the restitution coefficient as a function of the impact velocity [4,5]do not avoid the problem. Models using restitution coefficients rest on hypotheses that become invalid as the frequency of the impacts diverges: collisions with the floor are not truly instantaneous, and treating the ball as indeformable is highly questionable when the frequency of impacts is close to the resonant frequencies of the bouncing ball [6]. In our model of a bouncing ball we explicitly take into account the deformability of the body and we give up the notion of restitution coefficient, at least as a primitive concept; we still assume impacts to be instantaneous, but only from a microscopic point of view: a persistent contact with the floor is seen as a rapid sequence of instantaneous impacts. The idealized ball is seen as two point masses connected by a massless

246

dissipative spring: when it is not in contact with the floor, it is ruled by the following equations of motion

{ m?

= -gm my = -gm

+ k(y - z - L ) + v(?j- i) - k(y

-x

- L ) - v(G- i)

(1)

where m is the mass of the material points at z ( t ) and y ( t ) , L is the length at rest of the spring, k is its elastic constant, Y is a damping coefficient, and -g is the acceleration of gravity. All constants are positive. We indicate by t , the time when an impact occurs, i.e. x ( t n ) = 0. We define the time of flight Tn = tn+l - tn.

(2)

Impacts are modeled as an instantaneous elastic interaction obeying to the rule

q t ; ) = -?(ti) (3) where f(a*) means limt+ai f ( t ) . The special case x n ( t n ) = 0 may lead to a continuous contact of the lower point mass with the floor. We prove that our model is free from pathologies analogous to the inelastic collapse and compute the asymptotic expression for Tn and for the impact velocity. We also prove that contacts with zero velocity of the lower end of the ball are possible, but non-generic; finally we prove that starting from any initial condition, the system tends to the static equilibrium as

t

4

co.

Our results are fully exposed in [7]. References 1. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bzfurcations of Vector Fields, (Springer, Berlin/Heidelberg, 1983). 2. S. McNamara and W. R. Young, Phys. Fluids A , 4, 496-504, (1992). 3. N. Schoghofer and T. Zhou, Phys. Rev. E, 54, 5511-5515, (1996). 4. R. Ramirez, T. Poschel, N. V. Brilliantov, and T. Schwager, Phys. Rev. E 60, 4465-4472, (1999). 5. F. G. Bridges, A. Hatzes, and D. N. Lin, Nature 309,333-335, (1984). 6. J. Duran, Sands, Powders, and Grains: A n Introduction to the Physics of Granular Materials, (Springer, Berlin/Heidelberg, 1999). 7. A.M. Cherubini, G. Metafune, F. Paparella, On the Stopping Time of a Bounc-

ing Ball arXiv:0706.1978vl [math-ph], (2007).

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Discontinuous solutions for the Degasperis-Procesi equation G. M. Coclite

Department of Mathematics, University of Bari V i a E. Orabona 4, 70125 Bari, Italy E-mail: [email protected] K. H. Karlsen*

Centre of Mathematics for Applications ( C M A ) , University of Oslo P. 0. Box 1053, Blindern, N-0316 Oslo, Norway E-mail: [email protected]

Consider the Degasperis-Procesi equation ut

+

- ~ t z z 42121,

= 3UzUzz

+

~ ~ z z z ,

( 4).

E (0,Co) x

R,

(1)

which can be viewed as a shallow water approximation to the Euler equations. We are interested in the Cauchy problem for this equation, so we augment (1) with an initial condition uo:

u ( 0 , z )= uo(z),

zE

R.

(2)

Formally] (1) is equivalent to the hyperbolic-elliptic system Ut

+ U U , + P, = 0 ,

-P,,

+ P = (3/2)u2.

(3)

Since the Green's function of the operator u H u - uzz is e-Iz1/2 we have P ( t ,x) = P"(tl x) := e-I"-Y1u2(t,y)dy. Hence (3) can be written as a conservation law with a nonlocal flux function:

s,

A function u is a weak solution of (1)-(2) if it belongs to Lm(O,T ;L2(R)) for all T > 0 and (3) holds in in the sense of distributions on [O,m) x R. To prove existence we consider smooth solutions u, of the following fourth order viscous approximation of the Degasperis-Procesi equation (1): *K. H. Karlsen is supported by an Outstanding Young Investigators Award.

248

+

+

U E , t X X + 4%uE,, = 3uE,xuE>xx ~ E ~ E , X X X EU,,,, - EU,,,,,,. This equation can be written in the form of a viscous conservation law with a non-local flux: u,,t J, e - l ~ - Y l u z ( t ,y ) d y ] = w,,,,.

%,t -

+ [$ + a

Assuming that the initial data uo E L1(R) f?L"(R), we establish a series of E - uniform estimates, which implies that a subsequence of {u,},,,, converges strongly in Lro,(R+ x R),for any 1 I p < 00, and also in LP(R+ x IR), for any 1 I p < 2, to a limit function u that is a weak solution of the Degasperis-Procesi equation [ l ] . To assert that the weak solution is unique we would need to know somehow that the chain rule holds for our weak solutions. However, since we work in spaces of discontinuous functions, this is not true. Instead we replace the chain rule with an infinite family of entropy inequalities. Namely, we require that an admissible weak solution should satisfy the "entropy" inequality (P" solves the second equation in (3)) 77(U)t

+ q(u)x + d ( u ) P ; 5 0

in

q p , 00) x R),

(4)

for all convex C2 entropies 77 : R 4 R and corresponding entropy fluxes q : R -Ridefined by q / ( u ) = ~ ' ( u ) uWe . call a weak solution u that also satisfies (4)an entropy weak solution. We prove that the above mentioned weak solution, which is obtained as the limit of a sequence of viscous approximations with the artificial viscosity coefficients tending to zero, satisfies the entropy inequality (4), and thus is an entropy weak solution of the Degasperis-Procesi equation ( 1 ) - ( 2 ) . In addition the unique entropy weak solution of ( 1 ) - ( 2 ) satisfies the Oleinik type estimate

u,(t,z) I K ~ ( l + l / t ) , O < t I T , X E R ,

(5)

for some positive constant KT depending on T and on U O . An implication is that a shock wave in an entropy weak solution to the Degasperis-Procesi equation is admissible only if it jumps down in value (like the inviscid Burgers equation). In analogy with the theory for conservation laws we can replace the Kruzkov-type entropy inequalities (4) by an Oleinik-type estimate like ( 5 ) , namely there exists a unique weak solution of ( 1 ) satisfying (5) [2].

References 1. G. M. Coclite and K. H. Karlsen, J. f i n c t . Anal. 233 60 (2006). 2. G. M. Coclite and K. H. Karlsen, J. Dzfferential Equations 234 142 (2007).

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Variational Poisson-Nijenhuis structures for evolution PDEs Valentina Golovko Physics faculty, Lomonosov MSU, Moscow, Russia E-mail: [email protected] Variational Poisson-Nijenhuis structures on nonlinear PDEs are introduced and investigated. Relations between Schouten and Nijenhuis brackets with the Lie bracket of shadows of symmetries are established. This approach allows to construct a framework for the theory of nonlocal Poisson-Nijenhuis structures.

Keywords: Nonlocal Poisson-Nijenhuis structures; coverings.

1. Main result Consider a system of evolution equations & = { F = Ut

- f ( X ,

t , U , U 1 , . . . , u k ) = o},

where both u = ( u l , . . . ,urn)and f = (f',. . . , f") are vectors and ut = au/dt, U k = a k u / a x k . Let us extend & in two different ways. Namely, consider the e*-covering [3] L*& 4 E that is an analog of the cotangent bundle and the [-covering CE -+ & that is an analog of the tangent bundle. New nonlocal variables in both coverings will be treated as odd variables. Let us take a linear with respect to odd variables vector function ?-I = (El,. . . , ?-I") on the !*-covering and a linear with respect to odd variables vector function N = ( N 1 ., . . ,N")- on the [-covering. Assume that they satisfy the linearized the equations &(E)= 0 in the !*-covering and &((n/) = 0 in the [-covering, resp. In the terminology of the covering theory [5],?-I and N are shadows of symmetries of & in the !*- and !-coverings, resp. Recall [3] that to the shadow ?-I there corresponds an operator Ax in total derivatives that acts from symmetries of & to cosymmetries while to shadow N there corresponds a recursion operator R N . Assume that Ax is skew-adjoint. We are interested in the case where the pair ( A H R , N ) forms a Poisson-Nijenhuis structure, see [4]for the definition.

