Entire minimal parabolic trajectories: the planar anisotropic Kepler problem

Abstract

We continue the variational approach to parabolic trajectories introduced in our previous paper [5], which sees parabolic orbits as minimal phase transitions. We deepen and complete the analysis in the planar case for homogeneous singular potentials. We characterize all parabolic orbits connecting two minimal central configurations as free-time Morse minimizers (in a given homotopy class of paths). These may occur for at most one value of the homogeneity exponent. In addition, we link this threshold of existence of parabolic trajectories with the absence of collisions for all the minimizers of fixed-ends problems. Also the existence of action minimizing periodic trajectories with nontrivial homotopy type can be related with the same threshold.

28 pages, 4 figures

References in corpus (4)

• Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry
• Globally minimizing parabolic motions in the Newtonian N-body problem
• On the free time minimizers of the Newtonian N-body problem
• Entire Parabolic Trajectories as Minimal Phase Transitions

Cited by in corpus (10)

• On the free time minimizers of the Newtonian N-body problem
• Scattering parabolic solutions for the spatial N-centre problem
• Parabolic solutions for the planar $N$-centre problem: multiplicity and scattering
• Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem
• Metric cones, N-body collisions, and Marchal's lemma
• Parabolic orbits in Celestial Mechanics: a functional-analytic approach
• Symbolic dynamics for the anisotropic $N$-centre problem at negative energies
• Avoiding collisions under topological constraints in variational problems coming from celestial mechanics
• Periodic Solutions of the Planar N-Center Problem with topological constraints
• Uniqueness of hyperbolic Busemann functions in the Newtonian N-body problem