A Smooth Pseudo-Gradient for the Lagrangian Action Functional

@article{Abbondandolo2008ASP,
  title={A Smooth Pseudo-Gradient for the Lagrangian Action Functional},
  author={Alberto Abbondandolo and Matthias Schwarzy},
  journal={Advanced Nonlinear Studies},
  year={2008},
  volume={9},
  pages={597 - 623}
}
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… 
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