Gauge mechanics

Gauge Mechanics

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Gauge Mechanics

L. Mangiarotti University of Camerino, Italy

G. Sardanashvily Moscow State University, Russia

World Scientific

Singapore *New Jersey London 'HongKong

Published by World Scientific Publishing Co. Pte. Ltd P O Box 128, Farcer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

GAUGE MECHANICS Copyright © 1998 by World Scientific Publishing Co Pte. Ltd All rights reserved. This book, or parts thereof, may not he 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 permission from the Publisher.

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ISBN 981-02-3603-4

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Preface This book presents in a unified way modern geometric methods in analytical mechanics, based on the application of jet manifolds and connections. As is well known, the technique of Poisson and symplectic spaces provide the adequate Hamiltonian formulation of conservative mechanics. This formulation, however, cannot be extended to time-dependent mechanics subject to time-dependent transformations. We will formulate non-relativistic time-dependent mechanics as a particular field theory on fibre bundles over a time axis. The geometric approach to field theory is based on the identification of classical fields with sections of fibred manifolds. Jet manifolds provide the adequate mathe­ matical language for Lagrangian field theory, while the Hamiltonian one is phrased in terms of a polysymplectic structure. The 1-dimensional reduction of Lagrangian field theory leads us in a straightforward manner to Lagrangian time-dependent mechanics. At the same time, the canonical polysymplectic form on a momentum phase space of time-dependent mechanics reduces to the canonical exterior 3-form which plays the role similar to a symplectic form in conservative mechanics. With this canonical 3-form, we introduce the canonical Poisson structure and formulate Hamiltonian time-dependent mechanics in terms of Hamiltonian connections and Hamiltonian forms. Note that the theory of non-linear differential operators and the calculus of vari­ ations are conventionally phrased in terms of jet manifolds. On the other hand, jet formalism provides the contemporary language of differential geometry to deal with non-linear connections, represented by sections of jet bundles. Only jet spaces enable us to treat connections, Lagrangian and Hamiltonian dynamics simultaneously. In fact, the concept of connection is the main link throughout the book. Con­ nections on a configuration space of time-dependent mechanics are reference frames. Holonomic connections on a velocity phase space define non-relativistic dynamic equations which are also related to other types of connections, and can be writ­ ten as non-relativistic geodesic equations. Hamiltonian time-dependent mechanics deals with Hamiltonian connections whose geodesies are solutions of the Hamilton v

vi

PREFACE

equations. The presence of a reference frame, expressed in terms of connections, is the main peculiarity of time-dependent mechanics. In particular, each reference frame defines an energy function, and quantizations with respect to different reference frames are not equivalent. Another important peculiarity is that a Hamiltonian fails to be a scalar function under time-dependent transformations. As a consequence, many well-known con­ structions of conservative mechanics fail to be valid for time-dependent mechanics, and one should follow methods of field theory. At the same time, there is the essential difference between field theory and timedependent mechanics. In contrast with gauge potentials in field theory, connections on a configuration space of time-dependent mechanics fail to be dynamic variables since their curvature vanishes identically. Following geometric methods of field the­ ory, we obtain the frame-covariant formulation of time-dependent mechanics. By analogy with gauge field theory, one may speak about gauge time-dependent me­ chanics. In comparison with non-relativistic time-dependent mechanics, a configuration space of relativistic mechanics does not imply any preferable fibration over a time. To construct the velocity phase space of relativistic mechanics, we therefore use formalism of jets of submanifolds. At the same time, Hamiltonian relativistic me­ chanics is seen as an autonomous Hamiltonian system on the constraint space of relativistic hyperboloids. With respect to mathematical prerequisites, the reader is expected to be familiar with the basics of differential geometry of fibre bundles. For the convenience of the reader, several mathematical facts and notions are included as an Interlude, thus making our exposition self-contained.

Contents Preface

v

Introduction

1

1

Interlude: b u n d l e s , J e t s , C o n n e c t i o n s 1.1 1.2 1.3 1.4 1.5 1.6

Fibre bundles Multivector fields and differential forms Jet manifolds Connections Bundles with symmetries Composite fibre bundles

9 9 20 35 42 46 53

2

Geometry of Poisson Manifolds 2.1 Jacobi structure 2.2 Contact structure 2.3 Poisson structure 2.4 Symplectic structure 2.5 Presymplectic structure 2.6 Reduction of symplectic and Poisson structures 2.7 Appendix. Poisson homology and cohomology 2.8 Appendix. More brackets 2.9 Appendix. Multisymplectic structures

57 57 61 66 73 81 84 89 96 100

3

Hamiltonian Systems 3.1 Dynamic equations 3.2 Poisson Hamiltonian systems 3.3 Symplectic Hamiltonian systems 3.4 Presymplectic Hamiltonian systems 3.5 Dirac Hamiltonian systems 3.6 Dirac constraint systems

105 106 Ill 114 118 123 129

vii

CONTENTS

viii

3.7 3.8

Hamiltonian systems with symmetries Appendix. Hamiltonian field theory

133 139

4

Lagrangian time-dependent mechanics 4.1 Fibre bundles over K 4.2 Dynamic equations 4.3 Dynamic connections 4.4 Non-relativistic geodesic equations 4.5 Reference frames 4.6 Free motion equations 4.7 Relative acceleration 4.8 Lagrangian systems 4.9 Newtonian systems 4.10 Holonomic constraints 4.11 Non-holonomic constraints 4.12 Lagrangian conservation laws

151 152 159 162 170 175 179 182 186 195 206 211 218

5

Hamiltonian time-dependent mechanics 5.1 Canonical Poisson structure 5.2 Hamiltonian connections and Hamiltonian forms 5.3 Canonical transformations 5.4 The evolution equation 5.5 Degenerate systems 5.6 Quadratic degenerate systems 5.7 Hamiltonian conservation laws 5.8 Time-dependent systems with symmetries 5.9 Systems with time-dependent parameters 5.10 Unified Lagrangian and Hamiltonian formalism 5.11 Vertical extension of Hamiltonian formalism 5.12 Appendix. Time-reparametrized mechanics

227 228 231 242 247 248 262 269 271 274 282 285 296

6

Relativistic mechanics 6.1 Jets of submanifolds 6.2 Relativistic velocity and momentum phase spaces 6.3 Relativistic dynamics 6.4 Relativistic geodesic equations

299 299 303 307 311

CONTENTS

ix

Appendix A. Geometry of BRST mechanics

317

Appendix B. On quantum time-dependent mechanics

327

Bibliography

332

Index

347

Introduction The present book deals with first order mechanical systems, governed by the sec­ ond order differential equations in coordinates or the first order ones in coordinates and momenta. Our goal is the description of non-conservative mechanical systems subject to time-dependent transformations, including inertial and non-inertial frame transformations and phase transformations. Symplectic technique is well known to provide the adequate Hamiltonian for­ mulation of conservative (i.e., time-independent) mechanics where Hamiltonians are independent of time [2, 6, 72, 116, 126]. The familiar example is a mechanical sys­ tem whose momentum phase space is the cotangent bundle T'M of a configuration space M. This fibre bundle is provided with the canonical symplectic form Q = dpi A dq',

(1)

written with respect to the holonomic coordinates (g',p, =