“Leonard Pairs” in Classical Mechanics

Ġ = {G,H}. In particular, the DV F is called an integral if it has zero PB with the Hamiltonian {F,H} = 0. In this case F does not depend on t. In many problems of the classical mechanics DV form elegant algebraic structures which are closed with respect to PB. The Poisson structures with non-linear PB were discussed in [9] and [6]. Sklyanin introduced [9] the so-called quadratic Poisson algebra consisting of 4 DV S0, S1, S2, S3 such that PB {Si, Sk} is expressed as a quadratic function of the generators Si. The Sklyanin algebra appears quite naturally from theory of algebraic structures related to the Yang–Baxter equation in mathematical physics. Sklyanin also proposed to study general non-linear Poisson structures. Assume that there exists N dynamical variables Fi, i = 1, 2, . . . , N such that PB of these variables are closed in frames of the non-linear relations