Superintegrable Hamiltonian Systems: Geometry and Perturbations

Abstract

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. Mathematics Subject Classifications (2000):

Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System

“…1 right gives a pictorial representation of this structure, see e.g. [7], where the individual flowers are the coisotropic leaves, diffeomorphic to G, the petals are the invariant tori T r , the centers of the flowers are the coadjoint orbits of g * , and the meadow is a Weyl chamber, namely, the 'action space'. The picture is inaccurate in that it suggests that both fibrations J and π A are topologically trivial, while it is not necessarily so.…”

Perturbations of Superintegrable Systems

“…Since the Poisson bracket of two conserved quantities is again an integral of motion all these form a Lie algebra, and as the symplectic structure is non-degenerate such a Lie algebra is necessarily noncommutative once the dimension exceeds d. One therefore also speaks of non-commutative integrabililty [34]. The following result from [23] clarifies the semi-local situation (i.e. locally around a maximal torus).…”

Some ring-shaped potentials as a generalized 4-D isotropic oscillator. Periodic orbits

“…For a general view on this the reader may consult Fassò (2005) and references therein. Here, having established the connection among the 3-D and 4-D models Ferrer (2006), we focus on conditions for periodic orbits.…”