250

Theorem 1.1. [Z] Let the Hamiltonian operator AH and a recursion operator R N ) define a Poisson-Nijenhuis structure on the evolution equation &, while 'FI and N be the corresponding shadows in the e*- and l-covering over E , respectively. Then (i)

{X,X}= 0,

(ii)

{ N , N }= 0,

(iii)

{ X , N } = 0.

Remark 1.1. All the brackets in the theorem above are Lie brackets of shadows of symmetries [l].They are a generalization of the brackets for local symmetries. Remark 1.2. Having constructed a Poisson-Nijenhuis structure on & one can obtain a family of pair-wise compatible Hamiltonian operators Ai = RiA, i 2 0 , where A = AH an R = RN. Remark 1.3. The notion of Poisson-Nijenhuis structures can be generalized t o the case of nonlocal operators. Since t o a nonlocal Hamiltonian - operator A there corresponds a shadow 'FIA in some new covering r*:C*& -+ C*& over C'E and to a nonlocal recursion operator R there corresponds a shadow & in some new covering T : C& -+ C& over C&,properties (i)-(iii) from the theorem above can be taken as definitions for A t o be a Hamiltonian operator, R t o be a Nijenhuis one and for A and R to satisfy the compatibility condition.

-

Acknowledgments This work was supported in part by the NWO-RFBR grant 047.017.015 and RFBR-Consortium E.I.N.S.T.E.I.N. grant 06-01-92060.

References 1. V. Golovko, I. Krasil'shchik, A. Verbovetsky, Lie bracket for nonlocal shadows, Scientific Bulletin of MSTUCA, 91 (2007), 13-21 (in Russian). 2. V. Golovko, I. Krasil'shchik, A. Verbovetsky, Variational Poisson-Nijenhuis structures for partial digerential equations, (to appear). 3. P. Kersten, I. Krasil'shchik, A. Verbovetsky, Hamiltonian operators and t * coverings, J. Geom. and Phys. 50 (2004), 273-302. URL: arXiv:math.DG/ 0304245. 4. Y. Kosmann-Schwarzbach, F. Magri, Poisson-Nijenhuis structures, Ann. Inst. H. Poincark Phys. Thkor., 53 (1990), no. 1, 35-81. 5. I.S. Krasil'shchik, A.M. Vinogradov, Nonlocal trends i n the geometry of differential equations: symmetries, conservation laws, and Backlund transformations, Acta Appl. Math. 15 (1989) no. 1-2, 161-209.

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Analytic approximations of geometric maps and applications to KAM theory A . GonzBlez-Enriquez Dipartimento di Matematica ed Informatica Universita degli Studi d i Camerino Via Madonna delle Carceri, 62032 Camerino ( M C ) Italy. Email: alejandra.gonzalezOunicam.it R. de la Llave

Dept. Mathematics, University of Texas at Austin, 1 University Station, C1200 Austin, Texas 78712 USA. Email: 1laveomath.utexas. edu

Keywords: Smoothing; symplectic maps; volume-preserving maps; contact maps; KAM tori; uniqueness; bootstrap of regularity.

We show that finitely differentiable diffeomorphisms that are either symplectic, volume-preserving, or contact can be approximated with analytic diffeomorphisms that are, respectively, symplectic, volume-preserving, or contact. The method used here goes as follows. First, using the convolution operator with an analytic kernel, we define for t 1 1, a linear operator St, taking finitely differentiable functions into analytic ones. In general, it is not true that St[f]*il is equal to f*O, for a given form il. In the case that R is either an exact symplectic, exact volume or contact form, by using the deformation method [l] we prove that there exists a diffeomorphism pt, which is close to the identity, satisfying p: (St[f]*il) = f * R . Hence, given a finitely differentiable diffeomorphism f which is either symplectic, volumepreserving, or contact, for t sufficiently large, Tt[f] = St[f] o q t gives a symplectic, respectively, volume-preserving, or contact diffeomorphism approximating f . Furthermore, using the calculus of deformations [2],we prove that that if f is exact symplectic, then it is possible to construct analytic

252

approximating functions Tt [f]which are also exact. This method produces quantitative properties of the nonlinear operators Ttin terms of the degree of differentiability of f. More precisely, for t sufficiently large, Tt[f] is bounded uniformly, with respect to t , on some complex domains and the rate of convergence -in C'"-norms- of Tt[f] to f is given in terms of t and the degree of differentiability of f and t-l. Our main motivation to study the above problem is the existence of invariant tori for finitely differentiable symplectic maps which are not necessarily written in action angle-variables nor are perturbation of integrable systems, and for which the existence of an approximately invariant torus is known. Using the geometric approximation in the symplectic case, we prove some KAM results of 'a posteriori' type. More concretely, given a Diophantine frequency vector w , a finitely differentiable exact symplectic map f , and a parameterization of a maximal dimensional torus K that is finitely differentiable, approximately invariant, and that satisfies a nondegeneracy condition, we give an explicit condition on the size of the error f o K - K o R,, in finite differentiable norms, that guarantees the existence of an invariant torus for the map f (here R, represents the the translation on the torus by the vector w).We emphasize that K is not assumed to be either invariant or equal to (0,O) as it is done in some previous results [3-51. The key observation is that if f o K - K o T, is small in a finitely differentiable norm, then Tt[f]o St[K] - St[ K ]o T, is small in an analytic norm, for t sufficiently large. Using this fundamental fact and a local uniqueness result [B] we also obtain the bootstrap of regularity of of KAM tori for finitely differentiable symplectic maps.

References 1. J. Moser, Trans. Amer. Math. SOC.120, 286 (1965). 2. R. de la Llave, J. M. Marco and R. Moriybn, A n n . of Math. (2) 123, 537 (1986). 3. D. Salamon and E. Zehnder, Comment. Math. Helu. 64, 84 (1989). 4. D. Salamon, Math. Phys. Elec. Jour. 10, 1 (2004). 5. L. Chierchia, KAM lectures, in Dynamical systems, P a d I , Pubbl. Cent. Ric. Mat. Ennio Giorgi (Scuola Norm. Sup., Pisa, 2003) pp. 1-55. 6. A. Gonzdez-Enriquez and R. de la Llave, Analytic smoothing of geometric maps with applications to K A M theory, Preprint 07-135, h t t p ://urn.ma. utexas. edu/mp-arc (2007).

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A method to construct asymptotic solutions invariant under the renormalization group Masatomo Iwasa Department of Physics, Nagoya University, Nagoya 464-8602, Japan E-mail: [email protected]. ac.jp

1. The Renormalizatoin group method with Lie symmetry

The purpose of this paper is to see how to construct asymptotic solutions using Renormalizaton group method by means of Lie symmetry group. Due to limitation of space, we see only the application of the method to an example. More detailed and generalized procedure and the validity of this method are shown in another paper 111. We consider an application to the Rayleigh equation as follows: UI‘

+ u = & (d - p11 3 )

g,

,

9

where, u ( t ) E R,u’ = u” = and E is a small parameter. Our purpose is to construct an approximate solution of the system (1) for small E by incorporating an approximate Lie symmetry. Let Eq.(l) admits a Lie group transformation whose infinitesimal generator takes the form

x(t,u,

=

a, + ~ ( u, t ,ul)at + q ( t ,u,ul)au,

(2)

and let

x*= a, + [a, + Qa, + q(l)au/+ q(2)au,,

(3) be the prolongation of X. Because we wish to find the approximate symmetries to leading order, we only solve the following equation to determine each component of the vector field (2), = O(&).

1

(4)

254

Solving Eq.(4), we obtain a particular solution as follows:

Using the approximate infinitesimal genetator X, we construct a groupinvariant solution of the system (1). The group-invariant solution, u = U ( E ; C ) , is obtained by solving the so-called Lie equation

x*1. - 4 E ; t))l,=,(Ejt) = 0, x*{u’ u!(&; t)ll,=,(E;t) = 0, -

(6)

(7)

whose boundary condition is the solution of unperturbed system, i.e. U(&=

0, t ) = do)s.t.

do)’’+ do)= 0.

(8)

Since we are interested in the long-time behavior of the system, we consider the case of large t. Thus, the renormalization group equations (6) and (7) are approximated by

4

a,A = 2 (1 -

:),

aTCu = 0,

+

(9)

(10)

(g)

where A := (uz u”) , a := ;Log , T := E t . This renormalization group equation is equivalent to that obtained with renormalization group method proposed by the Illinois group [2].

2. C o n c l u d i n g remarks In the method presented here, the renormalization group equations appear as the main ingredient in the Lie group theory. This method does not require any perturbational analysis to determine approximate solutions of a perturbed system but, instead, approximate symmetries. Because the determining equation is always a linear system for the symmetry, it is easier to calculate an approximate symmetry than to obtain an approximate solution of the nonlinear system. Thus, the present method always remains within the framework of Lie group theory and completely frees us from the need to calculate naive perturbation solutions. References 1. M. Iwasa and K. Nozaki, Prog. Theor. Phys. 116 605 (2006). 2. L. Y . Chen, N. Goldenfeld and Y . Oono, Phys. Rev. E 54 376 (1996).

