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

TL;DR

This paper provides a comprehensive analysis of the gravitational three-body problem focusing on Jupiter and Saturn's Great Inequality, using heliocentric coordinates, Lagrange multipliers, and perturbation theory to explore nearly integrable Hamiltonian dynamics.

Contribution

It introduces a novel formulation of the three-body problem using Lagrange multipliers in heliocentric coordinates and develops a perturbation framework based on Delaunay variables and KAM theory.

Findings

Formulated equations of motion in two dimensions using polar angles and angular momenta.

Expressed the three-body Hamiltonian in Delaunay variables, showing near integrability.

Outlined a perturbation approach with KAM theory for future analysis.

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.