Global Aspects of Classical Integrable Systems

1. Front Matter

   Pages i-xvi

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2. The harmonic oscillator

   • Richard H. Cushman, Larry M. Bates

   Pages 1-36

3. Geodesics on S³

   • Richard H. Cushman, Larry M. Bates

   Pages 37-82

4. The Euler Top

   • Richard H. Cushman, Larry M. Bates

   Pages 83-146

5. The spherical pendulum

   • Richard H. Cushman, Larry M. Bates

   Pages 147-186

6. The Lagrange top

   • Richard H. Cushman, Larry M. Bates

   Pages 187-270

7. Back Matter

   Pages 271-435

   PDF

This book gives a complete global geometric description of the motion of the two di­ mensional hannonic oscillator, the Kepler problem, the Euler top, the spherical pendulum and the Lagrange top. These classical integrable Hamiltonian systems one sees treated in almost every physics book on classical mechanics. So why is this book necessary? The answer is that the standard treatments are not complete. For instance in physics books one cannot see the monodromy in the spherical pendulum from its explicit solution in terms of elliptic functions nor can one read off from the explicit solution the fact that a tennis racket makes a near half twist when it is tossed so as to spin nearly about its intermediate axis. Modem mathematics books on mechanics do not use the symplectic geometric tools they develop to treat the qualitative features of these problems either. One reason for this is that their basic tool for removing symmetries of Hamiltonian systems, called regular reduction, is not general enough to handle removal of the symmetries which occur in the spherical pendulum or in the Lagrange top. For these symmetries one needs singular reduction. Another reason is that the obstructions to making local action angle coordinates global such as monodromy were not known when these works were written.

• algebra
• classical mechanics
• hamiltonian mechanics
• manifold
• pendulum

"Ideal for someone who needs a thorough global understanding of one of these systems [and] who would like to learn some of the tools and language of modern geometric mechanics. The exercises at the end of each chapter are excellent. The book could serve as a good supplementary text for a graduate course in geometric mechanics."

--SIAM Review

• Mathematics Institute, University of Utrecht, Utrecht, The Netherlands

  Richard H. Cushman

• Department of Mathematics and Statistics, University of Calgary, Calgary, Canada

  Larry M. Bates

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