A Smooth Pseudo-Gradient for the Lagrangian Action Functional

By Alberto Abbondandolo, Matthias Schwarzy. 2008

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Abstract We study the action functional associated to a smooth Lagrangian function on the tangent 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 H1 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 suffcient to associate a Morse complex to the Lagrangian action functional.

Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4894a7babafc3e7e875b9a43026ed13f0252ff7b