Direct Methods of Variational Problems

The final solution for all kinds of variational problems boils down to the boundary value problem of solving the Euler equation. However, only in some special cases the analytical solution of the Euler equation can be found, in most cases, the analytical

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undamental Theories and Their Applications of the Calculus of Variations

Fundamental Theories and Their Applications of the Calculus of Variations

Dazhong Lao Shanshan Zhao •

Fundamental Theories and Their Applications of the Calculus of Variations

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Dazhong Lao School of Aerospace Engineering Beijing Institute of Technology Beijing, China

Shanshan Zhao Dandong Design Institute of Chemical Fibre Dandong, China

ISBN 978-981-15-6069-9 ISBN 978-981-15-6070-5 https://doi.org/10.1007/978-981-15-6070-5

(eBook)

Jointly published with Beijing Institute of Technology Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Beijing Institute of Technology Press. © Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are reserved by the Publishers, 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 publishers, 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 publishers 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 publishers remain 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

The calculus of variations is an important branch of applied mathematics, which deals with the extremum problem of a functional in the integral form. This branch has found wide applications to mathematical and physical sciences, as well as a great variety of engineering fields. Mr. Dazhong Lao, the first author of this book, is an Associate Professor with School of Aerospace Engineering, Beijing Institute of Technology, while Ms. Shanshan Zhao, the co-author of the book, is a Senior Engineer. Professor Lao has been engaged in studying the calculus of variations over the past three decades. His experience of teaching the calculus of variations has also exceeded 20 years. As early as 2004, he had his book entitled “Fundamentals of the Calculus of Variations” published by The Defense Industry Publish

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