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A smooth pseudo-gradient for the Lagrangian action functional. (English) Zbl 1185.37145
The authors consider the action functional of a Lagrangian system with quadratic growth in the velocity on the natural Hilbert manifold of absolutely continuous trajectories with square integrable derivative. In this setting, it is known that the action is \(C^1\) and twice Gateaux differentiable but in general not twice Frechet differentiable [see V. Benci, J. Differ. Equations 63, 135–161 (1986; Zbl 0605.58034)].
The regularity of the action functional becomes a touchy business when dealing with Morse Theory, where having a \(C^2\) functional would be desirable. In this paper, the authors are able to overcome this difficulty by showing that if the critical points of the action are non-degenerate, there exists a smooth Morse-Smale vector field for which the action is a Lyapunov function. This provides the construction of the Morse complex and gives several multiplicity results.

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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