Noether's theorems. Applications in mechanics and field theory PDF
Preview Noether's theorems. Applications in mechanics and field theory
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 inVariational Geometry ISBN978-94-6239-170-3 ISBN978-94-6239-171-0 (eBook) DOI 10.2991/978-94-6239-171-0 LibraryofCongressControlNumber:2016932506 ©AtlantisPressandtheauthor(s)2016 Thisbook,oranypartsthereof,maynotbereproducedforcommercialpurposesinanyformorbyany means, electronic or mechanical, including photocopying, recording or any information storage and retrievalsystemknownortobeinvented,withoutpriorpermissionfromthePublisher. Printedonacid-freepaper 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. LagrangiantheorygenerallyischaracterizedbyahierarchyofnontrivialNoether 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- forwardcorollaryoftheglobalvariationalformula.ItassociatestoanyLagrangian 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.,itisatotaldifferentialon-shell.Thisalsoisthecaseofcovariant Hamiltonian formalism on smooth fibre bundles seen as the particular Lagrangian one on phase Legendre bundles. vii viii Preface Lagrangianformalismonsmoothfibrebundlesandgradedbundlesprovidesthe 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 principalbundles,SUSYgaugetheory,gaugegravitationtheoryonnaturalbundles, topological Chern–Simons field theory and topological BF theory, exemplifying a reducible degenerate Lagrangian system. Ourbookaddressestoawideaudienceoftheoreticians,mathematicalphysicists and mathematicians. With respect to mathematical prerequisites, the reader is expectedtobefamiliarwiththebasicsofdifferentialgeometryoffibrebundles.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
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