Note on constants of motion in conformal mechanics associated with near horizon extremal Myers-Perry black holes

摘要

We investigate dynamics of probe particles moving in the near-horizon limit of (2N + 1)-dimensional extremal Myers Perry black hole (in the cases of N = 3, 4, 5) with arbitrary rotation parameters. Very recently it has been shown in [T. Hakobyan, A. Nersessian and M. M. Sheikh-Jabbari, Phys. Lett. B 772, 586 (2017)] that in the most general cases with non-equal nonvanishing rotational parameters the system admits separation of variables in N-dimensional ellipsoidal coordinates. We wrote down the explicit expressions of Liouville integrals of motion, given in the above-mentioned reference in ellipsoidal coordinates, in initial "Cartesian" coordinates in seven, nine and eleven dimensions, and found that these expressions hold in any dimension. Then, taking the limit where all of the rotational parameters are equal, we reveal that each of these N - 1 integrals of motion results in the Hamiltonian of the spherical mechanics of a (2N + 1)-dimensional MP black hole with equal nonvanishing rotational parameters.