Topological classification of the Goryachev integrable systems in the rigid body dynamics: non-compact case

Abstract

We study the topology of the Liouville foliation of the Goryachev integrable case in the rigid body dynamics which is a one-parameter family of completely integrable Hamiltonian systems with two degrees of freedom. For this problem P. E. Ryabov has found a real separation of variables with the aid of which he studied the phase topology of the Goryachev systems for positive values of the parameter. We solve the similar problem for negative values of the parameter. This case is of special interest because all the leaves of the Liouville foliation and the surfaces of constant energy turn out to be non-compact. The results are presented in the form of Fomenko invariants for all regular energy levels.