255

Utiyama's reduction method and infinitesimal symmetries of invariant Lagrangians Josef JanySka

Department of Mathematics and Statistics, Masaryk University JandEkovo ndm. Ba, 602 00 B m o , Czech Republic *E-mail: [email protected]

Let G be a Lie group (a gauge group) and F1 and F2 be gauge-natural bundle functors in the sense of Eck [l]defined on the category of principal G-bundles. An operator D : Fl -+ Fz is said to be a gauge invariant (natural) differential operator (NDO) if it is invariant w.r.t. the principal bundle authomorphisms 'p : P + P . Moreover, D is a natural operator iff it commutes with the Lie derivatives in the sense that L s D ( P ) ( a )= T D ( P ) ( L = a ) for any right G-invariant vector field E on the principal bundle P . Let us consider the afine bundle r : P C o n P -+ M of principal connections on any principal G-bundle P which is a gauge-natural bundle (GNB) of order (1,l) and consider an r-th order gauge-invariant (natural) Lagrangian L : J T P C o n P-+ A"T'M. Utiyama [6] proved that invariant 1-order Lagrangians of principal connection (gauge fields) r are invariant (0-order) Lagrangians of the curvature tensors R [ r ] i.e. , L ( j ' r ) = E(R[r]). This theorem is known as the Utiyama's theorem. Moreover, Utiyama proved a general version of the invariant interaction (the minimal coupling principle), i.e. any invariant 1order Lagrangian of particle fields @ (sections of vector bundles associated with P w.r.t. linear representations t of G) and l7 are invariant (0-order) Lagrangians of the covariant derivatives V@,i.e. L ( j i @ ,I?) = E(V@).Now, let E be a right G-invariant vector field on P . Then on PCconP we have the induced vector field PCon(E) and a Lagrangian L is natural if and only if the Lie derivative L ~ c ~ ~ (=~0,)where , ~ LPCon(2)' is the 1st jet lift of PCon(Z) and L is the Lie derivative. Let us assume classical (linear) connections A on M . Then A can be considered as principal connections on the principal GNB F r ' M + M of 1st order frames. Utiyama's results are then special versions of the famous

256

1st and 2nd reduction theorems (RT) of Schouten [5].Utiyama's results can be very simply generalized for NDO with values in a GNB of order ( 1 , O ) . Then the Utiyama's theorem is in fact the 1st RT for principal connections (gauge fields) and the Utiyama's invariant interaction is the 2nd RT (in minimal possible orders). From the higher-orders RT [2,5] we see that in order to generalize Utiyama's results to higher order, we have to consider covariant derivatives of the curvature tensor of I' and in this case we have to consider also classical connections (gravitational fields) on the base manifold. From the higher-order Utiyama-like theorem 131 and the higher-order Utiyama invariant interaction [4] we have Theorem: Let s , r 2 k - 1, s 2 r - 2. All natural Lagrangians of orders s in A, r in r and k in CP are of the form

L ( j s A , j T I ' )= E(c,V(s-l)RIA],V(T-')RII']) L ( j 5 A , j T I ' , j k @=) L(t',C, V("-')R[A], V('-')R[I'], V("@),

where 2 is a unique zero order natural Lagrangian, c is the set of the structure constants of G and e' is the set of constants given b y the linear representation f2 of G defining the bundle of particle fields. Example: Let ( M ,9 ) be a (pseudo-)Riemannian oriented manifold. A higher order Yang-Mills Lagrangian is the natural Lagrangian given by the invariant function

L

= gxlPl

gx2P2 gP1ul

d . . .g p r a T cad

e Cbe R a x l X z ;PI ;.. .;PT R b P ~Pz i01;...;UT

.

Corollary: Let C be a right Gh-invariant vector field on F r ' M and E be a right G-invariant vector field on P . Then an ( s ,r)-order Lagrangian L ( j s A , j r I ' )is natural if and only if L ~ ~ , l ( ~ ) S + p ~ o= n (0~. ) T C Acknowledgment Supported by MSM0021622409, GA 201/05/0523 (Czech Republic) and PRIN 2005/2007 "Leggi di conservazione e termodinamica in meccanica dei continui e in teorie di campo" (Italy). References 1. D. E. Eck, Mem. Amer. Math. SOC.33 No. 247 (1981). 2. J. JanySka, Diff. Geom. Appl. 20 177 (2004). 3. J. JanySka, Rep. Math. Phys. 58 93 (2006). 4. J. JanySka, Rep. Math. Phys. 59 63 (2007).

5. J. A. Schouten, Ricci calculus, Berlin-Gottingen 1954. 6. R. Utiyama, Phys. Rev. 101 1597 (1956).

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Stability of the tethered satellite system relative equilibria. Unrestricted problem 1. I. Kosenko' Engineering Mechanics Department, Moscow State University of Service, Pushkino district, Moscow region 141221, Russia * E-mail: kosenkoOccas. m

S. Ya. Stepanov Department of Mechanics, Dorodnicyn Computing Center of R A S Vavilov street, Moscow 119333, Russia E-mail: stepsj0ccas.m

Keywords: Tethered satellite system; unilateral constraint; Routh reduction; relative equilibria; Lyapunov stability.

A free fly of two material points Mo and M I representing an orbital station and a subsatellite correspondingly under the action of gravity of a fixed attracting point P supposed as a planet is under consideration. Denote masses of the station and the subsatellite by mo and ml. The points supposed to be connected by an ideal flexible, massless, and inextensible tether of constant length. For that reason the free fly is interrupted from time to time by the impacts with the unilateral constraint of the tether. The problem is considered in its unrestricted form: material endpoints, the orbital station and the sub-satellite, composing the tethered satellite system (TSS), move independently in field of gravity of the fixed attracting center P each thus performing a piecewise Keplerian motion. To represent the dynamics of the system outlined above we bound ourselves with the Lagrange dynamics description. Then the mechanical system becomes a natural one with discontinuities on velocities. The standard Routh procedure of reduction similar to the node elimination in the threebody problem is performed resulting in the amended potential U.Then the only potentially stable radially oriented relative equilibria are computed.

258

Having in mind an application of the theorem [l]on stability of the equilibrium for the Lagrange mechanical system with the unilateral constraints we follow the sequence of consideration such that first we suppose the constraint is tense and compute an equilibrium. Secondly we compute the tether tension, and in case of its positive value the equilibrium turns out to be physically possible. Finally we conclude that within limits of the tether admissible length the constraint is tense, and compute in the problem parameters plane, ml and the tether length, the configurations are stable in the Lyapunov sense using the condition for the amended potential in the vicinity of the equilibrium to have a strict maximum on all variables. In the calculation, computer algebra was applied. Then according to A. P. Ivanov’s [l]theorem for the configurations mentioned stability holds for the relative equilibrium in the case of purely elastic impacts. In addition, it is possible to conclude also that due to the existence of the Jacobi integral between the impacts the stability is conserved if the restitution coefficient at impact is less than one. In this case the amplitude of the bouncing over the constraint damps and the value of the integral constant decreases from impact to impact. The TSS limit motion corresponds to the case of the tensed constraint. Note finally that the result obtained can be useful to estimate the stability of the orbital large-scale constructs such as a space elevator. Treating our simplified model we can conclude the space elevator stability is possible if the mass ratio rnl/rno is small enough because the elevator tether has to be long enough. It is also easy to compute the upper limit for this ratio. Problems similar to one analysed here were considered in [Z-41 but only for the case of equal masses and without taking into account the unilateral nature of the tether constraint. The paper was prepared with the partial support of Russian Foundation for Basic Research, projects 06-01-90505, 05-01-00454, 88-6667.2006.1.

References 1. 2. 3. 4.

A. P. Ivanov, J . App. Math. Mech., 48, 523 (1984). V. V. Beletskii, 0. N . Ponomareva, Cosmic Research, 2 8 , 664 (1990). L.-S. Wang, S.-F. Chen, Cel. Mech. Dyn. Astron., 63,289 (1996). M. Krupa, A. Steindl, H. Troger, Meccanica, 35,353 (2000).

