A smooth pseudo-gradient for the Lagrangian action functional

Details A smooth pseudo-gradient for the Lagrangian action functional A smooth pseudo-gradient for the Lagrangian action functional

2008-12-23

arxiv.org/abs/0812.4364v2

Advanced Nonlinear Studies 9 (2009), 597-623

Mathematics

Dynamical Systems

27 pages, final version, as published

Scientific paper

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.

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