2023/07/10 by Jonathan Tot, Tot, Jonathan, S. R. Valluri +3
Engineering · Physics and Astronomy · #70F07 (Primary) 37J40 #Astro and Planetary Science #Classical Physics (physics.class-ph) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Spacecraft Dynamics and Control #Stellar, planetary, and galactic studies
paper · pdf · doi:10.48550/arxiv.2307.04810
openalex publication_date 2023/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ri, 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 θi, angular momenta ℓi and eccentricity vectors ei. The dynamical system is simplified considerably by change of variables to certain auxiliary vector fi=ri+ei. 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.