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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 $V$ thatdepend solely on relative (mutual) distances $r_{ij}=\mid {\bf r}_i - {\bfr}_j\mid$ between bodies. Following the proposal by J. L. Lagrange, in thecenter-of-mass frame we introduce the relative distances (complemented byangles) as generalized coordinates and show that the kinetic energy does notdepend on $d$, confirming results by Murnaghan (1936) at $d=2$ and vanKampen-Wintner (1937) at $d=3$, where it corresponds to a 3D solid body.Realizing $\mathbb{Z}_2$-symmetry $(r_{ij} \rightarrow -r_{ij})$ we introducenew variables $\rho_{ij}=r_{ij}^2$, which allows us to make the tensor ofinertia non-singular for binary collisions. In these variables the kineticenergy is a polynomial function in the $\rho$-phase space. The 3 body positionsform a triangle (of interaction) and the kinetic energy is$\mathcal{S}_3$-permutationally invariant wrt interchange of body positions andmasses (as well as wrt interchange of edges of the triangle and masses). Forequal masses, we use lowest order symmetric polynomial invariants of$\mathbb{Z}_2^{\otimes3} \oplus \mathcal{S}_3$ to define new generalizedcoordinates, they are called the {\it geometrical variables}. Two of them ofthe 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 thealgebraic lemniscate by Fujiwara et al where the problem becomesone-dimensional in the geometrical variables, and (III) the (an)harmonicoscillator.

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