Multiple brake orbits in \mathbf m-dimensional disks

2015/03/19 by R. Giambò, Roberto Giambò, Fabio Giannoni +6 · 1 citation
Mathematics · #Analytic and geometric function theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.DS

paper · pdf · doi:10.48550/arxiv.1503.05805

26 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1003.3846

arxiv created 2015/03/19 · openalex publication_date 2015/03/19 · arxiv updated 2015/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (M,g) be a (complete) Riemannian surface, and let Ω⊂ M be an open subset whose closure is homeomorphic to a disk. We prove that if ∂Ω is smooth and it satisfies a strong concavity assumption, then there are at least two distinct orthogonal geodesics in Ω=Ω\bigcup∂Ω. Using the results given in [6], we then obtain a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. In our proof we shall use recent deformation results proved in [7].

Citations

• Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and homeomorphic to the N-dimensional disk

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

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