Hamilton系统的大范围周期轨道的估计

Estimation of Large-Scale Periodic Orbits in Hamiltonian System

Hamilton系统的相轨道位于正则值所确定的等能曲面上,而系统的大范围周期轨道可以代表等能曲面的同调类,这些同调类一般非平凡。而等能曲面的拓扑性质又由相空间的拓扑性质和Hamilton函数的大尺度性质决定,用这两种性质估算了受外力的刚体运动的等能曲面的第1同调群的秩。用同伦论、同调论和Morse理论把已有证明中的不足之处加以改进,得出基本定理的新的证明。

The phase orbits of a Hamiltonian system are on the equi-energy＇level surface which is determined by the regular value. The large-scale periodic orbits of the system can represent the homology classes, which are generally non-trivial,on the equi-energy level surface and the topological properties of the equi-energy level surface are determined by that of the phase space and the large-scale properties of Hamiltonian function. These properties were used for estimation of the rank of the first homology group of the equi-energy level surfaces about the motion of a rigid body under external force. The proof was improved by the theory of differential topology and algebra topology.