Poisson Structures for Aristotelian Model of Three Body Motion

TL;DR

This paper constructs explicit Poisson structures for a three-body dynamical system, extending previous results to include new constants of motion and analyzing special interaction cases.

Contribution

It introduces explicit Poisson structures for both time-dependent and independent Hamiltonians of the three-body system, including new constants of motion and special interaction scenarios.

Findings

Derived explicit Poisson structures for the system.

Identified new constants of motion incorporating all parameters.

Analyzed special cases with non-interacting pairs of bodies.

Abstract

We present explicitly Poisson structures, for both time-dependent and time-independent Hamiltonians, of a dynamical system with three degrees of freedom introduced and studied by Calogero et al [2005]. For the time-independent case, new constant of motion includes all parameters of the system. This extends the result of Calogero et al [2009] for semi-symmetrical motion. We also discuss the case of three bodies two of which are not interacting with each other but are coupled with the interaction of third one.