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

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