Advances in Differential Equations

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

Rossella Bartolo and Antonio Masiello

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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

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

First available in Project Euclid: 23 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58E10: Applications to the theory of geodesics (problems in one independent variable)
Secondary: 34B15: Nonlinear boundary value problems 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.) 70H35


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.

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