Rotation function for integrable problems reducing to the Abel equations. Orbital classification of Goryachev–Chaplygin systems.

The rotation vector, one of the most important orbital invariants of integrable Hamiltonian systems with two degrees of freedom, is constructed using the rotation function (see [1]). A general theory of computation of the rotation functions for dynamical systems reducing to the Abel equations is developed. Using this theory, an explicit formula for the rotation function in the Goryachev–Chaplygin case in the dynamics of heavy rigid bodies is obtained. The orbital classification of the family of Goryachev–Chaplygin systems for different values of energy is presented. For this purpose the orbital Fomenko–Bolsinov invariant, which is the classifying object, is computed. The orbital non-equivalence of the Goryachev–Chaplygin flows on surfaces of equal energy corresponding to different energy levels is established as a result of an analytic study and computer analysis (together with S. Takahasi, Japan) . In addition, explicit formulae for the transition from the coordinate system on the Jacobian (the Abelian variables) to the Euler–Poisson coordinate system in the Goryachev–Chaplygin case are obtained and the covering of the Jacobian by the Liouville torus is studied. This can be used for finding an explicit solution to the Goryachev–Chaplygin problem in terms of two-dimensional theta functions.