Teodorescu P.P. Mechanical Systems, Volume III: Analytical Mechanics

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Mechanical Systems, Classical Models

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:

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Mechanical Systems,

Classical Models

Volume III:

123

Petre P. Teodorescu

Faculty of Mathematics,

University of Bucharest,

Romania

Analytical Mechanics

Romania

Prof. Dr. Petre P. Teodorescu

Str. Popa Soare 38

023984 Bucuresti 20

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

 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

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Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

ISSN 1559-7458 e-ISSN 1559-7466

Translated into English, revised and extended by Petre P. Teodorescu

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

Contents

Preface ix

18. Lagrangian Mechanics 1

1. Preliminary Results 2

1.1 Introductory Notions 2

1.2 Differential Principles of Mechanics 20

2. Lagrange’s Equations 45

2.1 Space of Configurations 46

2.2 Lagrange’s Equations of Second Kind 57

2.3 Transformations. First Integrals 65

3. Other Problems Concerning Lagrange’s Equations 83

3.1 New Forms of Lagrange’s Equations 83

3.2 Applications 98

19. Hamiltonian Mechanics 115

1. Hamilton’s Equations 115

1.1 General Results 115

1.2 Lagrange’s Brackets. Poisson’s Brackets 138

1.3 Applications 151

2. The Hamilton–Jacobi Method 160

2.1 General Results 160

2.2 Systems of Equations with Separate Variables 172

2.3 Applications 191

20. Variational Principles. Canonical Transformations 213

1. Variational Principles 213

1.1 Mathematical Preliminaries 214

1.2 The General Integral Principle 225

1.3 Hamilton’s Principle 231

1.4 Maupertuis’s Principle. Other Variational Principles 242

1.5 Continuous Mechanical Systems 254

2. Canonical Transformations 265

2.1 General Considerations. Conditions of Canonicity 265

2.2 Structure of Canonical Transformations. Properties 289

3. Symmetry Transformations. Noether’s Theorem. Conservation Laws 300

3.1 Symmetry Transformations. Noether’s Theorem 301

MECHANICAL SYSTEMS, CLASSICAL MODELS

3.2 Lie Groups 311

3.3 Space-Time Symmetries. Conservation Laws 319

21. Other Considerations on Analytical Methods in Dynamics of

Discrete Mechanical Systems

335

1. Integral Invariants. Ergodic Theorems 335

1.1 Integral Invariants of Order

2s 335

1.2 Invariants of First Order 341

1.3 Ergodic Theorems 354

2. Periodic Motions. Action-Angle Variables 356

2.1 Periodic Motions. Quasi-Periodic Motions 356

2.2 Action-Angle Variables 361

2.3 Adiabatic Invariance 365

3. Methods of Exterior Differential Calculus. Elements of

Invariantive Mechanics

368

3.1 Methods of Exterior Differential Calculus 368

3.2 Elements of Invariantive Mechanics 373

3.3 Applications 385

4. Formalisms in the Dynamics of Mechanical Systems 390

4.1 Formalisms in Spaces with

1s + Dimensions 390

4.2 Formalism in Spaces with

21s + or with 22s +

Dimensions

394

4.3 Notions on the Inverse Problem of Mechanics and the

Birkhoffian Formalism

397

5. Control Systems 402

5.1 Control Systems 402

5.2 Optimal Trajectories 407

1.1 General Considerations 411

1.2 Conditions of Holonomy. Quasi-co-ordinates.

Non-holonomic Spaces

421

2. Lagrange’s Equations. Other Equations of Motion 430

2.1 Motion of a Rigid Solid on a Fixed Surface 430

2.2 Lagrange’s Equations 436

2.3 Applications 443

2.4 Other Equations of Motion 467

3.2 Applications 482

4. Other Problems on the Dynamics of Non-holonomic Mechanical

Systems

491

4.1 Collisions 491

4.2 First Integrals of the Equations of Motion 498

22. Dynamics of Non-holonomic Mechanical Systems 411

1. Kinematics of Non-holonomic Mechanical Systems 411

3. Gibbs–Appell Equations 478

3.1 Gibbs–Appell Equations of Motion 478

Contents

vii

23. Stability and Vibrations 505

1. Stability of Mechanical Systems 505

1.1 Stability of Equilibrium 505

1.2 Stability of Motion 537

1.3 Applications 554

2. Vibrations of Mechanical Systems 566

2.1 Small Free Oscillations About a Stable Position of

Equilibrium

566

2.2 Small Forced Oscillations 597

2.3 Non-linear Vibrations 606

2.4 Applications 612

24. Dynamical Systems. Catastrophes and Chaos 629

1. Continuous and Discrete Dynamical Systems 630

1.1 Continuous Linear Dynamical Systems 630

1.2 Non-linear Differential Equations and Systems of

