Format:
1 Online-Ressource
ISBN:
9783034892575
,
9783764323363
Note:
This book is designed to expose from a general and universal standpoint a variety of methods and results concerning integrable systems of classical mechanics. By such systems we mean Hamiltonian systems with a finite number of degrees of freedom possessing sufficiently many conserved quantities (integrals of motion) so that in principle integration of the corresponding equations of motion can be reduced to quadratures, i.e. to evaluating integrals of known functions. The investigation of these systems was an important line of study in the last century which, among other things, stimulated the appearance of the theory of Lie groups. Early in our century, however, the work of H. Poincare made it clear that global integrals of motion for Hamiltonian systems exist only in exceptional cases, and the interest in integrable systems declined. Until recently, only a small number ofsuch systems with two or more degrees of freedom were known. In the last fifteen years, however, remarkable progress has been made in this direction due to the invention by Gardner, Greene, Kruskal, and Miura [GGKM 19671 of a new approach to the integration of nonlinear evolution equations known as the inverse scattering method or the method of isospectral deformations. Applied to problems of mechanics this method revealed the complete integrability of numerous classical systems. It should be pointed out that all systems of this kind discovered so far are related to Lie algebras, although often this relationship is not so simple as the one expressed by the well-known theorem of E. Noether
Language:
English
DOI:
10.1007/978-3-0348-9257-5
Author information:
Perelomov, Askolʹd M. 1935-
