Entire Parabolic Trajectories as Minimal Phase Transitions

2011/05/17 by Vivina Barutello, Susanna Terracini, Barutello, Vivina +3 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Quantum chaos and dynamical systems #math.DS

paper · pdf · doi:10.48550/arxiv.1105.3358

44 pages, 7 figures

arxiv created 2011/05/17 · openalex publication_date 2011/05/17 · arxiv updated 2015/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For the class of anisotropic Kepler problems in \RR^d∖\0\ with homogeneous potentials, we seek parabolic trajectories having prescribed asymptotic directions at infinity and which, in addition, are Morse minimizing geodesics for the Jacobi metric. Such trajectories correspond to saddle heteroclinics on the collision manifold, are structurally unstable and appear only for a codimension-one submanifold of such potentials. We give them a variational characterization in terms of the behavior of the parameter-free minimizers of an associated obstacle problem. We then give a full characterization of such a codimension-one manifold of potentials and we show how to parameterize it with respect to the degree of homogeneity.

Cited by

• Positive energy solutions in the anisotropic Kepler problem with homogeneous potential
• Parabolic Orbits for a Class of Singular Hamiltonian Systems with a Fixed Energy
• Hyperbolic Orbits for Restricted Three-body Problems with Fixed Energy

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