Contraction of superintegrable Hamiltonian systems (Q2737822)

The authors start from a class of maximally superintegrable (i.e.\ having \(2N-1\) independent integrals) Hamiltonian systems, associated with the groups \(\text{SO}(p,q), p+q=N+1\), which they have analysed in previous papers. The idea is to derive new superintegrable Hamiltonians from these by a process of contraction, meaning that the new systems are linked to a group obtained by an Inönü-Wigner contraction. Further attention is paid to an associated procedure of contraction of separable coordinates by which, for example, separable coordinates for the original superintegrable system turn into Cartesian or polar-type coordinates on a flat space after contraction. A unifying factor in this analysis is the use of the so-called Cayley-Klein scheme, which is explained in Section~2, along with the corresponding homogeneous spaces. Section~3 discusses the contraction of the Hamiltonians under consideration. Marsden-Weinstein reduction theory is used in Section~4, whereby specifically the reduction in affine coordinates is new. It is further shown that contraction and reduction in some sense commute. Section~5 focusses on the first integrals to show that the contracted systems are still superintegrable. The Hamilton-Jacobi equation is studied in detail in Section~6, while the final section contains a number of illustrative examples.