Superintegrable systems on 3 dimensional conformally flat spaces

Abstract

We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional quadratic first integrals, thus constructing a large class of superintegrable systems and the complete Poisson algebra of first integrals. We then use the isometries to reduce our systems to 2 degrees of freedom. For each isometry algebra we give a universal reduction of the corresponding general Hamiltonian. The superintegrable specialisations reduce, in this way, to systems of Darboux-Koenigs type, whose integrals are reductions of those of the 3 dimensional system. 1 2 n(n + 1) = 6 Killing vectors (when n = 3).The conformal algebra of this 3D metric has dimension 1 2 (n + 1)(n + 2) = 10 (when n = 3). A convenient

Adding potentials to superintegrable systems with symmetry

“…In [2], we considered subalgebras of g, having dimensions 2, 3 and 4. We showed that there were initially 15 inequivalent subalgebras of interest, but then reduced this to a list of seven subalgebras, given in table 3.…”

“…In our previous papers [1,2], we used the method introduced in [3] to construct kinetic energies , related to three-dimensional conformally flat spaces and having four or five independent first integrals. In both these papers, it is evident that an isometry group for the corresponding metric simplified the construction of a closed Poisson algebra of integrals.…”

“…Clearly, imposing more symmetries restricts the potential. In [2], we considered isometry algebras of dimension 2, 3 and 4, since there is a ‘gap phenomenon’ [4] which tells us that five symmetries imply six symmetries, in which case the space has constant curvature, which we wished to avoid.…”

“…In §2., we give our general notation regarding the 10-dimensional conformal algebra, used throughout. Section 3. gives a summary of important results from [2] and an outline of the general approach used in this paper. Following [2], we then present our results for individual isometry algebras, in sequence.…”

Quasi-bi-Hamiltonian structures and superintegrability: Study of a Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion

“…Second, it is well known that the two more important superintegrable systems are the harmonic oscillator and the Kepler problem and that, associated to them, there are four families of potentials with separability in two different coordinate systems in the Euclidean plane and that they are, therefore, superintegrable with quadratic in the momenta constants of motion (first studied in [20] and then by other authors as e.g. [17,19,22,23,28,38,35,40,42]). Two of them are related with the harmonic oscillator and they are not considered in this paper.…”