标题： 路径空间中的莫尔斯理论 显示英文标题

标题： Morse theory in path space

摘要： 我们考虑一个弯曲流形上的路径空间，在保守物理系统中引入一个具有恒定总能量的点粒子，以制定其作用泛函和测地线方程，并允许路径上出现断裂。 通过作用泛函的二阶变分，可以导出测地线偏离方程，并讨论弯曲流形上的雅可比场。 我们利用作用泛函及其物理意义研究路径空间的拓扑结构，定义了作用泛函的梯度、有界流能解空间以及与作用泛函临界点相关的模空间。 此外，我们在保守物理系统中的$n$-球面$S^{n}$上考虑粒子运动，明确讨论路径空间的模空间及其对应的同调群。 显示英文摘要

摘要： We consider the path space of a curved manifold on which a point particle is introduced in a conservative physical system with constant total energy to formulate its action functional and geodesic equation together with breaks on the path. The second variation of the action functional is exploited to yield the geodesic deviation equation and to discuss the Jacobi fields on the curved manifold. We investigate the topology of the path space using the action functional on it and its physical meaning by defining the gradient of the action functional, the space of bounded flow energy solutions and the moduli space associated with the critical points of the action functional. We also consider the particle motion on the $n$-sphere $S^{n}$ in the conservative physical system to discuss explicitly the moduli space of the path space and the corresponding homology groups.