Hamilton系统等能曲面之奇异同调群的秩估计

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

为了加深对等能曲面的拓扑结构的了解,利用正合同调序列及Morse不等式的方法估计了等能曲面一般维数奇异同调群的秩的上界.分别对等能曲面的0维、1维、2维奇异同调群的秩进行了估计,得出了估计不等式,依据0维、1维、2维的估计不等式,归纳出一般维数奇异同调群的秩的上界估计不等式,证明此归纳不等式成立,并将其运用到刚体运动的力学例子中,与前人研究结果对照,验证其正确性.

This paper intends to make us understand topological structures deeply, and it takes advantage of exact homology sequence and Morse inequalities to estimate the upper bound of the rank of q-dimensional singular homology group of energy level surface （q is an arbitrary nature number）. It estimates the ranks of 0-dimension, 1-dimension and 2-dimension homology groups, and guesses a formula for the rank of q-dimension singular homology group. It proves that the guess is right and applies it to an example of rigid body dynamics, comparing it with other schol- ars＇ conclusions. It succeeds in obtaining a new inequality which estimates the upper bound of the rank of q-dimension singular homology group of energy level surface.