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Globally minimizing parabolic motions in the Newtonian N-body problem E. Maderna Instituto de Matemdtica y Estadz'stica "Prof. Rafael Laguardia" Universidad de la Repdblica Herrera y Reissig 565, 12000 Montevideo (UY) E-mail : [email protected] A . Venturelli Laboratoire d'Analyse Non-line'aire et Ge'ometrie Universite' d 'Avignon 33, rue Louis Pasteur, 84000 Avignon, ( F R ) E-mail: Andrea. VenturelliOuni-avignon.fr

This note is a short version of [3]. We consider N positive masses in a euclidean space R d ,subject to a gravitational interaction, and we find some interesting solutions having a given asymptotic behaviour. The equation of motion of the N-body problem can be written

where mi denotes the mass and r'i E Rd the position of the i-th body. Since these equations are invariant by translation, we assume that the center of mass is fixed at the origin. The problem has a variational formulation, hence it is natural to search solutions that are minimizers of the lagrangian action. We search in particular solutions defined on an infinite interval [O, +m), starting from a given configuration at t = 0 and having a given asymptotic behaviour for t -+ 1-00. A solution t ++ ( F l , ...,F'N)( t ) of the N-body problem is said to be parabolic if the velocity of every body tends to zero as t + +oo. Let U be the N-body Newtonian potential and I the moment of inertia with respect to the center of mass. Given any configuration x, we denote 5 = x / m the normalized configuration. It is well known (see for instance [1,2]) that if t ++ x ( t ) is a parabolic solution, the normalized config-

260

uration Z ( t ) is asymptotic to the set of central configurations (i.e. critical points of 0 = I1/’U). Given a central configuration zo with I(x0) = 1, we say that a parabolic solution t H x ( t ) is asymptotic to xo if Z ( t ) -+ zo as t -+ +co. A central configuration xo is said to be minimizing if it is an absolute minimum of 0. We can now state our main result.

Main Theorem. Given any initial configuration xi and any minimizing normalized central configuration 2 0 , there exists a parabolic solution y : [O,+co[-+ (IRd)N starting f r o m zi at t = 0 and asymptotic t o zo for t -+ fco. This solution is a minimizer of the lagrangian action with fixed ends in every compact interval contained in [0,+co[and it as collision-free for t > 0. The parabolic solution y is obtained as limit of a sequence yn : [O,t,] + ( R d ) Nof minimizers connecting xi with a configuration homothetic to xo in time t,, and t , + +co. The proof of the Theorem is achieved by comparing the action of the N-body problem with the action of a Kepler problem. A basic tool is the minimizing property of the homothetic-parabolic solution with associated central configuration x ~ Our . Main Theorem has a natural interpretation in terms of McGehee coordinates and collision manifold. Indeed, in [1,4,5] it is shown that if xo is a central configuration with I(x0) = 1, the state (XO,~ 0 x 0 ) with vo = ( 2 U ( x 0 ) ~is/ a ~ critical point of the McGehee vector field in the collision manifold, and its stable set corresponds to parabolic solutions asymptotic to xo as t -+ +co. Therefore, we can formulate the Main Theorem by saying that the stable set of (20,~ 0 x 0 ) (for the McGehee vector field) projects on the whole configuration space, provided xo is a minimizing central configuration.

References 1. A . Chenciner, Collision Totales, Mouvements Complbtement Paraboliques et Re‘duction des Homothe‘ties dans le Problbme des n Corps, Regular and Chaotic Dynamics, Vol. 3, n03, pp. 93-105, (1998). 2. N . Hulkower, D. Saari, On the manifolds of total collapse orbits and of comEq. 41, no. 1, pletely parabolic orbits for the n-body problem, Journal Diff. 27-43, (1981). 3. E. Maderna, A. Venturelli, Globally Manimizing Parabolic Motions in the Newtonian N-Body Problem, preprint (2007). 4. R. McGehee, R p l e collisions an the collinear three-body problem, Inv. Math. 27, p p . 191-227, (1974). 5 . R. Moeckel Orbit Near Triple Collision in the Three-Body Problem, Indiana Univ. Math. Journ., vol. 32, no 2, pp. 221-240, (1983).

26 1

Geodesically equivalent flat bi-cofactor systems K. Marciniak

Department of Science and Technology Campus Norrkoping, Linkoping University 601-74 Norrkoping, Sweden E-mail: [email protected] We find a sufficient condition for a J-tensor J to generate from any given flat bicofactor system a multiparameter family of geodesically equivalent flat bi-cofactor systems.

1. Cofactor and bi-cofactor systems The system of Newton equations

on a pseudo-Riemannian manifold M equipped with the metric tensor g is called cofactor [l],[2], [3] if the force F = (Pi) attains the form

F = - (cofJ)-l VV (with V V = GdV, G = 9-l and with cofJ being the cofactor matrix for J so that J (cofJ) = (cofJ) J = (det J) I ) for some quasipotential V = V ( q ) where the (1,l)-tensor J is a J-tensor for the metric g or J,-tensor 141; this means that J is symmetric and satisfies the characteristic equation V hJ; = (ajSi d g j h ) for some 1-form a (this means that J is a conformal Killing tensor of trace type with ai = & t r J and with vanishing Nijenhuis torsion). Consequently, the system (1) is called bi-cofactor if the force F can be represented as a cofactor system in two distinct ways i.e. if

+

F = - ( cofJI)-' V V

= - (cofJZ)-l

VW

(2)

for two distinct J,-tensors Ji and for two quasipotentials V and W . The system (2) is called flat if the underlying metric is flat.

262

A fundamental feature of a cofactor system is the fact [4]that such a system is always geodesically equivalent to a potential Newton system. Based on this observation one can prove the following theorem. Theorem 1.1. Assume that the metric g has a third Jg-tensor J 3 and consider G3 = a3J3G (with u3 = det 5 3 ) as a new metric tensor (in contravariant form). Then (1) The metric 93 = GT1 is geodesically equivalent to g in the sense that

geodesics of g and g3 coincide as unparametrised curves on M . (2) In the new independent variable t 3 defined through dt3 = dt/u3 the bi-cofactor system (2) attains the form

where ( r ( 3 ) ) i k are Christoffel symbols of the metric 93 and where V ( 3 )= Gad. It can be moreover shown that both the tensor J 1 J T 1 and JZ J T 1 are Jg,-tensors so that 93 is the underlying metric of the reparametrized system ( 3 ) . The parameter t3 is the affine parameter associated with the metric 93. The system (3) is therefore geodesically equivalent to the system (2) meaning that they both share the same (unparametrized) trajectories and that their underlying metrics are geodesically equivalent. 2. Families of equivalent flat bi-cofactor systems

Suppose now that the underlying metric g is flat so that in some pseudoeuclidean coordinate system it attains the form g = diag(e1,. . . ,

E ~ with ) ~i =

fl.

(4)

It is then known (see [2], [4]) that the general form of a J,-tensor in these coordinates is Jij

= mqiqj

+ p"$ + pjqi +

+

+

p

(5)

where m, pi and yaj = yji are $ ( n l ) ( n 2) independent constants and where Jij = JkGkj. Suppose now that we deform metric g with the help of a Jg-tensor as in Theorem 1.1. We have then

263

Theorem 2.1. Assume that g is of the f o r m (4) and that J is a J,-tensor given by (5). Then for the equivalent metric g 3 given by G3 = (det J ) J G to be flat it is suficient that m = 0 and PTg

(COP) P =0

with p = (pi,. . . , ,On)T.

(6)

Proof. Since g is flat, by the classical result of Beltrami the equivalent metric g3 is of constant curvature and therefore their scalar curvatures N and ~3 are related by the formula XI( g 3 ) i j = wgij

-

Vifj

+ fifj

(7)

(see [5] p. 293) with f i being in our case f i = -&ai with a. - L o?q, k . Substituting this into (7) and performing contraction with G3 we obtain 1 vinj)].

n Thus, if

N =

0, a sufficient condition for

w3

to be zero is

Calculating this condition in our setting and using the fact (true for any 0 torsionless tensor J) that ( ~ d ( t r J= ) J T d a we arrive a t (6). Theorem 2.1 applied to a flat bi-cofactor system yields immediately a multiparameter family of geodesically equivalent flat bi-cofactor systems: given a flat bi-cofactor system, take (in pseudo-euclidean coordinates for g ) any Jg-tensor J with m = 0 and such that it satisfies (6) (J,-tensors with these properties form a i n ( n 3)-parameter family) and deform this system as in Theorem 1.1 to a new system with the metric 93 given by G3 = (det J)JG. This new system will then also be flat.

+

References 1. S. Rauch-Wojciechowski,K. Marciniak and H. Lundmark, J. Math. Phys. 40 (1999) no. 12, 6366-6398. 2. H. Lundmark, Stud. Appl. Math. 110 (2003) no. 3, 257-296. 3. M. Crampin and W. Sarlet, J. Math. Phys. 42 (2001), no. 9, 4313-4326. 4. S. Benenti, Acta Appl. Math. 87 (2005), no. 1-3, 33-91. 5 . J. A. Schouten, Ricci-calculus. Springer-Verlag Berlin-Gottingen-Heidelberg 1954.

