Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

No infinite spin for partial collisions converging to isolated central configurations on the plane

Anna Gierzkiewicz, Rodrigo G. Schaefer, Piotr Zgliczyński

Abstract

In the $n$-body problem, when a~cluster of bodies tends to a collision, then its normalized shape curve converges to the set of normalized central configurations, which has $SO(2)$ symmetry in the planar case. This leaves a possibility that the normalized shape curve tends to the circle obtained by rotation of some central configuration instead of a particular point on it. This is the \emph{infinite spin problem} which concerns the rotational behavior of total collision orbits in the $n$-body problem. The question also makes sense for partial collision. We show that the infinite spin is not possible if the limiting circle is isolated from other connected components of the set of normalized central configurations. Our approach extends the method from recent work for total collision by Moeckel and Montgomery, which was based on a combination of the center manifold theorem with Łojasiewicz inequality. To that we add a shadowing result for pseudo-orbits near normally hyperbolic manifold and careful estimates on the influence of other bodies on the cluster of colliding bodies.

No infinite spin for partial collisions converging to isolated central configurations on the plane

Abstract

In the -body problem, when a~cluster of bodies tends to a collision, then its normalized shape curve converges to the set of normalized central configurations, which has symmetry in the planar case. This leaves a possibility that the normalized shape curve tends to the circle obtained by rotation of some central configuration instead of a particular point on it. This is the \emph{infinite spin problem} which concerns the rotational behavior of total collision orbits in the -body problem. The question also makes sense for partial collision. We show that the infinite spin is not possible if the limiting circle is isolated from other connected components of the set of normalized central configurations. Our approach extends the method from recent work for total collision by Moeckel and Montgomery, which was based on a combination of the center manifold theorem with Łojasiewicz inequality. To that we add a shadowing result for pseudo-orbits near normally hyperbolic manifold and careful estimates on the influence of other bodies on the cluster of colliding bodies.
Paper Structure (21 sections, 34 theorems, 279 equations)

This paper contains 21 sections, 34 theorems, 279 equations.

Table of Contents

  1. Introduction
  2. Derivation of Moeckel and Montgomery's equations and McGehee regularization
  3. Equations of motion
  4. McGehee-like regularization
  5. Estimation for non-autonomous terms
  6. Convergence in the system after McGehee regularization
  7. Shadowing of perturbed orbits near normally hyperbolic manifolds
  8. Isolating segments
  9. General construction of a shadowing solution
  10. Shadowing partial collision orbit by an orbit of autonomous system \ref{["eq:r'-auto"]}-\ref{["eq:zw'-auto"]}
  11. No infinite spin
  12. Computation of $\dot{\omega}$
  13. Computation of $\frac{\partial \mathcal{L}}{\partial \omega}$
  14. Computation of $\frac{\partial \mathcal{L}}{\partial s}$
  15. Computation of $\frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \omega}$ and equation for $\dot{\omega}$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Suppose that $q(t)$ is a solution of $n$-body problem eq:second-order-formulation undergoing a partial collision as $t \to T$. Let $\mathcal{G}$ be a nontrivial cluster of colliding bodies. Suppose that its reduced and normalized configuration $\left[\hat{q}_\mathcal{G}(t) \right] \in S_{\#\mathcal{

Theorems & Definitions (68)

  • Definition 1.1
  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • proof
  • Remark 2.3: Angular momentum in various coordinates
  • Lemma 2.2
  • proof
  • Remark 2.4
  • ...and 58 more