Morse theory in path space

TL;DR

This paper applies Morse theory to analyze the topology of path space in curved manifolds, linking geometric, topological, and physical aspects of particle motion in conservative systems.

Contribution

It introduces a novel application of Morse theory to path spaces, exploring geodesic deviations, Jacobi fields, and the topology of moduli spaces in curved manifolds.

Findings

Topological structure of path space characterized via Morse theory.

Explicit analysis of particle motion on the n-sphere.

Identification of homology groups related to path space.

Abstract

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.