Noether's Theorems Applications in Mechanics and Field Theory
The book provides a detailed exposition of the calculus of variations on fibre bundles and graded manifolds. It presents applications in such area's as non-relativistic mechanics, gauge theory, gravitation theory and topological field theory with em
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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 variables 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 provides 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 straightforward 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 d
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