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The circular restricted three body problem, which considers the dynamics ofan infinitesimal particle in the presence of the gravitational interaction withtwo massive bodies moving on circular orbits about their common center of mass,is a very useful model for investigating the behavior of real astronomicalobjects in the Solar System. In such a system, there are five Lagrangianequilibrium points, and one important characteristic of the motion is theexistence of linearly stable equilibria at the two equilibrium points that formequilateral triangles with the primaries, in the plane of the primaries' orbit.We analyze the stability of motion in the restricted three body problem byusing the concept of Jacobi stability, as introduced and developed in theKosambi-Cartan-Chern (KCC) theory. The KCC theory is a differential geometricapproach to the variational equations describing the deviation of the wholetrajectory of a dynamical system with respect to the nearby ones. We obtain thegeneral result that, from the point of view of the KCC theory and of Jacobistability, all five Lagrangian equilibrium points of the restricted three bodyproblem are unstable.

Jacobi stability analysis of the classical restricted three body problem