Classical $n$-body system in geometrical and volume variables. I. Three-body case

TL;DR

This paper introduces a novel set of geometrical and volume variables for analyzing the classical 3-body problem with arbitrary degrees of freedom, simplifying the kinetic energy expression and revealing symmetries, with applications to gravity, choreographies, and oscillators.

Contribution

It develops a new coordinate framework based on symmetric polynomial invariants that simplifies the kinetic energy and captures symmetries in the 3-body system across different dimensions.

Findings

Kinetic energy becomes a polynomial in the new variables.

The framework applies to gravity, choreographies, and oscillators.

Symmetries are explicitly incorporated into the variables.

Abstract

We consider the classical 3-body system with $d$ degrees of freedom $(d>1)$ at zero total angular momentum. The study is restricted to potentials $V$ that depend solely on relative (mutual) distances $r_{ij}=\mid {\bf r}_i - {\bf r}_j\mid$ between bodies. Following the proposal by J. L. Lagrange, in the center-of-mass frame we introduce the relative distances (complemented by angles) as generalized coordinates and show that the kinetic energy does not depend on $d$, confirming results by Murnaghan (1936) at $d=2$ and van Kampen-Wintner (1937) at $d=3$, where it corresponds to a 3D solid body. Realizing $\mathbb{Z}_2$-symmetry $(r_{ij} \rightarrow -r_{ij})$ we introduce new variables $\rho_{ij}=r_{ij}^2$, which allows us to make the tensor of inertia non-singular for binary collisions. In these variables the kinetic energy is a polynomial function in the $\rho$-phase space. The 3 body positions form a triangle (of interaction) and the kinetic energy is $\mathcal{S}_3$-permutationally invariant wrt interchange of body positions and masses (as well as wrt interchange of edges of the triangle and masses). For equal masses, we use lowest order symmetric polynomial invariants of $\mathbb{Z}_2^{\otimes3} \oplus \mathcal{S}_3$ to define new generalized coordinates, they are called the {\it geometrical variables}. Two of them of the lowest order (sum of squares of sides of triangle and square of the area) are called {\it volume variables}. We study three examples in some detail: (I) 3-body Newton gravity in $d=3$, (II) 3-body choreography in $d=2$ on the algebraic lemniscate by Fujiwara et al where the problem becomes one-dimensional in the geometrical variables, and (III) the (an)harmonic oscillator.