Polynomial integrable systems

Abstract:

Introduction to theory of finite-dimensional integrable systems is briefly given. Notion of polynomial integrable system is introduced.

It is stated that

(i) any Calogero-Moser system is canonically-equivalent to a polynomial integrable system, its Hamiltonian and integrals are polynomials in momenta p and coordinates q.

(ii) for any Calogero-Moser model there exists a change of variables in which the potential in a rational function.

(iii) any Calogero-Moser model is equivalent to Euler-Arnold top in a constant magnetic field (gyroscope) with non-compact algebra gl(n,R) (for a classical Weyl group) with constant Casimir operators as a constraint.

(iv) A solution of celebrated 3-body elliptic Calogero model is presented in detail as an example.