Orthogonal Geodesic Chords, Brake Orbits and Homoclinic Orbits in Riemannian Manifolds

Examples with minimal number of brake orbits and homoclinics in annular potential regions

We use a geometric construction to exhibit examples of autonomous Lagrangian systems admitting exactly two homoclinics emanating from a nondegenerate maximum of the potential energy and reaching a regular level of the potential having the same value of the maximum point. Similarly, we show examples of Hamiltonian systems that admit exactly two brake orbits in an annular potential region connecting the two connected components of the boundary of the potential well. These examples show that the estimates proven in [R. Giambò, F. Giannoni, P. Piccione, Arch. Ration. Mech. Anal. 200, (2011) 691-72…

Multiple orthogonal geodesic chords and a proof of Seifert's conjecture on brake orbits

Using nonsmooth critical point theory, we prove the existence of at least N orthogonal geodesic chords in a class of Riemannian N-disk with strongly concave boundary. This yields a proof of a celebrated conjecture by H.Seifert on the number of brake orbits in a potential well of a natural Lagrangian/Hamiltonian system.

On the normal exponential map in singular conformal metrics

Brake orbits and homoclinics of autonomous dynamical systems correspond, via Maupertuis principle, to geodesics in Riemannian manifolds endowed with a metric which is singular on the boundary (Jacobi metric). Motivated by the classical, yet still intriguing in many aspects, problem of establishing multiplicity results for brake orbits and homoclinics, as done in [6, 7, 10], and by the development of a Morse theory in [8] for geodesics in such kind of metric, in this paper we study the related normal exponential map from a global perspective.

三维Reeb流中的同宿轨道、Reeb弦与良好Birkhoff截面

Homoclinic orbits, Reeb chords and nice Birkhoff sections for Reeb flows in 3D

本文证明在三维流形上，对一般接触形式，双曲周期轨道存在同宿连接，并由此构造包含给定轨道和Legendrian链环的Birkhoff截面，进而证明一般情形下每个Legendrian结有无穷多个Reeb弦。

Brake orbits for Hamiltonian systems of classical type via Finsler geodesics

We consider Hamiltonian functions of classical type, namely even and convex with respect to the generalized momenta. A brake orbit is a periodic solution of Hamilton's equations such that the generalized momenta are zero on two different points. Under mild assumptions, this paper reduces the multiplicity problem of the brake orbits for a Hamiltonian function of classical type to the multiplicity problem of orthogonal geodesic chords in a concave Finslerian manifold with boundary. This paper will be used for a generalization of a Seifert's conjecture about the multiplicity of brake orbits to Ha…

Functions on the sphere with critical points in pairs and orthogonal geodesic chords

Using an estimate on the number of critical points for a Morse-even function on the sphere $\mathbb S^m$, $m\ge1$, we prove a multiplicity result for orthogonal geodesic chords in Riemannian manifolds with boundary that are diffeomorphic to Euclidean balls. This yields also a multiplicity result for brake orbits in a potential well.