Multiple brake orbits in a potential well and a Seifert conjecture

In a celebrated paper published in 1948 on Math. Z., Seifert studied the ex istence of a brake orbit in a potential well homemorphic to the N-dimension al disk for a classical Hamiltonian system. He used a footnote of the same paper to suggest the possibility to prove the existence of at least N (geom etrically distinct) brake orbits. This result can be proved using the Maupe rtuis Principle, that relates the solutions of classical autonomous Hamilto nian systems with the geodesics in the so called Jacobi metric. The existen ce of at least N geometrically distinct brake orbits can be obtained by pro ving the existence of at least N geometrically distinct orthogonal geodesic chords on a Riemannian manifold with boundary, homeomorphic to the N-dimen sional disk, and satisfying a condition of strong concavity (with respect t o the Jacobi metric).