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.
Highlights
Integrable Hamiltonian systems are in d ≥ 2 degrees of freedom the exception rather than the rule
KAM theory is about integrable systems—under strong non-resonance conditions their quasi-periodic solutions persist under sufficiently small perturbations
In the minimally superintegrable case of d + 1 conserved quantities the perturbation analysis consists of three steps: compute a normal form that serves as intermediate system, study how the dynamics of the intermediate system structures the phase space (organized by the fast (d − 1)-tori, the relative equilibria) and check the necessary non-degeneracy conditions concerning both the superintegrable system and the reduced intermediate system
Summary
Integrable Hamiltonian systems are in d ≥ 2 degrees of freedom the exception rather than the rule. In the extreme case of a maximally superintegrable system one has to study the Hamiltonian system in d − 1 degrees of freedom that is obtained by averaging the perturbation along the unperturbed periodic orbits and reducing the acquired S1-symmetry. Normal forms provide the means to obtain intermediate systems, and the most obvious way is to normalize by averaging along the unperturbed dynamics In general such an intermediate system inherits n + 1 integrals of motion in involution and the perturbation is said to remove the degeneracy [3] in case of a (non-degenerate) integrable intermediate system.
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