Mathematical Aspects of Classical and Celestial Mechanics (Encyclopaedia of Mathematical Sciences (3))

1 Basic Principles of Classical Mechanics \* 2 The n-Body Problem \* 3 Symmetry Groups and Order Reduction \* 4 Variational Principles and Methods \* 5 Integrable Systems and Integration Methods \* 6 Perturbation Theory for Integrable Systems \* 7 Non-Integrable Systems \* 8 Theory of Small Oscillations \* 9 Tensor Invariants of Equations of Dynamics \* References.;Describes the fundamental principles, problems, and methods of classical mechanics. This book devotes its attention to the mathematical side of the subject. It aims to acquaint the reader with classical mechanics as a whole, in both its classical and its contemporary aspects. In this book we describe the basic principles, problems, and methods of cl- sical mechanics. Our main attention is devoted to the mathematical side of the subject. Although the physical background of the models considered here and the applied aspects of the phenomena studied in this book are explored to a considerably lesser extent, we have tried to set forth?rst and foremost the “working” apparatus of classical mechanics. This apparatus is contained mainly in Chapters 1, 3, 5, 6, and 8. Chapter 1 is devoted to the basic mathematical models of classical - chanics that are usually used for describing the motion of real mechanical systems. Special attention is given to the study of motion with constraints and to the problems of realization of constraints in dynamics. In Chapter 3 we discuss symmetry groups of mechanical systems and the corresponding conservation laws. We also expound various aspects of ord- reduction theory for systems with symmetries, which is often used in appli- tions. Chapter 4 is devoted to variational principles and methods of classical mechanics. They allow one, in particular, to obtain non-trivial results on the existence of periodic trajectories. Special attention is given to the case where the region of possible motion has a non-empty boundary. Applications of the variational methods to the theory of stability of motion are indicated. Describes the fundamental principles, problems, and methods of classical mechanics. This book devotes its attention to the mathematical side of the subject. It aims to acquaint the reader with classical mechanics as a whole, in both its classical and its contemporary aspects. This work describes the fundamental principles, problems, and methods of classical mechanics. The main attention is devoted to the mathematical side of the subject. The authors have endeavored to give an exposition stressing the working apparatus of classical mechanics. The book is significantly expanded compared to the previous edition. The authors have added two chapters on the variational principles and methods of classical mechanics as well as on tensor invariants of equations of dynamics. Moreover, various other sections have been revised, added or expanded. The main purpose of the book is to acquaint the reader with classical mechanics as a whole, in both its classical and its contemporary aspects. The book addresses all mathematicians, physicists and engineers. From the reviews of the previous editions: " ... The book accomplishes the goals it has set for itself. While it is not an introduction to the field, it is an excellent overview ..." "American Mathematical Monthly", November 1989 In This Text, The Author Constructs The Mathematical Apparatus Of Classical Mechanics From The Beginning, Examining All The Basic Problems In Dynamics, Including The Theory Of Oscillations, The Theory Of Rigid Body Motion, And The Hamiltonian Formalism. This Modern Approch, Based On The Theory Of The Geometry Of Manifolds, Distinguishes Iteself From The Traditional Approach Of Standard Textbooks. Geometrical Considerations Are Emphasized Throughout And Include Phase Spaces And Flows, Vector Fields, And Lie Groups. The Work Includes A Detailed Discussion Of Qualitative Methods Of The Theory Of Dynamical Systems And Of Asymptotic Methods Like Perturbation Techniques, Averaging, And Adiabatic Invariance. Newtonian Mechanics: Experimental Facts. Investigation Of The Equations Of Motion -- Lagrangian Mechanics: Variational Principles. Lagrangian Mechanics On Manifolds. Oscillations. Rigid Bodies. -- Hamiltonian Mechanics: Differential Forms. Symplectic Manifolds. Canonical Formalism. Introduction To Perturbation Theory -- Appendices. By V. I. Arnold. "This work describes the fundamental principles, problems, and methods of classical mechanics. The main attention is devoted to the mathematical side of the subject. The authors have endeavored to give an exposition stressing the working apparatus of classical mechanics. The book is significantly expanded compared to the previous edition. The authors have added two chapters on the variational principles and methods of classical mechanics as well as on tensor invariants of equations of dynamics. Moreover, various other sections have been revised, added or expanded." "The main purpose of the book is to acquaint the reader with classical mechanics as a whole, in both its classical and its contemporary aspects. The book addresses all mathematicians, physicists and engineers."--Jacket "In this text, the author constructs the mathematical apparatus of classical mechanics from the beginning, examining all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the Hamiltonian formalism. This modern approch, based on the theory of the geometry of manifolds, distinguishes iteself from the traditional approach of standard textbooks. Geometrical considerations are emphasized throughout and include phase spaces and flows, vector fields, and Lie groups. The work includes a detailed discussion of qualitative methods of the theory of dynamical systems and of asymptotic methods like perturbation techniques, averaging, and adiabatic invariance"--Publisher's description The main purpose of the book is to acquaint mathematicians, physicists and engineers with classical mechanics as a whole, in both its traditional and its contemporary aspects. As such, it describes the fundamental principles, problems, and methods of classical mechanics, with the emphasis firmly laid on the working apparatus, rather than the physical foundations or applications. Chapters cover the n-body problem, symmetry groups of mechanical systems and the corresponding conservation laws, the problem of the integrability of the equations of motion, the theory of oscillations and perturbation theory. For describing the motion of a mechanical system various mathematical models are used based on different "principles" - laws of motion. Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt. Includes Bibliographical References (p. [475]-506) And Indexes.