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. Since 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.

Morse theory for trajectories of Lagrangian systems on Riemannian manifolds with convex boundary / Bartolo, Rossella; Masiello, Antonio. - In: ADVANCES IN DIFFERENTIAL EQUATIONS. - ISSN 1079-9389. - STAMPA. - 2:4(1997), pp. 593-618.

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

Rossella Bartolo;Antonio Masiello
1997-01-01

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. Since 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.
1997
Morse theory for trajectories of Lagrangian systems on Riemannian manifolds with convex boundary / Bartolo, Rossella; Masiello, Antonio. - In: ADVANCES IN DIFFERENTIAL EQUATIONS. - ISSN 1079-9389. - STAMPA. - 2:4(1997), pp. 593-618.
Morse theory for trajectories of Lagrangian systems on Riemannian manifolds with convex boundary / Bartolo, Rossella; Masiello, Antonio. - In: ADVANCES IN DIFFERENTIAL EQUATIONS. - ISSN 1079-9389. - STAMPA. - 2:4(1997), pp. 593-618.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11589/1337
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