The Morse index theorem for mechanical systems with reflections

TL;DR

This paper establishes a Morse index theorem for mechanical systems where paths can reflect at a hypersurface, extending classical results to include reflections and various boundary conditions.

Contribution

It introduces a Morse index theorem applicable to paths with reflections at hypersurfaces, covering fixed and periodic boundary conditions.

Findings

Proves a Morse index theorem for reflected paths.

Extends classical Morse theory to systems with reflections.

Handles both interior and boundary reflections in manifolds.

Abstract

We prove a Morse index theorem for action functionals on paths that are allowed to reflect at a hypersurface (either in the interior or at the boundary of a manifold). Both fixed and periodic boundary conditions are treated.