On the Normal Form of the Hamiltonian in the Vicinity of Lagrange Points in the Restricted Elliptical Three-Body Problem

The planar restricted problem of three bodies moving under gravitational attraction is considered. The motions close to the Lagrange libration points are studied. The orbital eccentricity of the smaller of the two main attracting bodies and their mass ratio are chosen as problem parameters. A linear canonical transformation that is $2\pi$-periodic in true anomaly is obtained analytically up to the 4th degree of eccentricity inclusive, which reduces the Hamiltonian function of the linearized equations of perturbed motion to a real normal form corresponding to two harmonic oscillators independent of each other. The oscillations frequencies and the coefficients in the transformation are obtained explicitly in terms of the problem parameters.