Classical Mechanics Hamiltonian and Lagrangian Formalism
Formalism of classical mechanics underlies a number of powerful mathematical methods that are widely used in theoretical and mathematical physics. This book considers the basics facts of Lagrangian and Hamiltonian mechanics, as well as related topics, suc
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Alexei Deriglazov
Classical Mechanics Hamiltonian and Lagrangian Formalism
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Alexei Deriglazov Depto. de Matemática Universidade Federal de Juiz de Fora 36036-330 Juiz de Fora MG, Brazil [email protected]
ISBN 978-3-642-14036-5 e-ISBN 978-3-642-14037-2 DOI 10.1007/978-3-642-14037-2 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010932503 c Springer-Verlag Berlin Heidelberg 2010 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: eStudio Calamar S.L., Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Formalism of classical mechanics underlies a number of powerful mathematical methods, widely used in theoretical and mathematical physics [1–11]. In these lectures we present some selected topics of classical mechanics, which may be useful for graduate level students intending to work in one of the branches of a vast field of theoretical physics. Except for the last chapter, which is devoted to the discussion of singular theories and their local symmetries, the topics selected correspond to the standard course of classical mechanics. For the convenience of the reader, we have tried to make the material of different chapters as independent as possible. So, the reader who is familiar with Lagrangian mechanics can proceed to any one of Chaps. 3, 4, 5, 6, 7, 8 after reading the second chapter. In our presentation of the material we have tried, where possible, to replace intuitive motivations and “scientific folklore” by exact proofs or direct computations. To illustrate how classical-mechanics formalism works in other branches of theoretical physics, we have presented examples related to electrodynamics, as well as to relativistic and quantum mechanics. Most of the suggested exercises are directly related to the main body of the text. While in some cases the formalism is developed beyond the traditional level adopted in the standard textbooks on classical mechanics [12–14], the only mathematical prerequisites are some knowledge of calculus and linear algebra. In the frameworks of classical and quantum theories, the Hamiltonian and Lagrangian formulations each have advantages and disadvantages. Since
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