Normalization of the Levi-Civita Hamiltonian at a collinear Lagrange point

Abstract

The normalizations of the Hamiltonian of the circular restricted three-body problem at a collinear Lagrange equilibrium are used to compute approximations of its center and tube manifolds, as well as of all the dynamics in their neighbourhood. For small values of the reduced mass m the radius of convergence of any (even partial) normalization at L1,L2 is affected by the complex singularities of the gravitational potential energy of the closest primary body, i.e. the one with smallest mass. In this paper we investigate if regularizations with respect to the body of mass m improve the convergence of the normalizations. In particular, we consider the Hamiltonian describing the planar three-body problem in Levi-Civita regularizing variables, and we show that for a suitable interval of the value of the Hamiltonian larger than the value at the Lagrange equilibrium, the Levi-Civita Hamiltonian has a fictitious center-saddle equilibrium at which the Hamiltonian can be normalized. We find that, for a sample value of m corresponding to the Sun-Jupiter mass ratio, the normalized regularized Hamiltonian provides approximation of the center and of the tube manifolds of L1 up to an energy corresponding to a Lyapunov orbit of amplitude which is larger than the distance |1-m-xL1| of L1 from a complex singularity.