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

TL;DR

The paper introduces a natural criterion for decoupling the Jacobi equation in certain N-body problems, enabling simplified analysis of linear stability and extending known decompositions to new classes.

Contribution

It provides a simple, natural splitting criterion for the Jacobi equation applicable to various N-body problems, including homographic motions and isosceles three-body problems.

Findings

Applied to classical three-body problem, proved elliptic Lagrange solutions are linearly unstable under certain mass conditions.

Extended the Meyer-Schmidt decomposition to broader classes of N-body problems.

Provided a short proof of a stability theorem for elliptic Lagrange solutions.

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 $\mu=(m_1+m_2+m_3)^2/(m_1m_2+m_2m_3+m_1m_3)<27/8$ then the elliptic Lagrange solutions are linearly unstable for any value of the excentricity.