Mathematik in den Naturwissenschaften Leipzig A smooth pseudo - gradient for the Lagrangian action functional

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 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. Introduction This note is about the action functional SL(γ) = ∫ 1 0 L(t, γ(t), γ(t)) dt associated to a smooth Lagrangian function L on [0, 1]×TM , where M is a smooth n-dimensional manifold, and γ is a curve in M . The critical points of such a functional are the solutions of the Euler-Lagrange equation associated to L. If L has quadratic growth in the velocities (see condition (L1) in Section 2), the natural functional setting for studying the functional SL is the Hilbert manifold H([0, 1],M) of absolutely continuous paths in M with square integrable derivative, or the Hilbert submanifolds one gets by imposing the boundary conditions one is interested in. In fact, as Benci as showed in [8], this assumption implies that the action functional SL is continuously differentiable on H([0, 1],M) and, assuming also that L is strictly fiberwise convex and M is compact, that the Palais-Smale condition holds (see Section 2 for precise statements). These facts can be used to prove for instance that ifM is compact the Euler-Lagrange equation associated to a time-periodic L has at least as many periodic solutions as the Lusternik-Schnirelmann category of the free loop space of M (see [8]). Under this quadratic growth assumption on L, the functional SL is actually also twice Gateaux differentiable on H([0, 1],M), but it is twice Fréchét differentiable if and only if L is exactly quadratic in the velocities (more precisely, if for every (t, q) the function v 7→ L(t, q, v) is a polynomial of degree at most 2). In this case, SL is actually smooth. This fact is related to well-known facts about the differentiability of Nemitsky operators. Infinite dimensional Morse theory for the functional SL (see e.g. [9]), and in particular the Morse lemma, would require C 2 regularity. The aim of this note is to show that if the critical points of SL are non-degenerate, even if SL is not C, there exists a smooth (i.e. C) Morse-Smale vector field X on H([0, 1],M) for which SL is a Lyapunov function. The existence of such a vector field allows to construct the Morse In [1] and [5] it is erroneously stated that the action functional SL is indeed C . Actually, C regularity is sufficient in [1], which deals with Lusternik-Schnirelmann theory. The error in [5] was corrected in [4], by considering a smaller class of Lagrangian functions. The main result of this note implies that the original class of Lagrangian functions can also be considered in [5].