264

Symmetry group and symplectic structure for exotic particles in the plane L. Martina * Physics Department, Universitb del Salento and Sezione INFN d i Lecce. Lecce, 73100, Italy * E-mail: [email protected] Position coordinates of “exotic” particles, associated with the 2-parameter central extension of the planar Galilei group, do not commute. Realization of the model in solid state Physics, via Berry phase effects, and the relation to anyons are discussed.

Keywords: 2D-Galilei group; Berry Phase; Anyons.

It is well-known that the Galilei group, in d 2 3 space dimensions, only admits a 1-parameter central extension, generated by the physical mass, m. However, in the early seventies L6vy-Leblond 111 recognized that, owing to the commutativity of the planar rotation group 0 ( 2 ) , the Galilei group in the plane admits a second “exotic” extension parameter IE, connected to the non-commutativity of Galilean boost generators,

[KI’K2] = in.

(1)

This fact has long been considered a mere mathematical curiosity, until the construction of theoretical physical models carrying such an “exotic” structure. In particular [2] a mechanical model was constructed along the lines of Souriau’s “orbit” method. The Poisson structure is characterized by a non vanishing bracket of the planar coordinates IE

{x~,xz} = - = 8, m2 -

(2)

which can be generalized by including a “magnetic” contribution. Then, an exotic and charged particle interacts with an arbitrary external electromagnetic field, according to the equations of motion

m*$.- p i - emOEijEj, pi = eEi

+ eBcijxj

(m*= m ( l - eOB)). (3)

265

The novel features, crucial for physical applications, are i) the anomalous velocity term perpendicular to the electric field, making momentum and velocity non parallel, and ii) the egectiwe mass m*,coming from the interplay between the exotic structure and the magnetic field. Of course, only uniform external fields preserve Galilean symmetry, and an enlarged group involving space-time translations, rotations, boosts and electric linear SUperposition was found [4]. Its central extensions are provided by m, B and magnetic field B. By this approach a unified description of the combined system of particle plus fields can be given. In particular, the complete classification and the symplectic structure of the group coadjoint orbits was given in terms of the two Casimir functions of the algebra [5]. In particular, extrema of the Casimir functions signal minimal dimensional orbits, for which Hamiltonian reduction procedure is required. This leads to reduced symplectic coordinates, which are canonically conjugated as in (2), and momenta, which are completely determined by the Hall law. This phenomenon shows analogies with the quasi-particle motion in Quantum Hall Effect, which has only the electric potential in non commuting variables as Hamiltonian [3]. Again in 2D, one can extend the model [4,6] in order to include extra terms, taking into account the anomalies in the gyromagnetic factor, experimentally determined, with respect to the predicted value g = 2 in the context of the relativistic anyon theories. These models gave also a first physical meaning of the “exotic” constant, as the non relativistic limit s/c2 -+ m2B of a relativistic particle of spin s in the plane. For a deep analysis of this topic see Ref. [7]. A different interpretation of 6 comes from condensed matter Physics [8], where the “exotic” term has been identified with a Berry phase curvature in the semiclassical dynamics of a Bloch electron in a crystal structure. Recent developments include the Anomalous [9], Spin [lo] and Optical Ill] Hall Effects. These “Berry-related” semiclassical models exist in three (or higher) dimensions. Exotic Galilean symmetry is lost, but the basic structure is unchanged. In particular, in Ref. [12] it was proved that a very large class of systems, obtained by adiabatically approximating Bloch wave packets, are indeed classical Hamiltonian systems written in non canonical variables. Even if, accordingly to Darboux’s theorem, canonical variables could be found, this could be very difficult in general and useless in practice. To be concrete, in the presence of external, slowly modulated electromagnetic field, the noncommutative parameter is generically promoted to a vector-valued function of the lattice quasi-momentum 0 = O(k), called

266

the “dual”, or “reciprocal”, magnetic field . The Poisson structure is given by

(4) and the Hamiltonian is H = &O (k)+eV (r,t)-M (k).B(r,t). Both 0 and B have vanishing divergence, in the respective domains of definition, because of their curvature nature. The equations of motion for such a particle are given by

[& (k) - M (k) . B (r, t)] - k x O (k) , k = -e (r x B (r, t ) + E (r,t ) ) V, (M (k) . B (r,t ) ), r=v

k

+

(5)

where 80(k) is the energy band function and M (k) is the magnetic moment of the wave packet. It should be noticed here that k takes value on the first Brillouin zone, so the phase space has a fibered structure over a 3D torus. This implies that any integral of Oi (k) is a topological invariant, called the first Chern number, integer multiple of ( 2 ~ )This ~ . explains the quantization of the Anomalous conductivity. Analysis of the Lagrangian symmetries of the system (5) can be performed by resorting to the action functional on the phase space [12].

Acknowledgements Indebtedness is expressed to C. Duval, Z. Horvbth, P. Horvathy, M. Plyushchay and P. Stichel for joint researches, which are partially supported by Italian MUR and by INFN under the project LE41. References J.-M. L&y-Leblond, in Group Theory and Applications, Acad. Press (1972). C. Duval et al.,Phys. Lett. B 479,284 (2000); J. Phys. A 34,10097 (2001). R. B. Laughlin,Phys. Rev. Lett. 50 (1983), 1395. P. A. Horvgthy, L. Martina and P. Stichel Phys. Lett. B 615,87 (2005). L. Martina, Fundam. Appl. Math. 12 109 (2006). P. Horvgthy, L. Martina, P. Stichel, Mod. Phys.Lett. A 2 0 (2005), 1177. C. Duval and P. A. Horvtithy, Phys. Lett. B 457,306 (2002). M. C. Chang and Q. Niu, Phys. Rev. Lett. 75,1348 (1995). D. Culcer et al. Phys. Rev. B 68, 045327 (2003). S. Murakami et al. Science 301,1348 (2003). 11. V. S. Liberman et al. Phys. Rev. A 46,5199 (1992); M. Onoda et al. Phys. Rev. Lett. 93,083901 (2004); C. Duval et al. Journ. Geom. Phys. 57 925 (2007). 12. L. Martina et al., Theor. Math. Phys 151,792 (2007). 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

267

Some properties of the dynamics of two interacting particles in a uniform magnetic field D. Pinheiro' and R. S. Mackay" * Centro de Matemcitica da Universidade do Porto, Rua do Camp0 Alegre 687, 4169-007 Porto, Portugal E-mail: [email protected]

**Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom E-mail: [email protected] We summarise some properties of the dynamics of two interacting particles moving under the action of a uniform magnetic field. Keywords: Hamiltonian dynamics; Scattering map.

1. The planar problem In Ref. 1 we made a detailed study of the problem of the interaction of two charged particles moving in a plane under the effect of a uniform magnetic field. We assumed that the interaction between the particles was given by a potential depending on the distance between the two particles and that the magnetic field was orthogonal to the plane of motion. That problem can be formulated as a Hamiltonian system with four degrees of freedom. We made extensive use of the symmetries in that Hamiltonian system to obtain a reduction in the dimension of the problem to two degrees of freedom. In the special case of same sign charges with equal gyrofrequencies (equal ratio of charge to mass) or on some special submanifolds we proved that this system is integrable. We then specialized our analysis to the most physically interesting case of a Coulomb-like potential. Analysing the reduced systems and the associated reconstruction maps we provided a detailed description for the regimes of parameters and level sets of the conserved quantities where bounded and unbounded motion are possible and we identified the cases where close approaches between the two particles are possible. Furthermore, we identified

268

regimes where the system is non-integrable and contains chaos by proving the existence of invariant subsets containing a suspension of a non-trivial subshift. 2. The spatial problem

In Ref. 2 we look a t the same problem but with the particles now moving in R3. This system can be formulated as a Hamiltonian system with six degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotational symmetry we reduce this Hamiltonian system to one with three degrees of freedom; for certain values of the conserved quantities or choices of parameters, we obtain a system with two degrees of freedom. Furthermore, it contains the planar case as a subsystem. The reductions simplify the analysis of some properties of this system: we use the reconstruction map to obtain a classification for the dynamics in terms of boundedness of the motion and the existence of collisions. We also study the scattering map associated with this system in the limit of widely separated trajectories. In this limit and provided that the two particles gyrofrequencies are rationally independent we prove that the norms of the gyroradii of the particles are conserved during an interaction and that the interaction of the two particles is responsible for a rotation of the guiding centres around a fixed centre in the case of two charges whose sum is not zero and a drift of the guiding centres in the case of two charges whose sum is zero. We also analyse the case in which the two particles have rationally dependent gyrofrequencies, obtaining transfer of magnetic moment between the particles for low order rational dependence of the gyrofrequencies. Furthermore, we have made a numerical study of the scattering map without using the assumption that the two particles trajectories are widely separated. We observed regular behaviour for large energies and chaotic scattering for small positive energies.

