A criterion for orbital equivalence of integrable Hamiltonian systems in the vicinity of elliptic orbits. An orbital invariant in the Lagrange problem

In this paper we obtain a criterion for the continuous and smooth orbital equivalence of integrable Hamiltonian systems with $n$ degrees of freedom in the vicinity of compact elliptic orbits. Moreover, we construct a complete orbital invariant for a non-degenerate integrable Hamiltonian system with two degrees of freedom in a neighbourhood of an elliptic singular point, and propose a rule from which to compute this orbital invariant. The orbital invariant is computed for integrable Lagrange systems in rigid body dynamics. In this way we find an explicit decomposition of all Lagrange systems into classes of orbitally equivalent ones in the vicinity of equilibria.