Superintegrable Hamiltonian Systems: Geometry and Perturbations

Abstract Many and important integrable Hamiltonian systems are ‘superintegrable’, in the sense that there is an open subset of their 2d-dimensional phase space in which all motions are linear on tori of dimension n<d. A thorough comprehension of these systems requires a description which goes beyond the standard notion of Liouville–Arnold integrability, that is, the existence of an invariant fibration by Lagrangian tori. Instead, the natural object to look at is formed by both the fibration by the (isotropic) invariant tori and by its (coisotropic) polar foliation, which together form what in symplectic geometry is called a ‘dual pair’, or ‘bifoliation’, or ‘bifibration’. We review this geometric structure, relating it to the dynamical properties of superintegrable systems and pointing out its importance for a thorough understanding of these systems.

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