Topological and Geometrical Obstructions to Complete Integrability

Abstract

We start the analysis of the reasons for nonintegrable behavior of Hamiltonian systems with a discussion of relatively recently discovered “rough” topological obstructions to integrability. In [130] it was proved that a closed analytic surface with genus ≥ 2 cannot be the configuration space of an integrable analytic system. The reason is the existence of an infinite number of unstable periodic orbits on which the integrals are dependent. This result (unnoticed earlier due to the preference for local study of dynamical systems) was generalized in several directions. The proof of nonintegrability is based on variational methods and subtle results from the theory of singularities of analytic mappings.