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Noether's Theorems Applications in Mechanics and Field Theory Atlantis Studies in Variational Geometry Atlantis Studies in Variational Geometry Series Editors: Demeter Krupka · Huafei Sun Gennadi Sardanashvily Noether's Theorems Applications in Mechanics and Field Theory Atlantis Studies in Variational Geometry Volume 3 Series editors Demeter Krupka, University of Hradec Kralove, Hradec Kralove, Czech Republic Huafei Sun, Beijing Institute of Technology, Beijing, China More information about this series at http://www.atlantis-press.com Gennadi Sardanashvily Noether’s Theorems Applications in Mechanics and Field Theory Gennadi Sardanashvily Moscow State University Moscow Russia ISSN 2214-0700 ISSN 2214-0719 (electronic) Atlantis Studies in Variational Geometry ISBN 978-94-6239-170-3 ISBN 978-94-6239-171-0 (eBook) DOI 10.2991/978-94-6239-171-0 Library of Congress Control Number: 2016932506 © Atlantis Press and the author(s) 2016 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher. Printed on acid-free paper To my wife Aida Karamysheva Professor, molecule biologist Preface Noether’s first and second theorems are formulated in a very general setting of reducible degenerate Grassmann-graded Lagrangian theory of even and odd vari- ables on graded bundles. Lagrangian theory generally is characterized by a hierarchy of nontrivial Noether and higher-stage Noether identities and the corresponding gauge and higher-stage gauge symmetries which characterize the degeneracy of a Lagrangian system. By analogy with Noether identities of differential operators, they are described in the homology terms. In these terms, Noether’s inverse and direct second theorems associate to the Koszul–Tate graded chain complex of Noether and higher-stage Noether identities the gauge cochain sequence whose ascent gauge operator pro- vides gauge and higher-stage gauge symmetries of Grassmann-graded Lagrangian theory. If these symmetries are algebraically closed, an ascent gauge operator is generalized to a nilpotent BRST operator which brings a gauge cochain sequence into a BRST complex and provides the BRST extension of original Lagrangian theory. In the present book, the calculus of variations and Lagrangian formalism are phrased in algebraic terms of a variational bicomplex on an infinite order jet manifold that enables one to extend this formalism to Grassmann-graded Lagrangian systems of even and odd variables on graded bundles. Cohomology of a graded variational bicomplex provides the global solutions of the direct and inverse problems of the calculus of variations. In this framework, Noether’s direct first theorem is formulated as a straight- forward corollary of the global variational formula. It associates to any Lagrangian symmetry the conserved symmetry current whose total differential vanishes on-shell. Proved in a very general setting, so-called Noether’s third theorem states that a conserved symmetry current along any gauge symmetry is reduced to a superpotential, i.e., it is a total differential on-shell. This also is the case of covariant Hamiltonian formalism on smooth fibre bundles seen as the particular Lagrangian one on phase Legendre bundles. vii viii Preface Lagrangian formalism on smooth fibre bundles and graded bundles provides the comprehensive formulations both of classical field theory and nonrelativistic mechanics. Non-autonomous nonrelativistic mechanics is adequately formulated as partic- ular Lagrangian and Hamiltonian theory on a configuration bundle over the time axis. Conserved symmetry currents of Noether’s first theorem in mechanics are integrals of motion, but the converse need not be true. In Hamiltonian mechanics, Noether’s inverse first theorem states that all integrals of motion come from symmetries. In particular, this is the case of energy functions with respect to dif- ferent reference frames. The book presents a number of physically relevant models: superintegrable Hamiltonian systems, the global Kepler problem, Yang–Mills gauge theory on principal bundles, SUSY gauge theory, gauge gravitation theory on natural bundles, topological Chern–Simons field theory and topological BF theory, exemplifying a reducible degenerate Lagrangian system. Our book addresses to a wide audience of theoreticians, mathematical physicists and mathematicians. With respect to mathematical prerequisites, the reader is expected to be familiar with the basics of differential geometry of fibre bundles. We have tried to give the necessary mathematical background, thus making our exposition self-contained. For the sake of convenience of the reader, a number of relevant mathematical topics are compiled in appendixes. Moscow Gennadi Sardanashvily October 2015 Contents 1 Calculus of Variations on Fibre Bundles .................... 