Index theory in nonlinear analysis
This book provides detailed information on index theories and their applications, especially Maslov-type index theories and their iteration theories for non-periodic solutions of Hamiltonian systems. It focuses on two index theories: L-index theory (
- PDF / 3,542,748 Bytes
- 346 Pages / 439.42 x 683.15 pts Page_size
- 40 Downloads / 225 Views
Index theory in nonlinear analysis
Index theory in nonlinear analysis
Chungen Liu
Index theory in nonlinear analysis
123
Chungen Liu School of Mathematical and Information Science Guangzhou University Guangzhou, Guangdong, China
ISBN 978-981-13-7286-5 ISBN 978-981-13-7287-2 (eBook) https://doi.org/10.1007/978-981-13-7287-2 Mathematics Subject Classification: 37J45, 34C25, 53D12, 58J20 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
Foreword
This book contains three aspects of index theories with applications. From them, we choose one aspect to demonstrate how to use the L-index theory to study the multiplicity of brake orbits on a symmetric convex hypersurface. We begin from a famous Seifert conjecture about the number of brake orbits on a compact hypersurface in R2n .
1 Seifert Conjecture Let us recall the famous conjecture proposed by H. Seifert in his pioneer work [268] in 1948 concerning the multiplicity of brake orbits of certain Hamiltonian systems in R2n . We assume H ∈ C 2 (R2n , R) possesses the following form: H (p, q) =
1 A(q)p · p + V (q), 2
(1)
where p, q ∈ Rn , A(q) is a positive definite n×n symmetric matrix for any q ∈ Rn , A is C 2 , V ∈ C 2 (Rn , R) is the potential energy. The solution of the following problem for Hamiltonian system x˙ = J H (x), x = (p, q), τ p(0) = p( ) = 0. 2
(2) (3)
is called a brake orbit. Moreover, if h is the total energy of a brake orbit (q, p), ¯ i.e., H (p(t), q(t)) = h, then V (q(0)) = V (q(τ )) = h and q(t) ∈ ≡ {q ∈ 0 −I Rn |V (q) ≤ h} for all t ∈ R, where J = is the standard symplectic matrix I 0 and I is the n × n identity matrix. v
vi
Foreword
Data Loading...