Acknowledgments

D. Pinheiro’s research is supported by F’unda@io para a CiGncia e a Tecnologia and Centro de Matematica da Universidade do Porto. References 1. D. Pinheiro and R. S. MacKay, “Interaction of two charges in a uniform magnetic field: I. Planar problem”, Nonlinearity 19 (2006), 1713-1745. 2. D. Pinheiro and R. S. MacKay, “Interaction of two charges in a uniform magnetic field: 11. Spatial problem”, University of Warwick Preprint 2006.

269

On the dihedral n-body problem Alessandro Portaluri Dipartimento di Matematica, University of Milano-Bicocca, Via R. Cozti 53, 20125, Milano Italy. * E-mail: [email protected] www.matapp. unimib. it Consider n = 21 2 4 point particles with equal masses in space, subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group Dl, where Dl is the group of order 21 generated by two rotations of angle n around two secant lines in space meeting at an angle of n/1. By adding a homogeneous gravitational (Newtonian) potential one finds a special n-body problem with three degrees of freedom, in which all orbits have zero angular momentum. Keywords: n-body problem; McGehee coordinates; central configurations.

1. Brief description of the r e s u l t s In this short communication we summarise some of the results proved in a joint work with D.L. Ferrario [2] in which we investigated the qualitative behaviour of solutions of the dihedral symmetric n-body problem in space which is a special case of the full n-body problem. Briefly, for 1 2 1 let Dl c SO(3) be the dihedral group of order 21. The action of Dl on R3 induces an orthogonal action on the configuration space R6' of n = 21 point particles qi E R3.So, they form a (possibly degenerate and non-regular) antiprism in space (and they are vertices of two symmetric parallel 1-gons). Because of the symmetry, the masses will remain in such a configuration for all time. Hence we have a system with only three degrees of freedom. For 1 = 2, the four bodies are a t vertices of a tetrahedron, and the problem has been studied in a series of papers by Delgado and Vidal [1,5]. However in this situation we are dealing with just three types than can arise: a planar regular 21-gon1 a regular I-gonal prism and a I-gonal anti-prism which can be visualized together with the asymptotic collision sets in Fig. 1. Furthermore, by using McGehee coordinates [3] together with some topological argument in the same vein as in Moeckel [4], we prove the existence

270 co (two 1-adic collisions)

prism/

,

I tantiprism

binary collisions) 00

( I binary

Fig. 1. Central configurations in the upper-half fundamental domain

of some heteroclinic connections in the total collision manifold and between these central configurations and connections between central configurations and the asymptotic collision sets. Taking into account the existence of these connecting orbits we are able to establish some global results on the behaviour of the dynamical system. In particular all of these results reveal a chaotic behaviour for collision orbits to total collapse in the sense that all these orbits are extremely sensitive to small changes in initial conditions. Indeed, after passing close to total planar or prism type central configuration, the system emerges with arbitrarily large kinetic energy; moreover in every neighborhood of every planar or prism type collision orbit there are anti-prism collision orbits as well as orbits which avoid collision and emerge from a neighborhood of the singularity in any of the non-regular n-gonal or n-prism type configuration.

References 1. J. Delgado and C. Vidal, The tetrahedral Cbody problem. J . Dynam. Differential Equations 11, 4 (1999), 735-780. 2. D. L. Ferrario and A. Portaluri, On the dihedral n-body problem. Preprint (2007). 3. R. McGehee, Triple collision in the collinear three-body problem. Invent. Math. 27 (1974), 191-227. 4. R. Moeckel, Orbits near triple collision in the three-body problem. Indiana Univ. Math. J . 32, 2 (1983), 221-240. 5. C. Vidal, The tetrahedral 4-body problem with rotation. Celestial Mech. Dynam. Astronom. 7f,1 (1998/99), 15-33.

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Lax representation for integrable OAEs 0. Rajas*, P. van der Kamp and G.R.W. Quispel Department of Mathematics, La Trobe University, Victoria, 3086, Australia *E-mail: [email protected]. edu. au We derive a Lax-representation for integrable maps (or OAEs) obtained by travelling wave reductions from integrable PAEs with Lax pairs.

Keywords: Discrete integrable systems; Lax representation

1. Outline

It is believed that all integrable systems possess a Lax representation. OAEs can be obtained from PAEs by travelling wave reductions. Suppose the Lax representation of the PAE is known. Then the question arises: do integrable maps obtained in this way posses a Lax representation? If yes, how does one obtain it? This paper provides the answer to the above question positively for so called (q,p)-travelling wave reductions introduced in [l].One of the Lax-matrices for the OAE coincides with the monodromy matrix, which is obtained by taking a product of PAE Lax-matrices along the (q,p)staircase. 2. L a x representation

Consider an integrable PAE on a two-dimensional lattice, namely f i , m = f('~i,m,~1+1,,,~i,m+i,~i+l,~+i; ~ i = ) 0, with l , m E Z and cyi parameters. This equation has a Lax representation if there are matrices L , M , N depending on a spectral parameter k such that L i , m M < ~ - M ~ ~ , m L ~ ,=m + l fi,mNi,m, in which f i , m does not depend on k , and Nl,m is nonsingular on the equationa. Similarly, an OAE f n = f(v,, un+l,. . . , v,+,; ai) = 0, with n , a E Z and Q, parameters, has a Lax representation if there are "Note that the right hand side vanishes for solutions of the equation and is set to 0 by many authors.

272

matrices C,M , N depending on a spectral parameter k such that M,Cn Cn+1M, = f,N,. Right-multiplying by -M;' we obtain the invariance of C n , i.e., trC,+1 - trC, = fnhn,where A, = -trN,M;l. The coefficients in the expansion in powers of the spectral parameter of the trace of the monodromy matrix give integrals of the mapping. A P A E can be reduced to an OAE through a travelling wave reduction by the ansatz u, = ~ 1 via, the similarity variable n = ql p m , where q , p are coprimes and l,m are the independent lattice variables. It is clear that this relationship induces the periodicity condition ul+p,m-g = ~ l l , which ~ , allows us to solve the initial value problem on the OAE. The staircase method provides a way of generating invariants for OAEs obtained in this way. The monodromy matrix C, is defined to be the ordered product of Lax-matrices along a standard staircase [l].For that purpose it is useful to introduce the following

+

Definition 2.1. Given p , q , 1 E N, let Q i be a matrix such that

j=O

j=a

Now we can state our main result, explicit formulae for the Lax-matrices of (q,p)-reductions in terms of the PAE Lax-matrices. Theorem 2.1. Given q , p E N with g c d ( q , p ) = 1 and 1 < q 5 p , there are integers s, s-l E [0,q ) such that p E s mod q and ss-l E 1 mod q. Thus, the Lux representation for the OAE arising through a (q,p)-travelling wave reduction is given by

i=l

where mi

=

i=s-'+1

~i mod 4 and N, = M,L,lNnMnCn

Acknowledgments

This research has been funded by MASCOS. O.R. acknowledges support from the EIPRS and LTUPRS scholarships. References 1. G.R.W. Quispel, H.W. Capel, V.G. Papageorgiou and F.W. Nijhoff, Physica A 173,243 (1991). 2. Peter H. van der Kamp, 0. Rojas and G.R.W. Quispel, J . Phys. A : Math. Theor. 40, in press (2007).

~

273

P s e u d o r o t a t i o n a l S p e c t r a of Molecules and I s o p a r a m e t r i c G e o m e t r y A. R. Rutherford

IRMACS Centre, Simon Fraser University Burnaby, British Columbia V 5 A 1S6, Canada Email: sandyrOinacs.sfu. ca We consider the problem of calculating the pseudorotational energy levels associated with the molecular Jahn-Teller effect for electronic triplets. We show that the energy levels are related to the spectra of Laplace-Beltrami operators on line bundles over the leaves of Cartan’s isoparametric foliation of S4. These spectra are computed explicitly using the Peter-Weyl theorem and harmonic analysis on line bundles. Keywords: Isoparametric foliation; Harmonic analysis; Molecular pseudorotations; Jahn-Teller effect.

