Super-integrable Calogero-type systems admit maximal number of Poisson structures

Abstract

We present a general scheme for constructing the Poisson structure of superintegrable dynamical systems of which the rational Calogero-Moser system is the most interesting one. This dynamical system is 2N dimensional with 2N − 1 first integrals and our construction yields 2N − 1 degenerate Poisson tensors that each admit 2(N − 1) Casimirs. Our results are quite generally applicable to all super-integrable systems and form an alternative to the traditional bi-Hamiltonian approach.

Superintegrable Systems, Multi-Hamiltonian Structures and Nambu Mechanics in an Arbitrary Dimension

“…In this context, our geometric proofs of Jacobi identity and of compatibility condition for alternative Poisson structures are different but validate and, in a sense, are complementary to that of Ref. [9].…”

“…The Calogero-Moser system is one of the four nD systems which are known to be maximally superintegrable for any finite integer n [9,14,16]. The other three systems are the Kepler-Coulomb problem, harmonic oscillator with rational frequency ratios, and Winternitz system.…”

“…Then, the equations of Nahm system in the theory of static SU(2) monopoles were realized in this formulation [2,6]. Connection between Nambu mechanics and so-called superintegrable systems has been the subject of some recent studies [7,8,9,10].…”

Applications of Nambu Mechanics to Systems of Hydrodynamical Type II

“…Among the notable examples are the SU (n) isotropic harmonic oscillator and the SO(4) Kepler problem. Most recently Gonera and Nutku [22] showed that the Nambu structure can be extracted also from the rational Calogero-Moser system [23].…”

Oscillatory nonequilibrium Nambu systems: the canonical-dissipative Yamaleev oscillator

“…Certain electrodynamic [5,[11][12][13][14][15] and biochemical [16,17] problems have been re-formulated using the multi-Hamiltonian approach of Nambu mechanics. Multi-Hamiltonian oscillators generalizing the harmonic oscillator [18,19], chiral models [8] and the Calogero-Moser system [20,21] have been studied in the context of Nambu mechanics. While one key benefit of Nambu mechanics is that it provides us with a principled way to construct evolution equations from invariants of motion, this benefit also limits the applicability of Nambu's multi-Hamiltonian approach to systems that features such invariants.…”