Research On The Estimation Of Rank Of Homology Group Of Energy Level Surface On Hamiltonian Systems

The author of the paper [1] estimates how many types of large-scale periodic orbits of a energy level surface exist in a nonlinear mechanical system. This estimation can be transformed to estimate the upper bound of the rank of the first homology group of energy level surface according to the fundamental group, Hurewicz theory and some topology properties of energy level surface (see [1]). In this paper we take advantage of methods which belong to [1] to extend some conclusions in the paper [1], that is to say, we estimate the upper bound of the rank of q-dimensional singular homology group (q from 0 to 2 n ? 1, 2 n ? 1= the dimension of energy level surface).Firstly, we estimate again the rank of first homology group so as to improve accuracy and feasibility.Secondly, the ranks of 0-dimensional singular homology group and 2-dimensional singular homology group have been also assessed then we guess a formula for the rank of q-dimensional singular homology group.Finally, we prove that the guess is right and apply it to an example of rigid body dynamics comparing with predecessors'conclusions.The innovation of this paper is a new formula which estimate the upper bound of the rank of q-dimensional singular homology group of the energy level surface. Here our tools are exact homology sequences and the Morse inequalities.