标题： 一种针对某些类$N$-体问题的雅可比方程的自然分解 显示英文标题

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

摘要： 我们考虑几个$N$-体问题。 主要结果是对于这些类别的问题解耦雅可比方程的一个非常简单和自然的准则。 如果$E$是一个欧几里得空间，且 $N$体问题的势函数 $U(x)$是定义在 $E^N$中一个开子集上的 $C^2$函数，那么给定运动 $x(t)$沿着雅可比方程写为 $\ddot J=HU_x(J)$，其中 $E^N$的自同态 $HU_x$表示势函数关于质量内积的二阶导数。 我们的分解特别适用于由中心构型引起的分式运动的情况。它允许我们推导出在相空间中欧拉-拉格朗日流的线性化已知的Meyer-Schmidt分解，该分解二十年前提出用于研究平面$N$体问题的相对平衡。然而，我们的分解原理适用于许多其他类型的$N$体问题，例如等腰三体问题，在这个问题中Sitnikov证明了振荡运动的存在性。作为第一个具体应用，对于经典的三体问题，我们给出了Y. Ou定理的一个简单而简短的证明，该定理保证如果质量满足$\mu=(m_1+m_2+m_3)^2/(m_1m_2+m_2m_3+m_1m_3)<27/8$，则对于偏心率的任何值，椭圆拉格朗日解都是线性不稳定的。 显示英文摘要

摘要： 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.