Mechanical Systems, Classical Models

by Teodorescu, Petre P.

• ISBN13: 9789048127634
• ISBN10: 9048127637
• Format: Hardcover
• Copyright: 2009-09-01
• Publisher: Springer Verlag

Summary

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 such as Lagrangian and Hamiltonian mechanics, the Hamilton-Jacobi method, and a study of systems with separate variables are thoroughly discussed. Also included are variational principles and canonical transformations, integral invariants and exterior differential calculus, and particular attention is given to non-holonomic mechanical systems.The author explains in detail all important aspects of the science of mechanics, regarded as a natural science, and shows how they are useful in understanding important natural phenomena and solving problems of interest in applied and engineering sciences.Professor Teodorescu has spent more than fifty years as a Professor of Mechanics at the University of Bucharest and this book relies on the extensive literature on the subject as well as the author's original contributions.

Table of Contents

Preface	p. ix
Lagrangian Mechanics	p. 1
Preliminary Results	p. 2
Introductory Notions	p. 2
Differential Principles of Mechanics	p. 20
Lagrange's Equations	p. 45
Space of Configurations	p. 46
Lagrange's Equations of Second Kind	p. 57
Transformations. First Integrals	p. 65
Other Problems Concerning Lagrange's Equations	p. 83
New Forms of Lagrange's Equations	p. 83
Applications	p. 98
Hamiltonian Mechanics	p. 115
Hamilton's Equations	p. 115
General Results	p. 115
Lagrange's Brackets. Poisson's Brackets	p. 138
Applications	p. 151
The Hamilton-Jacobi Method	p. 160
General Results	p. 160
Systems of Equations with Separate Variables	p. 172
Applications	p. 191
Variational Principles. Canonical Transformations	p. 213
Variational Principles	p. 213
Mathematical Preliminaries	p. 214
The General Integral Principle	p. 225
Hamilton's Principle	p. 231
Maupertuis's Principle. Other Variational Principles	p. 242
Continuous Mechanical Systems	p. 254
Canonical Transformations	p. 265
General Considerations. Conditions of Canonicity	p. 265
Structure of Canonical Transformations. Properties	p. 289
Symmetry Transformations. Noether's Theorem. Conservation Laws	p. 300
Symmetry Transformations. Noether's Theorem	p. 301
Lie Groups	p. 311
Space-Time Symmetries. Conservation Laws	p. 319
Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems	p. 335
Integral Invariants. Ergodic Theorems	p. 335
Integral Invariants of Order 2s	p. 335
Invariants of First Order	p. 341
Ergodic Theorems	p. 354
Periodic Motions. Action-Angle Variables	p. 356
Periodic Motions. Quasi-Periodic Motions	p. 356
Action-Angle Variables	p. 361
Adiabatic Invariance	p. 365
Methods of Exterior Differential Calculus. Elements of Invariantive Mechanics	p. 368
Methods of Exterior Differential Calculus	p. 368
Elements of Invariantive Mechanics	p. 373
Applications	p. 385
Formalisms in the Dynamics of Mechanical Systems	p. 390
Formalisms in Spaces with s + 1 Dimensions	p. 390
Formalism in Spaces with 2s + 1 or with 2s + 2 Dimensions	p. 394
Nations on the Inverse Problem of Mechanics and the Birkhoffian formalism	p. 397
Control Systems	p. 402
Control Systems	p. 402
Optimal Trajectories	p. 407
Dynamics of Non-holonomic Mechanical Systems	p. 411
Kinematics of Non-holonomic Mechanical Systems	p. 411
General Considerations	p. 411
Conditions of Holonomy. Quasi-co-ordinates. Non-holonomic Spaces	p. 421
Lagrange's Equations. Other Equations of Motion	p. 430
Motion of a Rigid Solid on a Fixed Surface	p. 430
Lagrange's Equations	p. 436
Applications	p. 443
Other Equations of Motion	p. 467
Gibbs-Appell Equations	p. 478
Gibbs-Appell Equations of Motion	p. 478
Applications	p. 482
Other Problems on the Dynamics of Non-Holonomic Mechanical Systems	p. 491
Collisions	p. 491
First Integrals of the Equations of Motion	p. 498
Stability and Vibrations	p. 505
Stability of Mechanical Systems	p. 505
Stability of Equilibrium	p. 505
Stability of Motion	p. 537
Applications	p. 554
Vibrations of Mechanical Systems	p. 566
Small Free Oscillations About a Stable Position of Equilibrium	p. 566
Small Forced Oscillations	p. 597
Non-linear Vibrations	p. 606
Applications	p. 612
Dynamical Systems. Catastrophes and Chaos	p. 629
Continuous and Discrete Dynamical Systems	p. 630
Continuous Linear Dynamical Systems	p. 630
Non-linear Differential Equations and Systems of Non-linear Differential Equations	p. 648
Discrete Linear Dynamical Systems	p. 675
Elements of the Theory of Catastrophes	p. 682
Ramifications	p. 683
Elementary Catastrophes	p. 689
Periodic Solutions. Global Bifurcations	p. 697
Periodic Solutions	p. 698
Global Bifurcations	p. 707
Fractals. Chaotic Motions	p. 712
Fractals	p. 712
Chaotic Motions	p. 723
Bibliography	p. 739
Subject Index	p. 759
Name Index	p. 765
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