Classical n-body system in volume variables II: Four-body case

Abstract

It is evident that the positions of four bodies in [Formula: see text]-dimensional space can be identified with vertices of a tetrahedron. Square of volume of the tetrahedron, weighted sum of squared areas of four facets and weighted sum of squared edges are called the volume variables. A family of translation-invariant potentials which depend on volume variables alone had been considered as well as solutions of the Newton equations which solely depend on volume variables. For the case of zero angular momentum [Formula: see text], the corresponding Hamiltonian, which describes these solutions, is derived. Three examples are studied in detail: (I) the (super)integrable four-body closed chain of harmonic oscillators for [Formula: see text] (the harmonic molecule); (II) a generic, two volume variable-dependent potential whose trajectories possess a constant moment of inertia ([Formula: see text]) and (III) the four-body anharmonic oscillator for [Formula: see text]. This work is the second of the sequel: the first one [A. M. Escobar-Ruiz, R. Linares, A. V. Turbiner and W. Miller Jr., Int. J. Mod. Phys. A 36, 2150140 (2021)] was dedicated to study the three-body classical problem in volume variables.