A smooth pseudo-gradient for the Lagrangian action functional

TL;DR

This paper introduces a smooth pseudo-gradient for the Lagrangian action functional, enabling the construction of a Morse complex despite the functional's lack of twice differentiability, under generic non-degeneracy conditions.

Contribution

It develops a method to associate a Morse complex to the Lagrangian action functional using a smooth pseudo-gradient, overcoming differentiability challenges.

Findings

The action functional acts as a Lyapunov function for a Morse-Smale vector field.

A Morse complex can be constructed for the Lagrangian action functional.

The approach applies under generic non-degeneracy assumptions.

Abstract

We study the action functional associated to a smooth Lagrangian function on the cotangent bundle of a manifold, having quadratic growth in the velocities. We show that, although the action functional is in general not twice differentiable on the Hilbert manifold consisting of H^1 curves, it is a Lyapunov function for some smooth Morse-Smale vector field, under the generic assumption that all the critical points are non-degenerate. This fact is sufficient to associate a Morse complex to the Lagrangian action functional.