Mathematical Methods of Classical Mechanics

1. Front Matter

   Pages i-x

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2. Newtonian Mechanics

   1. Front Matter

      Pages 1-1

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   2. Experimental facts

      • V. I. Arnold

      Pages 3-14

   3. Investigation of the equations of motion

      • V. I. Arnold

      Pages 15-52

3. Lagrangian Mechanics

   1. Front Matter

      Pages 53-53

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   2. Variational principles

      • V. I. Arnold

      Pages 55-74

   3. Lagrangian mechanics on manifolds

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      Pages 75-97

   4. Oscillations

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      Pages 98-122

   5. Rigid bodies

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      Pages 123-159

4. Hamiltonian Mechanics

   1. Front Matter

      Pages 161-161

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   2. Differential forms

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      Pages 163-200

   3. Symplectic manifolds

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      Pages 201-232

   4. Canonical formalism

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      Pages 233-270

   5. Introduction to perturbation theory

      • V. I. Arnold

      Pages 271-300

5. Back Matter

   Pages 301-464

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Many different mathematical methods and concepts are used in classical mechanics: differential equations and phase ftows, smooth mappings and manifolds, Lie groups and Lie algebras, symplectic geometry and ergodic theory. Many modern mathematical theories arose from problems in mechanics and only later acquired that axiomatic-abstract form which makes them so hard to study. In this book we construct the mathematical apparatus of classical mechanics from the very beginning; thus, the reader is not assumed to have any previous knowledge beyond standard courses in analysis (differential and integral calculus, differential equations), geometry (vector spaces, vectors) and linear algebra (linear operators, quadratic forms). With the help of this apparatus, we examine all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the hamiltonian formalism. The author has tried to show the geometric, qualitative aspect of phenomena. In this respect the book is closer to courses in theoretical mechanics for theoretical physicists than to traditional courses in theoretical mechanics as taught by mathematicians.

• Hamiltonian
• Lie
• Mathematica
• Newtonian mechanics
• algebra
• calculus
• classical mechanics
• differential equation
• dynamical systems
• dynamics
• equation
• invariant
• manifold
• mechanics
• oscillation

Second Edition

V.I. Arnol’d

Mathematical Methods of Classical Mechanics

"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

• Department of Mathematics, University of Moscow, Moscow, USSR

  V. I. Arnold

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