Solvability in classical mechanics and algebraic Heun observables

TL;DR

This paper constructs a classical analog of the algebraic Heun operator linked to bispectral pairs, revealing that the dynamics governed by this operator involve elliptic functions, with concrete examples illustrating the general property.

Contribution

It introduces a classical analog of the algebraic Heun operator for bispectral pairs and shows its Hamiltonian dynamics involve elliptic functions, supported by explicit examples.

Findings

Classical algebraic Heun operator constructed for bispectral pairs

Dynamics governed by the operator involve elliptic functions

Concrete examples demonstrate the general property

Abstract

We construct a classical analog of the algebraic Heun operator $W$ associated to any bispectral pair of the operators $X$ and $Y$. We show that the dynamics of the classical variables $X$ or $Y$ is governed by elliptic functions if the classical $W$ is taken as the Hamiltonian. We demonstrate this general property by concrete examples of classical dynamical systems.