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Positive energy solutions in the anisotropic Kepler problem with homogeneous potential

Guowei Yu

Abstract

We study positive energy solutions of the anisotropic Kepler problem with homogeneous potential. First some asymptotic property of positive energy solutions is obtained, as time goes to infinity. Afterwards, we prove the existence of hyperbolic solutions with given initial configuration and asymptotic behavior, when time goes to positive or negative infinity, and in the planar case, the existence of bi-hyperbolic solutions with given asymptotic behaviors, when time goes to both positive and negative infinities, under various conditions.

Positive energy solutions in the anisotropic Kepler problem with homogeneous potential

Abstract

We study positive energy solutions of the anisotropic Kepler problem with homogeneous potential. First some asymptotic property of positive energy solutions is obtained, as time goes to infinity. Afterwards, we prove the existence of hyperbolic solutions with given initial configuration and asymptotic behavior, when time goes to positive or negative infinity, and in the planar case, the existence of bi-hyperbolic solutions with given asymptotic behaviors, when time goes to both positive and negative infinities, under various conditions.

Paper Structure

This paper contains 6 sections, 26 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Asymptotic analysis of positive energy solutions
  3. Hyperbolic solutions with given initial configuration
  4. Bi-hyperbolic solutions for anisotropic Kepler problem
  5. rule out collision for fixed-time local minimizers
  6. Bi-hyperbolic solution for Gutzwiller's anisotropic Kepler problem

Key Result

Proposition 1.1

BFT08 Given a solution $x(t) =r(t) s(t)$ with $r(t) \to 0$, when $t \to t_0^{\pm}$, then there exist positive constants $\kappa_{\pm}$, such that the following asymptotic results hold, as $t \to t_0^{\pm}$,

Theorems & Definitions (60)

  • Proposition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Definition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • ...and 50 more