Global Aspects of Classic Integrable Systems - Hardcover

Cushman, Richard; Bates, Larry

Hardcover

ISBN 10: 3764354852 ISBN 13: 9783764354855

Publisher: Birkhauser Verlag AG, 1997

Synopsis

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.

About this title

Review

"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

Synopsis

Providing a complete description of the geometry of the energy momentum mapping of five classical integrable systems, this text can be used as a textbook for graduate courses on classical mechanics or global analysis.

Edition details

• Publisher: Birkhauser Verlag AG
• Publication date: 1997
• Language: English
• ISBN 10: 3764354852
• ISBN 13: 9783764354855
• Binding: Hardcover
• Number of pages: 435