From Singular Point Analysis to Rigorous Results on Integrability: a Dream of S. Kowalevskaya

Abstract

A Hamiltonian system with n degrees of freedom is called integrable if the system possesses n independent first integrals in involution. Solution of integrable systems is generically quasi-periodic and no chaos appears. Therefore to decide whether a given system is integrable or not is one of the most fundamental questions in the study of Hamiltonian dynamical systems. Although there is no guarantee that the ultimate criterion does exist, there have been a considerable progress in this direction, in the last decade. The candidate is sometimes called the singular point analysis, which has its origin in the work of S. Kowalevskaya in 1888.