Perturbations of Superintegrable Systems

Abstract

A superintegrable system has more integrals of motion than degrees d of freedom. The quasi-periodic motions then spin around tori of dimension n < d. Already under integrable perturbations almost all n-tori will break up; in the non-degenerate case the resulting d-tori have n fast and d − n slow frequencies. Such d-parameter families of d-tori do survive Hamiltonian perturbations as Cantor families of d-tori. A perturbation of a superintegrable system that admits a better approximation by a non-degenerate integrable perturbation of the superintegrable system is said to remove the degeneracy. In the minimal case d = n + 1 this can be achieved by means of averaging, but the more integrals of motion the superintegrable system admits the more difficult becomes the perturbation analysis.

Hamiltonian and reversible systems with smooth families of invariant tori

“…However, one cannot call the system (9) superintegrable for l > k ≥ 1 and d = s + 2l. Besides the existence of M > N independent first integrals, the definition of a superintegrable Hamiltonian system with N degrees of freedom (see the papers [21,27,34] and references therein; superintegrable systems are also known as properly degenerate or non-commutatively integrable systems) includes other requirements, for instance, that there be N integrals pairwise in involution among the M integrals under consideration (we have only s + l < N integrals in involution, e.g., u 1 , . .…”

Singular reduction of resonant Hamiltonians

“…Putting in (9) I 1 , I 2 in terms of J and h (using (26) and the identity h = q I 1 + p I 2 ) and writing θ 1 , θ 2 as functions of ψ the invariants a j 's are given by…”

“…it has more than n independent integrals [19,20,37]. Actually our system has 2n − 1 independent integrals, thus it is called maximally super-integrable [25,26]. In this latter case one has to study the reduced Hamiltonian system with n − 1 degrees of freedom obtained after averaging a given perturbation along the periodic solutions of the unperturbed system [53,61], then reducing by the acquired S 1 -symmetry.…”