Mechanical Systems, Classical Models: Volume III: Analytical Mechanics

780 Pages·2009·8.87 MB·English

by  Prof. Dr. Petre P. Teodorescu (auth.)

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MATHEMATICAL AND ANALYTICAL
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Volume III: Analytical Mechanics
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Petre P. Teodorescu
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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

ISSN1559-7458 e-ISSN1559-7466
ISBN978-90-481-2763-4 e-ISBN978-90-481-2764-1
DOI10.1007/978-90-481-2764-1
SpringerDordrechtHeidelbergLondonNewYork
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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
v
vi 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 s +1 Dimensions 390
4.2 Formalism in Spaces with 2s +1 or with 2s +2
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
22. Dynamics of Non-holonomic Mechanical Systems 411
1. Kinematics of Non-holonomic Mechanical Systems 411
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. Gibbs–Appell Equations 478
3.1 Gibbs–Appell Equations of Motion 478
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
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

Bibliography 739

Subject Index 759

Name Index 765
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
ix
x 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 P.P. Teodorescu
7 January 2009
Chapter 18
Lagrangian Mechanics

The analytical methods of calculation have been introduced to can study in a
systematic and unitary form the discrete mechanical systems (with a finite number of
particles or with a finite number of degrees of freedom); thus, appears the denomination
of “analytical mechanics”, besides that of “theoretical mechanics” (in the English,
German and Russian literature) and that of “rational mechanics” (in the French or
Italian literature). In “physical mechanics” stress is put on the experimental aspect,
while in “technical mechanics” on the possibility to apply in technics the results
obtained. In all these cases, the object of study is the same: mechanics, as a science of
nature.
In this order of ideas, we consider a discrete mechanical system S , formed by n
particles P (or a finite number of rigid solids), of position vectors r , subjected to the
i i
action of given forces F ,i = 1,2,...,n (the resultants of all external and internal
i
given forces which act upon the particle P ) and to certain links L ,k = 1,2,...,n .
i k L
The basic problem which is put consists in the determination of the motion (position and
velocity) of the mechanical system S in the time interval [t ,t ] (eventually, [t ,∞)) if
0 1 0
F = F (r ,r(cid:5) ;t), L = L (r ,r(cid:5) ;t), r (t ) = r0
i i j j k k j j i 0 i
and
v (t ) = v0, i = 1,2,...,n , k = 1,2,...,n
i 0 i L
are given. A distinction between external and internal forces is not necessary; in
exchange, the classification of the constraints and their representation in a finite or
differential form, as well as their elimination from the calculation (when it is possible)
have a particular importance. The elimination of the constraints and of the constraint
forces from the calculation allows to obtain equations of motion which contain only the
position vectors and their derivatives with respect to time. In general, one admits only
frictionless constraints, which – obviously – restrains much the sphere of the problems
which can by treated.
Euler, in 1736, and Maclaurin, in 1742, have written the first treatises of mechanics
in which they pass from purely geometric methods to methods based on differential
calculus (Euler, L., 1736). The brothers Jacob I Bernoulli and Jean I Bernoulli, Daniel
Bernoulli, the son of the latter one, Alexis-Claude Clairaut, Jacob Herman and Pierre-
Louis-Moreau de Maupertuis have made important steps, considering many problems
and enouncing various “principles”, which allowed to put in equation the problems of
mechanics and to give their solution in a unitary form. “A special ability was always
necessary to put in evidence the forces which have to be taken into consideration, so
P.P. Teodorescu, Mechanical Systems, Classical Models, 1

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