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

## TL;DR

This paper studies the motion of particles near extremal Myers-Perry black holes in higher dimensions, revealing how integrals of motion simplify in the equal rotation limit and relate to spherical mechanics.

## Contribution

It explicitly expresses Liouville integrals of motion in Cartesian coordinates for 7, 9, and 11 dimensions and connects these to spherical mechanics in the equal rotation case.

## Key findings

- Liouville integrals expressed in Cartesian coordinates for specific dimensions.
- Integrals reduce to spherical mechanics Hamiltonians in the equal rotation limit.
- Results hold for arbitrary rotation parameters in the general case.

## Abstract

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 arXiv:1703.00713v1 [hep-th] that in the most general case with nonequal 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 arXiv:1703.00713v1 [hep-th] 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.