Great Inequality of Jupiter and Saturn I: The Planetary Three Body Problem, Heliocentric development by Lagrange multipliers, Perturbation Theory Formulation

Abstract

In this paper, we undertake to present a self-contained and thorough analysis of the gravitational three body problem, with anticipated application to the Great Inequality of Jupiter and Saturn. The analysis of the three body Lagrangian is very convenient in heliocentric coordinates with Lagrange multipliers, the coordinates being the vector-sides $\vec{r}_i,\,i=1,2,3$ of the triangle that the bodies form. In two dimensions to begin with, the equations of motion are formulated into a dynamical system for the polar angles $\theta_i$, angular momenta $\ell_i$ and eccentricity vectors $\vec{e}_i$. The dynamical system is simplified considerably by change of variables to certain auxiliary vector $\vec{f}_i=\hat{r}_i+\vec{e}_i$. We then begin to formulate the Hamiltonian perturbation theory of the problem, now in three dimensions. We first give the geometric definitions for the Delaunay action-angle variables of the two body problem. We express the three body Hamiltonian in terms of Delaunay variables in each sector $i=1,2,3$, revealing that it is a nearly integrable Hamiltonian. We then present the KAM theory perturbative approach that will be followed in future work, including the modification that will be required because the Hamiltonian is degenerate.