The stability of the Lagrange triangular point and a theorem of Arnold

Abstract

Deprit and Deprit-Bartholome [5] computed the normal form for the Hamiltonian of the planar restricted three body problem at the Lagrange equilateral triangle equilibrium point up to terms of order 4. This normalization was carried out for all mass ratios p smaller than Routh’s critical mass ratio p1 = i( 1 - J&/9) except for two values p2 and Pi. At p2 and ,u~ the ratio of the linearized frequencies is 1: 2 and 1 : 3, respectively, and so low order resonance terms appear. This normalization was carried out in order to apply the KAM theory to prove the stability of this equilibrium point. From the coefficients of the normal form they computed a quantity D, (defined below) which when non-zero establishes the stability of the equilibrium point. They found that D, # 0 for 0 -CP < p,, ,u # p2, p,and p(,. The value ,u, does not correspond to resonance but is simply a point where the quantity D4 changes sign. In this paper we shall establish the full stability of the Lagrange equilibrium point in the planar restricted three body problem even in the case when ,U = P,. We do this by computing by machine the normal form up to terms of order 6 and then applying a theorem of Arnold [3] which establishes the stability of the equilibrium even in degenerate cases. Russman [lo] announced a theorem which implies that the equilibrium is isoenergetic stable when p = p,. By “isoenergetic stable” he means that the system is stabie only on the energy surface defined by the energy at the equilibrium itself. Unfortunately there is not even an indication of the proof. Arnold announced his theorem in a Doklady note with only a sketch of a proof. The sketch indicates that his proof requires completely redoing all