### Morse theory for trajectories of Lagrangian systems on Riemannian manifolds with convex boundary

#### Abstract

In this paper we relate the set of the orbits of a second-order Lagrangian systems joining two points on an open set with convex boundary of a Riemannian manifold with the topological structure of the open set. Such relations are obtained by developing a Morse Theory for the action integral of the Lagrangian system. Because of the presence of the boundary, the action integral does not satisfy the Palais-Smale condition. We perturb the action integral with a family of smooth functionals, satisfying the Palais-Smale condition. The Morse Relations for the action integral are obtained as limit of the Morse Relations of the perturbing functionals. A relation between the Morse index and the energy of the orbits as critical points of the action integral is obtained.

#### Article information

Source
Adv. Differential Equations, Volume 2, Number 4 (1997), 593-618.

Dates
First available in Project Euclid: 23 April 2013

Mathematical Reviews number (MathSciNet)
MR1441858

Zentralblatt MATH identifier
1023.58500

#### Citation

Bartolo, Rossella; Masiello, Antonio. Morse theory for trajectories of Lagrangian systems on Riemannian manifolds with convex boundary. Adv. Differential Equations 2 (1997), no. 4, 593--618. https://projecteuclid.org/euclid.ade/1366741150