Mechanical Systems, Classical Models Volume III: Analytical Mechanic

This third volume completes the Work Mechanical Systems, Classical Models. The first two volumes dealt with particle dynamics and with discrete and continuous mechanical systems. The present volume studies analytical mechanics. Topics like Lagrangian and

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MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING Series Editor

Alan Jeffrey

The importance of mathematics in the study of problems arising from the real world, and the increasing success with which it has been used to model situations ranging from the purely deterministic to the stochastic, in all areas of today's Physical Sciences and Engineering, is well established. The progress in applicable mathematics has been brought about by the extension and development of many important analytical approaches and techniques, in areas both old and new, frequently aided by the use of computers without which the solution of realistic problems in modern Physical Sciences and Engineering would otherwise have been impossible. The purpose of the series is to make available authoritative, up to date, and self-contained accounts of some of the most important and useful of these analytical approaches and techniques. Each volume in the series will provide a detailed introduction to a specific subject area of current importance, and then will go beyond this by reviewing recent contributions, thereby serving as a valuable reference source.

For further volumes: http://www.springer.com/series/7311

Mechanical Systems, Classical Models Volume III: Analytical Mechanics

by

Petre P. Teodorescu Faculty of Mathematics, University of Bucharest, Romania

123

Prof. Dr. Petre P. Teodorescu Str. Popa Soare 38 023984 Bucuresti 20 Romania

Translated into English, revised and extended by Petre P. Teodorescu All rights reserved © EDITURA TEHNICĂ, 2002 This translation of “Mechanical Systems, Classical Models” (original title: Sisteme mecanice.Modele clasice, Published by: Ed. Tehnicá, Bucuresti, Bucharest, Romania, 1984-2002), First Edition, is published by arrangement with EDITURA TEHNICĂ, Bucharest, ROMANIA

ISSN 1559-7458 e-ISSN 1559-7466 ISBN 978-90-481-2763-4 e-ISBN 978-90-481-2764-1 DOI 10.1007/978-90-481-2764-1 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2007943834 c Springer Science+Business Media B.V. 2009  No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Contents

Preface

ix

18.

Lagrangian Mechanics 1. Preliminary Results 1.1 Introductory Notions 1.2 Differential Principles of Mechanics 2. Lagrange’s Equations 2.1 Space of Configurations 2.2 Lagrange’s Equations of Second Kind 2.3 Transformations. First Integrals 3. Other Problems Concerning Lagrange’s Equations 3.1 New Forms of Lagrange’s Equations 3.2 Applications

1 2 2 20 45 46 57 65 83 83 98

19.

Hamiltonian Mechanics 1. Hamilton’s Equations 1.1 General Results