Linear $r$-matrix algebra for classical separable systems

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.