Linear $r$-matrix algebra for classical separable systems

Details Linear $r$-matrix algebra for classical separable systems Linear $r$-matrix algebra for classical separable systems

1993-06-29

arxiv.org/abs/hep-th/9306155v1

J.Phys. A27 (1994) 567-578

Physics

High Energy Physics

High Energy Physics - Theory

15 pages

Scientific paper

10.1088/0305-4470/27/2/038

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.

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