Classical $n$-body system in volume variables. II. Four-body case

TL;DR

This paper explores four-body classical systems in higher-dimensional spaces using volume variables, deriving Hamiltonians and analyzing specific integrable and anharmonic cases, extending previous three-body work.

Contribution

It introduces a volume-variable framework for four-body systems, derives Hamiltonians for zero angular momentum, and studies specific integrable and anharmonic examples.

Findings

Hamiltonian for four-body systems with zero angular momentum derived.

Identified integrable and anharmonic four-body solutions in volume variables.

Extended volume-variable approach from three-body to four-body systems.

Abstract

It is evident that the positions of 4 bodies in $d>2$ 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 is considered as well as solutions of the Newton equations which solely depend on volume variables. For the case of zero angular momentum $L=0$ the corresponding Hamiltonian, which describes these solutions, is derived. Three examples are studied in detail: (I) the (super)integrable 4-body closed chain of harmonic oscillators for $d>2$ (the harmonic molecule), (II) a generic, two volume variable dependent potential whose trajectories possess a constant moment of inertia ($d>1$), and (III) the 4-body anharmonic oscillator for $d \geq 1$. This work is the second of the sequel: the first one [IJMPA 36, No. 18 (2021)] was dedicated to study the 3-body classical problem in volume variables.