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A natural decomposition of the Jacobi equation for some classes of $N$-body problems

Renato Iturriaga, Ezequiel Maderna

TL;DR

This work develops a natural Hessian-based decomposition of the Jacobi equation for certain $N$-body problems under the mass inner product, enabling a decoupling into invariant subspaces. A central splitting lemma shows that Jacobi fields split along invariant subspaces, leading to a Meyer-Schmidt-type decomposition for planar central configurations and their homographic motions. The paper derives a simple analytic instability criterion for elliptic Lagrange solutions, matching Ou's result via a direct hyperbolicity argument on the essential part of the linearized flow. It further analyzes the orthogonal complement to equilateral configurations in the 3-body problem using a complex mass inner product to obtain explicit Hessian expressions and a signature condition that reproduces the $oxed{ ext{mu}<27/8}$ threshold.

Abstract

We consider several $N$-body problems. The main result is a very simple and natural criterion for decoupling the Jacobi equation for some classes of them. If $E$ is a Euclidean space, and the potential function $U(x)$ for the $N$-body problem is a $C^2$ function defined in an open subset of $E^N$, then the Jacobi equation along a given motion $x(t)$ writes $\ddot J=HU_x(J)$, where the endomorphism $HU_x$ of $E^N$ represents the second derivative of the potential with respect to the mass inner product. Our splitting in particular applies to the case of homographic motions by central configurations. It allows then to deduce the well known Meyer-Schmidt decomposition for the linearization of the Euler-Lagrange flow in the phase space, formulated twenty years ago to study the relative equilibria of the planar $N$-body problem. However, our decomposition principle applies in many other classes of $N$-body problems, for instance to the case of isosceles three body problem, in which Sitnikov proved the existence of oscillatory motions. As a first concrete application, for the classical three-body problem we give a simple and short proof of a theorem of Y. Ou, ensuring that if the masses verify $μ=(m_1+m_2+m_3)^2/(m_1m_2+m_2m_3+m_1m_3)<27/8$ then the elliptic Lagrange solutions are linearly unstable for any value of the excentricity.

A natural decomposition of the Jacobi equation for some classes of $N$-body problems

TL;DR

This work develops a natural Hessian-based decomposition of the Jacobi equation for certain -body problems under the mass inner product, enabling a decoupling into invariant subspaces. A central splitting lemma shows that Jacobi fields split along invariant subspaces, leading to a Meyer-Schmidt-type decomposition for planar central configurations and their homographic motions. The paper derives a simple analytic instability criterion for elliptic Lagrange solutions, matching Ou's result via a direct hyperbolicity argument on the essential part of the linearized flow. It further analyzes the orthogonal complement to equilateral configurations in the 3-body problem using a complex mass inner product to obtain explicit Hessian expressions and a signature condition that reproduces the threshold.

Abstract

We consider several -body problems. The main result is a very simple and natural criterion for decoupling the Jacobi equation for some classes of them. If is a Euclidean space, and the potential function for the -body problem is a function defined in an open subset of , then the Jacobi equation along a given motion writes , where the endomorphism of represents the second derivative of the potential with respect to the mass inner product. Our splitting in particular applies to the case of homographic motions by central configurations. It allows then to deduce the well known Meyer-Schmidt decomposition for the linearization of the Euler-Lagrange flow in the phase space, formulated twenty years ago to study the relative equilibria of the planar -body problem. However, our decomposition principle applies in many other classes of -body problems, for instance to the case of isosceles three body problem, in which Sitnikov proved the existence of oscillatory motions. As a first concrete application, for the classical three-body problem we give a simple and short proof of a theorem of Y. Ou, ensuring that if the masses verify then the elliptic Lagrange solutions are linearly unstable for any value of the excentricity.

Paper Structure

This paper contains 18 sections, 15 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The mass inner product and the Hessian endomorphism
  3. Reduction of the center of mass
  4. The second derivative and the Hessian
  5. The Jacobi equation
  6. The splitting lemma for the Jacobi equation
  7. Applications to the N-body problem
  8. The Meyer-Schmidt decomposition
  9. Instability of elliptic homographic motions
  10. Proof of Theorem \ref{['thm:StrongND->unstable']}
  11. On the Theorem of Hu and Ou
  12. The orthogonal to the space of equilateral triangles
  13. The complex mass inner product
  14. The space of positive and centered equilateral triangles
  15. The Hessian operator
  16. ...and 3 more sections

Key Result

Theorem 1

If $\mu<27/8$ then the essential part of the Meyer-Schmidt decomposition of any elliptic Lagrangian solution is hyperbolic.

Figures (1)

  • Figure 1: In red is the unit circle, along with the three cube roots of unity. In light blue is the negatively oriented and centered triangle $S=(r_1,r_2,r_3)$ which is orthogonal, for the mass inner product, to all positive equilateral triangles.

Theorems & Definitions (35)

  • Theorem : Ou Ou, 2014
  • Lemma 1.1
  • Definition 1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem : Hu and Ou HuOu, 2016
  • Definition 2: The mass inner product
  • Definition 3: Configurations of total collision
  • Definition 4: Centered configurations
  • Definition 5: The Hessian for the mass inner product
  • ...and 25 more