Research on the Estimation of Rank of Homology Group of Energy Level Surface on Hamiltonian Systems 

Author  QinTao 
Tutor  GuZhiMing 
School  Nanjing University of Aeronautics and Astronautics 
Course  Applied Mathematics 
Keywords  energy level surface rank of qdimensional homology group exact homology sequence Morse inequality 
CLC  O19 
Type  Master's thesis 
Year  2007 
Downloads  9 
Quotes  0 
The author of the paper [1] estimates how many types of largescale 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 qdimensional 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 0dimensional singular homology group and 2dimensional singular homology group have been also assessed then we guess a formula for the rank of qdimensional 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 qdimensional singular homology group of the energy level surface. Here our tools are exact homology sequences and the Morse inequalities.