  The circular restricted three body problem, which considers the dynamics of
an infinitesimal particle in the presence of the gravitational interaction with
two massive bodies moving on circular orbits about their common center of mass,
is a very useful model for investigating the behavior of real astronomical
objects in the Solar System. In such a system, there are five Lagrangian
equilibrium points, and one important characteristic of the motion is the
existence of linearly stable equilibria at the two equilibrium points that form
equilateral triangles with the primaries, in the plane of the primaries' orbit.
We analyze the stability of motion in the restricted three body problem by
using the concept of Jacobi stability, as introduced and developed in the
Kosambi-Cartan-Chern (KCC) theory. The KCC theory is a differential geometric
approach to the variational equations describing the deviation of the whole
trajectory of a dynamical system with respect to the nearby ones. We obtain the
general result that, from the point of view of the KCC theory and of Jacobi
stability, all five Lagrangian equilibrium points of the restricted three body
problem are unstable.
