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Rotation function for integrable problems reducing to the Abel equations. Orbital classification of Goryachev-Chaplygin systems. (English. Russian original) Zbl 0858.70010

Sb. Math. 186, No. 2, 271-296 (1995); translation from Mat. Sb. 186, No. 2, 105-128 (1995).
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. 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.
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.

MSC:

70H05 Hamilton’s equations
70E15 Free motion of a rigid body
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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