Petar Topalov

Consider a Riemannian metric on a surface, and let the geodesic flow of the metric have a second integral that is a third degree polynomial in momenta. Then we can naturally construct a vector field on the surface. We show that the vector field preserves the volume of the surface, and therefore is a Hamiltonian vector field. As examples we treat the Goryachev–Chaplygin top, the Toda lattice and the Calogero–Moser system, and construct their global Hamiltonians. We show that the simpliest choice of Hamiltonian leads to the Toda lattice.

We suggest a simple approach for obtaining integrals of Hamiltonian systems if there is known a trajectorian map of two Hamiltonian systems. An explicite formila is given. As an example, it is proved that if on a manifold are given two Riemannian metrics which are geodesically equivalent then there is a big family of integrals. Our theorem is a generalization of the well-known Painleve–Liouville theorems.

In this paper we investigate the Poincare map in the regular neighbourhood of a critical leaf of the Liouville foliation of an integrable Hamiltonian system with two degrees of freedom. It was proved in [3], that for an arbitrary surface transversal to the trajectories, the Poincare map is a one-time-map along the flow of some Hamiltonian, which is defined on the considering surface (this Hamiltonian is called "the Poincare Hamiltonian"). In the paper [4] it was proved that for every transversal surface the Poincare map is a restriction to the surface of some smooth function, which is defined on the regular neighbourhood of the critical leaf.

We show that an invariant surface allows to construct the Jacobi vector field along a geodesic line and construct the formula for the normal part of the Jacobi field. If a geodesic line is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic line is hyperbolic) than we can construct the fundamental solution of Jacobi equation $\ddot{u}=-K(t)u$. That was done for quadratically integrable geodesic flows.