Classical Integrable Systems

Abstract

The completely integrable system in the Liouville sense is a challenging topic. A zigzag has been experienced in the progress of its recognition. The ideal goal of the early classical mechanics is to integrate the equations of motion explicitly. All the efforts made in this connection have culminated in the discovery of a series of completely integrable systems, such as Jacobi’s integration of the equations for geodesics on an ellipsoid, Kovalevski’s study of the motion of some kind of tops and so on. However, at the end of the 19th century, Poincaré and others realized that most Hamiltonian systems are not integrable, and pointed out that the famous three-body problem is nonintegrable. Moreover, it was found that the integrability is destroyed under small perturbations of the Hamiltonian. Thus the importance of the integrability came under suspicion, and the integrable systems were regarded as rare exceptions, possessing no generic property. Since then, in the study of dynamical systems the stress has been laid on the qualitative theory, whereas little attention has been paid to the integrable theory.