1 1.1 Infinite Order Jet Formalism . 1 1.2 Variational Bicomplex . 7 1.2.1 Cohomology of the Variational Bicomplex . 9 1.3 Lagrangian Formalism . 12 2 Noether’s First Theorem ............................... 17 2.1 Lagrangian Symmetries. 17 2.2 Gauge Symmetries: Noether’s Direct Second Theorem . 20 2.3 Noether’s First Theorem: Conservation Laws. 23 3 Lagrangian and Hamiltonian Field Theories ................. 27 3.1 First Order Lagrangian Formalism . 27 3.2 Cartan and Hamilton–De Donder Equations. 30 3.3 Noether’s First Theorem: Energy-Momentum Currents . 32 3.4 Conservation Laws in the Presence of a Background Field . 34 3.5 Covariant Hamiltonian Formalism . 36 3.6 Associated Lagrangian and Hamiltonian Systems . 41 3.7 Noether’s First Theorem: Hamiltonian Conservation Laws. 47 3.8 Quadratic Lagrangian and Hamiltonian Systems . 49 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics......... 59 4.1 Geometry of Fibre Bundles over R..................... 60 4.2 Lagrangian Mechanics. Integrals of Motion . 63 4.3 Noether’s First Theorem: Energy Conservation Laws . 67 4.4 Gauge Symmetries: Noether’s Second and Third Theorems . 71 4.5 Non-autonomous Hamiltonian Mechanics . 73 4.6 Hamiltonian Conservation Laws: Noether’s Inverse First Theorem . 80 4.7 Completely Integrable Hamiltonian Systems . 84 ix x Contents 5 Global Kepler Problem................................. 93 6 Calculus of Variations on Graded Bundles .................. 103 6.1 Grassmann-Graded Algebraic Calculus . 103 6.2 Grassmann-Graded Differential Calculus . 106 6.3 Differential Calculus on Graded Bundles . 109 6.4 Grassmann-Graded Variational Bicomplex. 121 6.5 Grassmann-Graded Lagrangian Theory . 127 6.6 Noether’s First Theorem: Supersymmetries . 129 7 Noether’s Second Theorems ............................. 135 7.1 Noether Identities: Reducible Degenerate Lagrangian Systems . 136 7.2 Noether’s Inverse Second Theorem . 145 7.3 Gauge Supersymmetries: Noether’s Direct Second Theorem . 148 7.4 Noether’s Third Theorem: Superpotential . 152 7.5 Lagrangian BRST Theory . 155 8 Yang–Mills Gauge Theory on Principal Bundles .............. 163 8.1 Geometry of Principal Bundles . 163 8.2 Principal Gauge Symmetries . 171 8.3 Noether’s Direct Second Theorem: Yang–Mills Lagrangian . 173 8.4 Noether’s First Theorem: Conservation Laws. 175 8.5 Hamiltonian Gauge Theory . 177 8.6 Noether’s Inverse Second Theorem: BRST Extension . 179 9 SUSY Gauge Theory on Principal Graded Bundles ............ 183 10 Gauge Gravitation Theory on Natural Bundles ............... 189 10.1 Relativity Principle: Natural Bundles . 189 10.2 Equivalence Principle: Lorentz Reduced Structure . 191 10.3 Metric-Affine Gauge Gravitation Theory . 194 10.4 Energy-Momentum Gauge Conservation Law . 197 10.5 BRST Gravitation Theory . 199 11 Chern–Simons Topological Field Theory .................... 201 12 Topological BF Theory................................. 207 Glossary ............................................... 211 Appendix A: Differential Calculus over Commutative Rings ......... 213 Appendix B: Differential Calculus on Fibre Bundles............... 227 Contents xi Appendix C: Calculus on Sheaves ............................ 259 Appendix D: Noether Identities of Differential Operators ........... 271 References ............................................. 281 Index ................................................. 287 Introduction Noether’s theorems are well known to treat symmetries of Lagrangian systems. Noether’s first theorem associates to a Lagrangian symmetry the conserved sym- metry current whose total differential vanishes on-shell. The second ones provide the correspondence between Noether identities and gauge symmetries of a Lagrangian system. We refer the reader to the brilliant book of Yvette Kosmann-Schwarzbach [84] for the history and references on this subject. Our book aims to present Noether’s theorems in a very general setting of reducible degenerate Grassmann-graded Lagrangian theory of even and odd vari- ables on graded bundles. We however start with even Lagrangian formalism on smooth fibre bundles (Chap. 2), and focus ourselves especially on first-order Lagrangian theory (Chap. 3) because the most physically relevant models (nonrelativistic mechanics, gauge field theory, gravitation theory, etc.) are of this type. Lagrangian theory of even (commutative) variables on an n-dimensional smooth manifold X conventionally is formulated in terms of fibre bundles over X and jet manifolds of their sections [15, 53, 108, 133, 143], and it lies in the framework of general technique of nonlinear differential equations and operators [25, 53, 85]. This formulation is based on the categorial
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