Solvable and/or integrable many-body models on a circle

TL;DR

This paper discusses various many-body models confined to a circle, highlighting their integrability and solvability, including some new models and reinterpretations of existing ones, using algebraic and analytical techniques.

Contribution

It introduces new and reinterpret existing many-body models on a circle that are integrable and solvable through algebraic and analytical methods.

Findings

Models are integrable with explicit constants of motion.

Some models are solvable by algebraic operations and quadratures.

The paper presents both new models and reinterpretations of known models.

Abstract

Various many-body models are treated, which describe $N$ points confined to move on a plane circle. Their Newtonian equations of motion ("accelerations equal forces") are integrable, i. e. they allow the explicit exhibition of $N$ constants of motion in terms of the dependent variables and their time-derivatives. Some of these models are moreover solvable by purely algebraic operations, by (explicitly performable) quadratures and, finally, by functional inversions. The techniques to manufacture these models are not new; some of these models are themselves new; others are reinterpretations of known models.