The Jahn-Teller theorem [3] implies that for a molecule with the symmetry of a nonlinear point group, the minimal energy submanifold will not contain the symmetric configuration of the nuclei. This is a quantum mechanical phenomena, which is counterintuitive from a classical or semiclassical perspective. The molecular Hamiltonian may be decomposed into electronic and nuclear degrees of freedom using the Born-Oppenheimer approximation. Corresponding to each electronic multiplet is a Born-Oppenheimer Hamiltonian, the spectrum of which gives the vibronic energy levels associated with that multiplet. We focus on electronic triplets, which occur for molecules with tetrahedral, octahedral, or icosahedral symmetry. Projecting the electronic part of the molecular Hamiltonian onto the subspace associated with the electronic triplet gives an element of Herm(S,R), the vector space of 3 x 3 real, symmetric matrices. On this vector space, we define the inner product ( A , B) = t r A B. The effective potential of the Born-Oppenheimer Hamiltonian is given by the eigenvalues of the electronic Hamiltonian. Therefore, we foliate the unit 4-dimensional sphere of traceless matrices in Herm(3,R) by their eigenvalues. If we or-

2 74

der the three eigenvalues by XI 5 XZ 5 X3, then the leaves, S(b), in the E [0,1]. For b E ( O , l ) , the leaves are foliation are labelled by b = homeomorphic to the 3-dimensional real, complete flag manifold. The focal submanifolds of the foliation are S(0) and S(l),which correspond to the Veronese embedding of RP(2) into S4.The minimal energy submanifold for the molecule corresponds to one of the leaves of this foliation. The principal curvatures of the leaf $(b) are

Note that for fixed b, the principal curvatures - although not the principal curvature directions - are constant. Therefore, this foliation is Cartan's isoparametric foliation [I]of S4. The Born-Oppenheimer Hamiltonians act on sections of the three nontrivial eigenspace line bundles over S(b). We denote the line bundle associated with X i by X,. Following the the nomenclature for electronic doublets, we refer to molecular excitations arising from the free motion of the molecular configuration on the minimal energy submanifold as pseudorotations [2,4]. To calculate the spectra of pseudorotations, we must determine the harmonic sections of the bundles X,. The universal covering space of S(b) is homeomorphic to S 3 . However, viewing it as S p ( l ) , the Laplace-Beltrami operator with respect to the lift of the metric from S(b) has the form

where i, j , and k are the unit quaternions, which generate sp(1). Using this form of the Laplace-Beltrami operator and the Peter-Weyl Theorem, we construct a complete basis of harmonic functions in terms of homogeneous polynomials in two complex variables. We determine the action of the deck transformations of the universal cover on the harmonic functions. Then, for each of the three line bundles, we identify the transformation properties required for harmonic functions to push down to give harmonic sections of Xi(b). This allows us to obtain explicit formulae for the spectra of the Laplace-Beltrami operator on each of these line bundles [ 5 ] .The flow with respect to b of the spectra are shown in Figure 1.

275

60

60

50

50

40

40

30

30

20

20

10

10

0

L 0.2

0.4

0.6

0.8

0 0.2

0.4

0.6

0.8

Fig, 1. The plot on the left shows the spectral flow of the Laplace-Beltrami operator on the line bundle X3(b), for b E ( 0 , l ) . As b ---t 0, the spectrum flows to the spectrum over RP(2). The plot on the right shows the spectral flow for Xz(b). The spectral flow for Xl(b) is given by reflecting the graph for X3(b) about b = 1 2'

References 1. E. Cartan, Annali d i Mat., 17,177-191 (1938). 2. G. DelacrBtaz, E. R. Grant, R. L. Whetten, L. Wostel, and J. W. Zwanziger, Phys. Rev. Lett., 56, 2598-2601 (1986). 3. H. A. J a h n and E. Teller, Proc. Roy. SOC.(London), A161,220-235 (1937). 4. H. C. Longuet-Higgens, U. Opik, M. H. L. Pryce, a n d R. A. Sack, Proc. Roy. SOC.(London), A244, 1-16 (1958). 5 . A. R . Rutherford, Pseudorotations in Molecules: Electronic Orbital Thplets, IRMACS preprint (2007).

276

Darboux integrable hyperbolic PDE in the plane of generic Type: a classification by means of Cartan Tensor and Maple ll@ F. Strazzullo Department of Mathematics and Statistics, Utah States University, Logan, Utah-84341, USA E-mail: francescot3cc.u~~. edu These pages are an account of a research project aiming to classify hyperbolic PDE in the plane of generic type which are Darboux integrable. Here we focus on some symbolic computation tools, the Fivevariables package the author developed, which work in the environment DifferentialGeometry of Maple11 and that can be used to produce a variety of examples. Keywords: Darboux integrability; hyperbolic exterior differential systems; Cartan Tensor; computer algebra system.

1. Extended abstract

According to [l],every Darboux integrable hyperbolic PDE in the plane can be realized as the symmetry reduction by a 3 dimensional group G of the direct sum of two “side” general EDS in in 5 variables which admit G as symmetry group. Therefore we can get a classification of Darboux integrable hyperbolic PDE in the plane by means of a classification of rank-3 Pfaffian systems over a 5-dimensional manifold which admit a 3-dimensional symmetry group. The classification of general rank-3 Pfaffian systems in five variables was the subject of Cartan’s famous five variables paper [2]. Cartan proved that to every such a system it is canonically associated a homogeneous polynomial 31of degree 4 in two variables, called Cartan Tensor. If two general EDS’s I and J are equivalent, then their Cartan tensors 31and 3Jhave the same types of roots: therefore the roots type of 31characterizes I . We must notice that if FIand 3~have the same roots type, I and J need not to be equivalent (unless 3 1 = 0). Thus a general rank-3 EDS an five variables must be of one of the following 6 Types: four simple roots, infinitely many roots, one root of multiplicity 4, one triple root and one simple root, two double roots and finally one double root and

277

two simple roots. Cartan’s results were resumed by Stormark in [3] with a modern approach, but a clear computation of the Cartan tensor was given by Hsiao in [4]:starting from this latter paper we were able to write a Maple module, that we called Fivevariables and which can run under the Maple 11 environment DifferentialGeometry. Using Five Variables we can now find the (roots) type of a general EDS, reducing really lengthly computations to matter of seconds. We got a list of normal forms for general EDS’s, depending on some generic functions subject to differential constraints: here we are considering only the case related to the final examples, where the “side” general Pfaffian systems I1 and 1 2 are equivalent and their symmetry group is abelian. Their normal form is Ii = {dbi - ui dai, dci - Hi dai, dui - vi dai}, while the Darboux integrable hyperbolic PDE in the plane has normal form =

+

+

+

(dA k d u 1 - Id,,, v2 -dB %du1 - z d u 2 , -dC 2 d u l - %du2}, where Hi = H i ( u i , ui) are such that Hivivi# 0 for i = 1 , 2 . Example 1.1 (Type 2). For Hi = v i l the side systems li are of Type 2: their Cartan Tensor is identically 0 and their symmetry algebra is the exceptional Lie algebra 82. W e obtain the P D E in the plane

r

= s 2 / t - 3t3

which has a 9 dimensional non-semisimple and non-solvable symmetry algebra. Example 1.2 (Type 3). For Hi = uivi’ the side systems Ii are of Type 3: their Cartan Tensor has a root of multiplicity 4 and they admit a 7dimensional solvable symmetry algebra. W e obtain the P D E in the plane

r

= s 2 / t - tanh-l ( t / s 2 )

that has a 7-dimensional solvable symmetry algebra. References 1. I. M. Anderson and M. Fels and P. Vassiliou, Superposition formulas for Darboux Integrable Exterior Differential Systems, (preprint, 2005). 2. E. Cartan, Les systbmes de Pfaff Ci cinq variables et les e‘quations aux dtrivbes partielles du second ordre, (Ann. Sci. Ecole Norm. Sup. (3), 27, 1910). 3. 0. Stormark, Lie’s Structural Approach to PDE Systems, (Cambridge University Press, Cambridge - United Kingdom, 2000). 4. S. H. C. Hsiao, On Cartan-Stemberg’s example of the reduction of a GStructure 11-III., (Tamakang J. Math., ( l l ) ,2, 1980).

.

278

Towards global classifications: a diophantine approach Peter H. van der Kamp

Department of Mathematics, La Dobe University, Victoria, 3086, Australia E-mail: [email protected]. a u

An open problem is the global classification, up to linear transformations, of polynomial two-component integrable equations (ut, vt) = K , ut = au, + Kil,'-Zl + . . . + KPil-ir + . . . Vt =

bv,

+

Ki-ir+l,iT'+l+ . . . + Kt-is,is . .

