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

We consider the classical three-body system with [Formula: see text] degrees of freedom at zero total angular momentum. The study is restricted to potentials that depend solely on relative (mutual) distances between bodies. Following proposal by J. L. Lagrange, in center-of-mass frame we introduce (complemented angles) as generalized coordinates and show kinetic energy does not text], confirming results Murnaghan (1936) van Kampen–Wintner (1937) where it corresponds a 3D solid body. Realizing text]-symmetry new variables which allows us make tensor inertia nonsingular for binary collisions. In these polynomial function text]-phase space. positions form triangle (of interaction) text]-permutationally invariant respect interchange body masses (as well edges masses). For equal masses, use lowest order symmetric invariants define coordinates, they are called geometrical variables. Two them (sum squares sides square area) volume It shown only (i) those mass-dependent alone (ii), Hamilton’s equations motion can be considered being relatively simple. three examples some detail: (I) Newton gravity (II) choreography algebraic lemniscate Fujiwara et al., problem becomes one-dimensional (III) (an)harmonic oscillator.