Non-linear Differential Equations

648

1.3 Discrete Linear Dynamical Systems 675

2. Elements of the Theory of Catastrophes 682

2.1 Ramifications 683

2.2 Elementary Catastrophes 689

3. Periodic Solutions. Global Bifurcations 697

3.1 Periodic Solutions 698

3.2 Global Bifurcations 707

4. Fractals. Chaotic Motions 712

4.1 Fractals 712

4.2 Chaotic Motions 723

Subject Index 759

Name Index 765

Bibliography 739

Preface

All phenomena in nature are characterized by motion. Mechanics deals with the

objective laws of mechanical motion of bodies, the simplest form of motion.

In the study of a science of nature, mathematics plays an important rôle. Mechanics

is the first science of nature which has been expressed in terms of mathematics, by

considering various mathematical models, associated to phenomena of the surrounding

nature. Thus, its development was influenced by the use of a strong mathematical tool.

As it was already seen in the first two volumes of the present book, its guideline is

precisely the mathematical model of mechanics. The classical models which we refer to

are in fact models based on the Newtonian model of mechanics, that is on its five

principles, i.e.: the inertia, the forces action, the action and reaction, the independence

of the forces action and the initial conditions principle, respectively. Other models, e.g.,

the model of attraction forces between the particles of a discrete mechanical system, are

part of the considered Newtonian model. Kepler’s laws brilliantly verify this model in

case of velocities much smaller then the light velocity in vacuum.

Mechanics has as object of study mechanical systems. The first two volumes of this

book dealt with particle dynamics and with discrete and continuous mechanical

systems, respectively. The present one deals with analytical mechanics. We put in

evidence the Lagrangian and the Hamiltonian mechanics, where the study of first

integrals plays a very important rôle. The Hamilton–Jacobi method is widely

considered, as well as the study of systems with separate variables. We mention also a

thorough study of variational principles and canonical transformations. The symmetry

transformations, including Noether’s theorem, lead to conservation laws. Integral

invariants and exterior differential calculus are also included. A particular attention has

been given to non-holonomic mechanical systems. Problems of stability and vibrations

have been also considered in the frame of Lagrangian and Hamiltonian mechanics. The

study of dynamical systems leads to catastrophes, bifurcations and chaos.

One presents some applications connected to important phenomena of the nature and

one gives also the possibility to solve problems presenting interest from technical,

engineering point of view. In this form, the book becomes – we dare say – a unique

outline of the literature in the field; the author wishes to present the most important

aspects related with the study of mechanical systems, mechanics being regarded as a

science of nature, as well as its links to other sciences of nature. Implications in

technical sciences are not neglected.

Concerning the mathematical tool, the five appendices in the first volume give the

book an autonomy with respect to other works, special previous mathematical

knowledge being not necessary. The numeration of the chapters follows that of the first

MECHANICAL SYSTEMS, CLASSICAL MODELS

two volumes, to which one makes reference for various results (theorems, formulae

etc.).

I am grateful to Mărgărit Baubec, Ph.D., for his valuable help in the presentation of

this book. The excellent cooperation with the team of Springer, Dordrecht, is gratefully

acknowledged.

The book covers a wide number of problems (classical or new ones) as one can see

from its contents. It uses the known literature, as well as the original results of the

author and his more than 50 years experience as Professor of Mechanics at the

University of Bucharest. It is devoted to a large circle of readers: mathematicians

(especially those involved in applied mathematics), physicists (particularly those

interested in mechanics and its connections), chemists, biologists, astronomers,

engineers of various specialities (civil, mechanical engineers etc., who are scientific

researchers or designers), students in various domains etc.

Bucharest, Romania

7 January 2009

P.P. Teodorescu