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, such as canonical transformations, integral invariants, potential motion in geometric setting, symmetries, the Noether theorem and systems with constraints. While in some cases the formalism is developed beyond the traditional level adopted in the standard textbooks on classical mechanics, only elementary mathematical methods are used in the exposition of the material. The mathematical constructions involved are explicitly described and explained, so the book can be a good starting point for the undergraduate student new to this field. At the same time and where possible, intuitive motivations are replaced by explicit proofs and direct computations, preserving the level of rigor that makes the book useful for the graduate students intending to work in one of the branches of the vast field of theoretical physics. To illustrate how classical-mechanics formalism works in other branches of theoretical physics, examples related to electrodynamics, as well as to relativistic and quantum mechanics, are included.
Classical Mechanics Alexei Deriglazov Classical Mechanics Hamiltonian and Lagrangian Formalism 123 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 c