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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
91/2008

A smooth pseudo-gradient for the Lagrangian action functional

Alberto Abbondandolo and Matthias Schwarz

Abstract

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.

Received:
Dec 23, 2008
Published:
Dec 23, 2008
MSC Codes:
58E05
Keywords:
Lagrangian action functional, infinite-dimensional Morse theory

Related publications

inJournal
2009 Repository Open Access
Alberto Abbondandolo and Matthias Schwarz

A smooth pseudo-gradient for the Lagrangian action functional

In: Advanced nonlinear studies, 9 (2009) 4, pp. 597-623