Abstract: The large-scale periodic orbits of a nonlinear mechanics system can represent the homology classes, which are generally non-trivial, of the energy level surface and the topology properties of an energy level surface are determined by the that of the phase space and the large-scale properties of the Hamiltonian. These properties are used for estimate of the rank of first homology group of energy level surfaces in the paper.

Key words: large-scale periodic motion, homology classes, Morsc inequalities