The linear stability of Lagrange points in polar coordinates

Abstract

In a system of two primary gravitational bodies, there are five equilibrium points known as Lagrange points. The Lagrange points exhibit widely different stability properties, such that a satellite placed at the Sun–Earth L1 or L2 points needs course corrections every few weeks, whereas Jupiter's Trojan asteroids are naturally stable on timescales of the order of 109 years. We analyze the stability of all Lagrange points in the circular restricted three-body problem using a matrix formulation expressed in polar coordinates. We examine the eigenvalues and eigenvectors, both of which are important for a full understanding of satellite motion near these points. Trajectories in the linearized polar representation exhibit distortions absent in the linearized Cartesian analysis, a result of geometric curvature effects that are not retained in the Cartesian formulation at first order. We include a set of questions and answers to support classroom lessons for the development of intuition about motion near Lagrange points. A set of supplementary notes provides supporting mathematical details of derivations and calculations described in the main text.