Critical Point Theory for Lagrangian Systems
Lagrangian systems constitute a very important and old class in dynamics. Their origin dates back to the end of the eighteenth century, with Joseph-Louis Lagrange’s reformulation of classical mechanics. The main feature of Lagrangian dynamics is its varia
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Series Editors Hyman Bass Joseph Oesterlé Yuri Tschinkel Alan Weinstein
Marco Mazzucchelli
Critical Point Theory for Lagrangian Systems
Marco Mazzucchelli Penn State University Department of Mathematics University Park, PA 16802 USA
ISBN 978-3-0348-0162-1 e-ISBN 978-3-0348-0163-8 DOI 10.1007/978-3-0348-0163-8 Springer Basel Dordrecht Heidelberg London New York Library of Congress Control Number: 201194 1493 Mathematics Subject Classification (2010): 70S05, 37J45, 58E05 © Springer Basel AG 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained. Printed on acid-free paper
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Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Lagrangian and Hamiltonian Systems 1.1 The formalism of classical mechanics . . . . . . . . . . . . . . . . . . . . . 1.2 Tonelli systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Action minimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 7 11
2 The 2.1 2.2 2.3
29 35 42
Morse Indices in Lagrangian Dynamics The Morse index and nullity . . . . . . . . . . . . . . . . . . . . . . . . . . . Bott’s iteration theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A symplectic excursion: the Maslov index . . . . . . . . . . . . . . . . . .
3 Functional Setting for the Lagrangian Action 3.1 Hilbert manifold structures for path spaces . . 3.2 Topological properties of the free loop space . 3.3 Convex quadratic-growth Lagrangians . . . . . 3.4 Regularity of the action functional . . . . . . . . 3.5 Critical points of the action functional . . . . . 3.6 The mean action functional in higher periods
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50 54 59 63 70 77
4 Discretizations 4.1 Uniqueness of the action minimizers . . . . . . . . . . . . . 4.2 The broken Euler-Lagrange loop spaces . . . . . . . . . . 4.3 The discrete action functional . . . . . . . . . . . . . . . . . 4.4 Critical points of the discrete action . . . . . . . . . . . . . 4.5 Homotopic approximation of the action sublevels . . . 4.6 Multiplicity of periodic orbits with prescribed period . 4.7 Discretizations in higher period . . . . . . . . . . . . . . . .
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80 86 89 91 99 103 106
5 Local Homology and Hilbert Subspaces 5.1 The abstract result . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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