Topological Invariant of Integrable Hamiltonian System on Cone Located in a Potential Field

The theory of topological classification of integrable Hamiltonian systems with two degrees of freedom due to Fomenko and his school‎. ‎On the basis of this theory we give a topological Liouville classification of the integrable Hamiltonian systems with two degrees of freedom‎. ‎Essentially‎, ‎to an integrable system with two degrees of freedom which is restricted to a nonsingular 3-dimensional iso-energy manifold. Fomenkochr('39')s theory ascribes in an effective way a certain discrete invariant which has the structure of a graph with numerical marks‎. ‎This invariant‎, ‎which is called the marked molecule or the Fomenko-Zieschang invariant‎, ‎gives a full description (up to Liouville equivalence) of the Liouville foliation for the system.The topological classification of integrable Hamiltonian systems corresponding to the Liouville equivalence in potential fields on surfaces of revolution for surfaces that is diffeomorphic with 2-dimensional sphere, contains a wide classes of mechanical systems that describes the motion of a particle on a 2-dimensional sphere with revolution metric, which has been studied.In this paper, the topology of non-singular iso-energy surfaces for a Hamiltonian system with two degrees of freedom on a cone located in a potential field is described. Also, the method of finding the topological invariant of integrable Hamiltonian systems is extended from compact case to non-compact rotating surfaces.