We consider three classical integrable Hamiltonian systems: the Euler case in rigid body dynamics, the Jacobi problem about geodesics on the ellipsoid and the Chaplygin case in dynamics of a rigid body in fluid. It was proved by A. V. Bolsinov and A. T. Fomenko [1] that the first two systems are topologically orbitally equivalent, i.e. there exists a homeomorphism of the phase manifolds mapping the oriented trajectories of the first system to those of the second one. Now we prove that the Chaplygin system is orbitally equivalent to the Euler and Jacobi systems [2]. More precisely, the following statement holds.

Theorem. For any "large" value of energy of the Chaplygin system there exists a unique Euler system and a unique Jacobi system topologically orbitally equivalent to the given Chaplygin system. For any other regular energy value of the Chaplygin system there exists a one-parameter family of the Euler systems topologically orbitally equivalent to the given Chaplygin system. In the case of "small" energies this orbital isomorphism is smooth.

References

1. A. V. Bolsinov and A. T. Fomenko. Orbital classication of the geodesicows on two-dimensional ellipsoids. The Jacobi problem is orbitally equivalentto the integrable Euler case in rigid body dynamics. Funkts. Analiz i egoPrilozh., 29 (1995), No. 3, P. 115 (in Russian).

2. A. T. Fomenko and S. S. Nikolaenko. The Chaplygin case in dynamicsof rigid body in fluid is orbitally equivalent to the Euler case in rigid body dynamics and to the Jacobi problem on geodesics on the ellipsoid. J. Geometry and Physics (to appear).