Dynamical Systems III

1. Front Matter

   Pages I-XIV

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2. Basic Principles of Classical Mechanics

   • Vladimir I. Arnold

   Pages 1-48

3. The n-Body Problem

   • Vladimir I. Arnold

   Pages 49-77

4. Symmetry Groups and Reduction (Lowering the Order)

   • Vladimir I. Arnold

   Pages 78-106

5. Integrable Systems and Integration Methods

   • Vladimir I. Arnold

   Pages 107-137

6. Perturbation Theory for Integrable Systems

   • Vladimir I. Arnold

   Pages 138-211

7. Nonintegrable Systems

   • Vladimir I. Arnold

   Pages 212-250

8. Theory of Small Oscillations

   • Vladimir I. Arnold

   Pages 251-273

9. Back Matter

   Pages 274-294

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This work describes the fundamental principles, problems, and methods of elassical mechanics focussing on its mathematical aspects. The authors have striven to give an exposition stressing the working apparatus of elassical mechanics, rather than its physical foundations or applications. This appara­ tus is basically contained in Chapters 1, 3,4 and 5. Chapter 1 is devoted to the fundamental mathematical models which are usually employed to describe the motion of real mechanical systems. Special consideration is given to the study of motion under constraints, and also to problems concerned with the realization of constraints in dynamics. Chapter 3 is concerned with the symmetry groups of mechanical systems and the corresponding conservation laws. Also discussed are various aspects of the theory of the reduction of order for systems with symmetry, often used in applications. Chapter 4 contains abrief survey of various approaches to the problem of the integrability of the equations of motion, and discusses some of the most general and effective methods of integrating these equations. Various elassical examples of integrated problems are outlined. The material pre­ sen ted in this chapter is used in Chapter 5, which is devoted to one of the most fruitful branches of mechanics - perturbation theory. The main task of perturbation theory is the investigation of problems of mechanics which are" elose" to exact1y integrable problems.

• dynamical systems
• integrable system
• integration
• mechanics
• n-body problem

From the reviews: "... As an encyclopaedia article, this book does not seek to serve as a textbook, nor to replace the original articles whose results it describes. The book's goal is to provide an overview, pointing out highlights and unsolved problems, and putting individual results into a coherent context. It is full of historical nuggets, many of them surprising. ... The examples are especially helpful; if a particular topic seems difficult, a later example frequently tames it. The writing is refreshingly direct, never degenerating into a vocabulary lesson for its own sake. 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, Nov. 1989 "This is a book to curl up with in front of a fire on a cold winter's evening. ..." SIAM Reviews, Sept. 1989

• Steklov Mathematical Institute, Moscow, USSR

  Vladimir I. Arnold

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