Linear $r$-matrix algebra for classical separable systems

TL;DR

This paper develops a Lax representation and an r-matrix algebra for a hierarchy of classical integrable systems with polynomial potentials, providing explicit separation of variables and discussing quantization.

Contribution

It introduces a novel Lax representation with a variable-dependent r-matrix for separable Hamiltonian systems, advancing integrability and quantization methods.

Findings

Constructed a Lax pair for the hierarchy of systems.

Derived a dynamical r-matrix satisfying a Yang-Baxter equation.

Provided explicit separation equations and action variables.

Abstract

We consider a hierarchy of the natural type Hamiltonian systems of $n$ degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of $2\times 2$ matrices for the whole hierarchy and construct the associated linear $r$-matrix algebra with the $r$-matrix dependent on the dynamical variables. A Yang-Baxter equation of dynamical type is proposed. Using the method of variable separation we provide the integration of the systems in classical mechanics conctructing the separation equations and, hence, the explicit form of action variables. The quantisation problem is discussed with the help of the separation variables.