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MiS Preprint

A smooth pseudo-gradient for the Lagrangian action functional

Alberto Abbondandolo and Matthias Schwarz


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.

MSC Codes:
Lagrangian action functional, infinite-dimensional Morse theory

Related publications

2009 Repository Open Access
Alberto Abbondandolo and Matthias Schwarz

A smooth pseudo-gradient for the Lagrangian action functional

In: Advanced nonlinear studies, 9 (2009) 4, pp. 597-623