A New Approach for Euler-Lagrange Orbits on Compact Manifolds with Boundary

Consider a compact manifold with boundary, homeomorphic to the N-dimensional disk, and a Tonelli Lagrangian function defined on the tangent bundle. In this paper, we study the multiplicity problem for Euler-Lagrange orbits that satisfy the conormal boundary conditions and that lay on the boundary only in their extreme points. In particular, for suitable values of the energy function and under mild hypotheses, if the Tonelli Lagrangian is reversible then the minimal number of Euler-Lagrange orbits with prescribed energy that satisfies the conormal boundary conditions is N. If L is not reversible, then this number is two.

[1]  Alberto Abbondandolo.  Lectures on the free period Lagrangian action functional , 2013, 1309.0149.

[2]  H. Seifert.  Periodische Bewegungen mechanischer Systeme , 1948 .

[3]  Guangcun Lu.  Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds , 2012, 1212.2078.

[4]  R. Giambò,et al.  Multiple Brake Orbits and Homoclinics in Riemannian Manifolds , 2011 .

[5]  L. Asselle.  On the existence of Euler-Lagrange orbits satisfying the conormal boundary conditions , 2015, 1507.05883.

[6]  A. Masiello,et al.  On the existence of geodesics on stationary Lorentz manifolds with convex boundary , 1991 .

[7]  R. Giambò,et al.  Multiple brake orbits in $$m$$m-dimensional disks , 2015 .

[8]  E. Caponio,et al.  Convex domains of Finsler and Riemannian manifolds , 2009, 0911.0360.

[9]  P. Majer,et al.  On the effect of the domain on the number of orthogonal geodesic chords , 1997 .

[10]  R. Giambò,et al.  Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles , 2018, Calculus of Variations and Partial Differential Equations.

[11]  R. Giambò,et al.  Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and homeomorphic to the N-dimensional disk , 2010, 1003.3846.

[12]  A multiplicity result for Euler–Lagrange orbits satisfying the conormal boundary conditions , 2020, Journal of Fixed Point Theory and Applications.

[13]  R. Giambò,et al.  On the Multiplicity of Orthogonal Geodesics in Riemannian Manifold With Concave Boundary. Applications to Brake Orbits and Homoclinics , 2009 .

[14]  A. Canino.  Periodic solutions of Lagrangian systems on manifolds with boundary , 1991 .

[15]  G. Contreras.  The Palais-Smale condition on contact type energy levels for convex Lagrangian systems , 2003, math/0304238.

[16]  Alan Weinstein,et al.  Periodic Orbits for Convex Hamiltonian Systems , 1978 .