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

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 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 $(r \rightarrow -r)$ we introduce new variables $\rho=r^2$, which allows us to make the tensor of inertia non-singular. 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}. It is shown that for potentials, which depend on geometrical variables only (i) and those which depend on mass-dependent volume variables alone (ii), the Hamilton's equations of motion look amazingly simple. In the case (ii) all trajectories are mass-independent! We study three examples in some detail: (I) 3-body Newton gravity in $d=3$, (II) 3-body choreography on the algebraic lemniscate by Fujiwara et al, and (III) the (an)harmonic oscillator.