A smooth pseudo-gradient for the Lagrangian action functional
Alberto Abbondandolo and Matthias Schwarz
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Submission date: 23. Dec. 2008
published in: Advanced nonlinear studies, 9 (2009) 4, p. 597-623
Keywords and phrases: Lagrangian action functional, infinite-dimensional Morse theory
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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 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.