Hamiltonian and Lagrangian Flows on Center Manifolds with Applicatio
The theory of center manifold reduction is studied in this monograph in the context of (infinite-dimensional) Hamil- tonian and Lagrangian systems. The aim is to establish a "natural reduction method" for Lagrangian systems to their center manifolds. Nona
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Alexander Mielke
Hamiltonian and Lagrangian Flows on Center Manifolds with Applications to Elliptic Variational Problems
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Author Alexander Mielke Mathematisches Institut A lJniversitlit Stuttgart Pfaffenwaldring 57 W-7000 Stuttgart 80, FRG
Mathematics Subject Classification (1991) : 58F05, 70H30, 35J50, 34C30, 57S20, 73C50, 76B25
ISBN 3-540-54710-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54710-X Springer-Verlag New York Berlin Heidelberg 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 any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 46/3140-543210 - Printed on acid-free paper
Fur Barbel, Annemarie, Lisabet und Frieder
Preface It is the aim of this work to establish connections between three fields which seem only loosely related from the usual point of view. These fields are described by the following terms: Hamiltonian and Lagrangian systems, center manifold reduction, and elliptic variational problems. All three topics have had a period of fast development within the last two decades; and the interrelations have grown considerably. Here we want to consider just one facet at the intersection of all three fields, namely the implications of center manifold theory to the study of variational problems. The main tool for the analysis is the Hamiltonian point of view. The original motivation for this work derives from my interest in Saint-Venant's problem. Having in mind the center manifold approach and hearing about the Galerkin or projection method to derive rod models, I thought it worthwile to study the connections between these two reduction procedures. However, it soon turned out that the tools to be developed involved fairly general ideas in Hamiltonian and Lagrangian system theory. I realized that many of the necessary results are known but spread over many sources or hidden in very abstract clothing. Often, the special needs for our objective were not directly covered. Thus, the plan evolved to write the abstract Part I on Hamiltonian and Lagrangian systems as self-contained as possible. [ made a controversial decision concerning the use of methods and notations from differential geometry. Since many applied researchers are not familiar with differential forms and coordinate free analysis on manifolds, I avoided these tools as much as possible. On the one hand the center ma
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