+

. .. ,

(1)

where ij E { - 1,0, l},the K;j are nonzero polynomials, of degree i + 2 - k in the variables (uo=u,u1=u,, . . .) and of degree j + k - 1 in (vo=v, v1=w2,. . .), and the dots refer to terms of higher degree. The word global means we aim for a complete description at any order n E N. A pair of functions S is called an approximate symmetry of (1) if the Lie-bracket [ K ,S] vanishes modulo cubic or higher degree terms [I]. And the existence of infinitely many approximate symmetries is taken as the definition of approximate integrability. We contribute to the above mentioned problem by globally classifying approximately integrable equations (1). Note that an equation may have infinitely many approximate symmetries, but fail to have any symmetries. That problem involves conditions of higher degree and is left open. In the symbolic method, the Kil-2 are transformed into polynomials in just two symbols. The action of the Lie-derivative on homogeneous quadratic parts of S is represented by two (related) sets of polynomials ( k = l , 2 ) in the two symbols, of total degree n: [KO!',S z ~ ~ ] , = @ ~ [ a :[2]. b]S~ We denote the tuple Ki'"-al , . . . , K,'-i"'i" by K ' . Suppose afb. Then an approximate symmetry exists iff the tuple S' = Gm[c:d]K1/Gn[a:b]consist of polynomials (with the right symmetry properties). A necessary condition for equation (1) to be approximately integrable, is the existence of a proper tuple H dividing infinitely many G,[c:d], including m=n for c/d=a/b. Using the Skolem-Mahler-Lech theorem, results

279

on diophantine equations involving roots of unity obtained by Beukers [3], and an algorithm of C .J. Smyth, we have classified the divisors of infinitely many G-tuples, with s = 1 , 2 , together with the orders m E N and values c / d E Q: at which they appear [4]. Using these results one can write down the set of divisors of infinitely many (7-tuples with 2 < s < 7. The main point we stress here is that this knowledge is also suficient. Denote by m ( H ) the set of orders m such that there exist c, d E Q: for which H I Gm[c:d]and no G-function vanishes. And denote by F ' I, the set of proper tuples H with infinite m ( H ) ,whose smallest element is n. Lemma: Let H E 1-1, and a , b E Q: such that H I G n [ ~ : b Let ] . P E 7-1, be the tuple of highest degree such that H I P and m ( H ) = m ( P ) . Then equation (1) is approximately integrable, with symmetries at orders m E m ( H ) , i f Gn[a:b]/Pdivides K ' . The equation is in a hierarchy of order m < n iff there is a divisor Q E 71, of P such that P / Q divides K 1 . Given 7-1, for all n E N,using this lemma, one can write down all approximately integrable equations (1) that are not in a lower hierarchy. Example: Take K 1 = K,1'2,Ki,0. With r # -1 the tuple H = (1+ y ) ( y - r)(yr - l),z(a: 1 r ) ( r x 1 r ) is a divisor of Gm[c:d] = G;l 1 2 [c:d](l,y),G~$[c:d](z,l)w h e n m E { 3 , 5 , 7 ,...}, c / d = ( l f r " ) / ( l +

+ +

+ +

r ) m . It has maximal degree, H = P E 'FI3. Hence any third order system of this type is approximately integrable. To write down minimal conditions on K 1 such that the equation is not in a lower hierarchy one should look for Q E 7-Im 1 spatial dimensions (64) G. Pucacco: Separation of variables on the hyperbolic plane (65) F. Pugliese: O n a special class of Monge-Ampere equations (66) 0 . Ragnisco: Integrable models on curved space from q-algebras: equations of motion and their solution (67) G. Rastelli: Decomposition of scalar potentials of natural Hamiltonians into integrable and perturbative terms. A naive approach (68) S. Rauch: Separation of potential and quasi-potential Newton equations (69) 0 . Rojas: Closed f o m expression for integrals of s G map (70) A. Rutherford: Pseudorotational Spectra of Molecules and Isoparametric Geometry

294

(71) G . Saccomandi: A general reduction method for finite amplitude elastic waves (72) V. Salnikov: T h e dynamics of the triple pendulum: various approaches to non-integrability (73) T. Salnikova: Periodic solutions of one variant of the bounded threedimensional three- body problem (74) V. Samokhin: O n deformations conserving a conservation law (75) J . Sanders: An addition formula for nilpotent normal forms (76) P.M. Santini: T h e dispersionless Kadomtsev-Petviashvili equation: Cauchy problem, long-time behavior and wave breaking (77) D. Saunders: Homogeneous Variational Systems (78) C . Scimiterna: Multiscale expansion of the lattice potential K d V and of its symmetries on functions of infinite order (79) F. Strazzullo: Darboux Integrable Hyperbolic PDE’s in the Plane of Generic Type (80) M.A. Teixeira: Singularities of non-smooth dynamical systems (81) P. Tempesta: *** (82) S. Terracini: Singularities and collisions of generalized solutions t o the N-body problem (83) J. Tolksdorf: Diruc Type First Order Differential Operators as a Natural Square Root of Gravity and Yang-Mills Gauge Theories (84) E. Valdinoci: Periodic and quasiperiodic motions i n the m a n y body planetary problem (85) A. Vanderbauwhede: Continuation of doubly-symmetric solutions i n reversible systems (86) P. van der Kamp: Towards Global Classifications: a Diophantine A p proach (87) A. Venturelli: Globally Minimizing Parabolic Solutions for the Newtonian N-body Problem (88) F. Verhulst: Emergence and break-up of invariant manifolds in a parametric P D E (89) Ren. Vitolo: T h e Hopf-saddle-node bifurcation for fixed points of 3Ddiffeomorphisms: a dynamical i n v e n t o v (90) S . Walcher: T h e Michaelis-Menten equation and Murphy’s law (91) P. Winternitz: Superintegrable systems i n quantum mechanics (92) I. Yehorchenko: Relative Invariants of Lie Algebras: Construction and Applications (93) B. Zhilinskii: Generalization of Hamiltonian monodromy. Q u a n t u m manifestations

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Previous SPT conferences and proceedings

0

Workshop SPT96 - Torino, 15-20 December 1996

[l]

0

Conference SPT98 - Roma 16-22 December 1998

[2]

0

Conference SPT2001 - Cala Gonone, 6-13 May 2001

0

Conference SPT2002

-

Cala Gonone, 19-26 May 2002 [5]

0

Conference SPT2004

-

Cala Gonone, 30 May

-

[3,4]

6 June 2004

[6,7]

References 1. D. Bambusi and G. Gaeta eds., “Symmetry and Perturbation Theory”, Quaderni GNFM-CNR, Firenze 1997. 2. A. Degasperis and G. Gaeta eds., “Symmetry and Perturbation Theory SPT98”, World Scientific, Singapore 1999. 3. G. Gaeta ed., Special issue on “Symmetry and Perturbation Theory”, Acta Applicandae Mathematicae vol. 70 1:3 (2002). 4. D. Bambusi, M. Cadoni and G. Gaeta eds., “Symmetry and Perturbation Theory - SPT 2001”, World Scientific, Singapore 2001. 5. S. Abenda, G. Gaeta and S. Walcher eds., ‘Symmetry and Perturbation Theory SPT 2002”, World Scientific, Singapore 2003. 6. G. Gaeta, B. Prinari, S. Rauch-Wojciechowski and S. Terracini eds., “Symmetry and Perturbation Theory - SPT 2004”, World Scientific, Singapore 2005. 7. G. Gaeta ed., Special issue on “Symmetry and Perturbation Theory”, Acta Applicandae Mathematicae vol. 87 1:3 (2005). ~

Symmetry And Perturbation Theory: Proceedings of the International Conference SPT 2007 Otranto, Italy 2-9 June 2007

Symmetry And Perturbation Theory: Proceedings of the International Conference SPT 2007 Otranto, Italy 2-9 June 2007

Symmetry And Perturbation Theory: Proceedings of the International Conference SPT 2007 Otranto, Italy 2-9 June 2007

Symmetry And Perturbation Theory: Proceedings Of The International Conference SPT 2004 Cala Genone, Italy, 30 May вЂ“ 6 June 2004

Symmetry and Perturbation Theory: Proceedings of the International Conference on SPT 2002

Symmetry and perturbation theory: proceedings of the international conference SPT 2002, Cala Gonone, Sardinia, Italy, 19-26 May 2002

Symmetry and perturbation theory : proceedings of the international conference SPT 2002, Cala Gonone, Sardinia, Italy, 19-26 May 2002

User Modeling 2007: 11th International Conference, UM 2007, Corfu, Greece, July 25-29, 2007, Proceedings

User Modeling 2007: 11th International Conference, UM 2007, Corfu, Greece, July 25-29, 2007, Proceedings