Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariant manifolds of degenerate tori and double parabolic orbits to infinity in the (n+2)-body problem

Immaculada Baldomá, Ernest Fontich, Pau Martín

Abstract

There are many interesting dynamical systems in which degenerate invariant tori appear. We give conditions under which these degenerate tori have stable and unstable invariant manifolds, with stable and unstable directions having arbitrary finite dimension. The setting in which the dimension is larger than one was not previously considered and is technically more involved because in such case the invariant manifolds do not have, in general, polynomial approximations. As an example, we apply our theorem to prove that there are motions in the $(n+2)$-body problem in which the distances among the first $n$ bodies remain bounded for all time, while the relative distances between the first $n$-bodies and the last two and the distances between the last bodies tend to infinity, when time goes to infinity. Moreover, we prove that the final motion of the first $n$ bodies corresponds to a KAM torus of the $n$-body problem.

Invariant manifolds of degenerate tori and double parabolic orbits to infinity in the (n+2)-body problem

Abstract

There are many interesting dynamical systems in which degenerate invariant tori appear. We give conditions under which these degenerate tori have stable and unstable invariant manifolds, with stable and unstable directions having arbitrary finite dimension. The setting in which the dimension is larger than one was not previously considered and is technically more involved because in such case the invariant manifolds do not have, in general, polynomial approximations. As an example, we apply our theorem to prove that there are motions in the -body problem in which the distances among the first bodies remain bounded for all time, while the relative distances between the first -bodies and the last two and the distances between the last bodies tend to infinity, when time goes to infinity. Moreover, we prove that the final motion of the first bodies corresponds to a KAM torus of the -body problem.
Paper Structure (43 sections, 32 theorems, 463 equations, 1 figure)

This paper contains 43 sections, 32 theorems, 463 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Parabolic invariant tori with stable and unstable invariant manifolds
  3. Degenerate tori and homogeneous functions
  4. Double parabolic orbits to infinity in the $(n+2)$-body problem
  5. Structure of the paper
  6. Invariant manifolds of normally parabolic invariant tori
  7. Notation and a small divisors lemma
  8. Notation
  9. Diophantine vectors and small divisors lemmas
  10. Results for maps
  11. Set up and hypotheses
  12. Main results
  13. Consequences of Theorems \ref{['thm:posterioriresult']} and \ref{['thm:approximationmaps']} for maps
  14. A conjugation result for attracting parabolic tori
  15. The case when all eigenvalues of the linearization of the transversal dynamics to the torus are roots of 1
  16. ...and 28 more sections

Key Result

Theorem 1.1

If $m_n$ and $m_{n+1}$ are small enough but both different from $0$, with the smallness condition only depending on $M_0 = \sum_{k=0}^{n-1} m_k$, the following holds.

Figures (1)

  • Figure 1: Tending to collinear and equilateral configurations.

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7: A posteriori result
  • Theorem 2.8: Construction of the approximations
  • Theorem 2.9: Existence of the stable manifold
  • ...and 54 more