Integrable systems on the sphere, ellipsoid and hyperboloid

Abstract

Affine transformations in Euclidean space generates a correspondence between integrable systems on cotangent bundles to the sphere, ellipsoid and hyperboloid embedded in R n . Using this correspondence and the suitable coupling constant transformations we can get real integrals of motion in the hyperboloid case starting with real integrals of motion in the sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic and sextic polynomials in momenta.

Non-Hermitian superintegrable systems

“…Finally, the present work provides further evidence for the conjecture that superintegrable models can be obtained from reductions of free particles in higher-dimensional spaces, it also holds for non-Hermitian Hamiltonians. In this sense, the existence of non-Hermitian integrable systems with higher-order conserved quantities and their relation to other reductions is also intriguing, see [65,72,73] and references therein.…”

“…Real superintegrable systems in the two-dimensional sphere s 2 1 + s 2 2 + s 2 3 = 1 have been widely studied in the past [14][15][16][17][18][19][20][21][22][23]65], in particular, a complete analysis of the symmetry reduction method to obtain them was done in [9]. They studied the set of conserved quantities together with the separability of the Hamilton-Jacobi equation.…”