  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.
