English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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

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