The Drach superintegrable systems

Abstract

Cubic invariants for two-dimensional degenerate Hamiltonian systems are considered by using variables of separation of the associated Stäckel problems with quadratic integrals of motion. For the superintegrable Stäckel systems the cubic invariant is shown to admit new algebro-geometric representation that is far more elementary than the all the known representations in physical variables. A complete list of all known systems on the plane which admit a cubic invariant is discussed.

Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion

“…[14][15][16][17][18][19] Much less is known about integrable and superintegrable systems with third-and higher-order integrals of motion. In 1935, Drach found ten different potentials in a complex Euclidian plane, allowing a third-order integral of motion in classical mechanics.…”

“…The third-order one is their Poisson commutator. [19][20][21][22] In 1984, Hietarinta 23 showed that in the case of integrals that are third-or higher-order polynomials in momenta, quantum and classically integrable potentials may not coincide.…”

Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold

“…In the general case of a superintegrable system the integrals are not necessarily quadratic functions of the momenta, but rather polynomial functions of the momenta. The case of the systems with a quadratic and a cubic integral of motion are studied by Tsiganov [31,32].…”

Untitled

“…As an example we consider natural Hamiltonians with pseudo-Euclidean metric [5] and reproduce one Drach system which is not superintegrable Stäckel system with quadratic integrals of motion [22]. Only for this Drach system the separated variables was unknown.…”

“…In this section we consider the normalized Poisson tensors 22) where P ′ has the form (3.11). Such tensors induce formal ωN structure on M, but their eigenvalues λ i will be a priori functionally dependent functions.…”