The classical spin triangle as an integrable system

TL;DR

This paper explicitly solves the dynamics of a classical three-spin system with Heisenberg interaction, deriving action-angle variables and analyzing special cases, including time-dependent magnetic fields.

Contribution

It provides an explicit calculation of the time evolution and action-angle variables for the classical spin triangle, including special and limiting cases.

Findings

Explicit formulas for spin dynamics using elliptic functions

Validation through numerical comparison

Analysis of special and limit cases

Abstract

The classical spin system consisting of three spins with Heisenberg interaction is an example of a completely integrable mechanical system. In this paper we explicitly calculate its time evolution and the corresponding action-angle variables. This calculation is facilitated by splitting the six degrees of freedom into three internal and three external variables, such that the internal variables evolve autonomously. Their oscillations can be explicitly calculated in terms of the Weierstrass elliptic function. We test our results by means of an example and comparison with direct numerical integration. A couple of special cases is analyzed where the general theory does not apply, including the aperiodic limit case for special initial conditions. The extension to systems with a time-depending magnetic field in a constant direction is straightforward.