  This work discusses the motion of extended test bodies as described by the
Mathisson-Papapetrou-Dixon (MPD) equations in the pole-dipole-quadrupole
approximation. We focus on the case that the quadrupole is solely induced by
the spin of the body which is assumed to move on a circular equatorial orbit in
a Kerr background. To fix the center of mass of the MPD body we use two
different spin supplementary conditions (SSCs): the Tulczyjew-Dixon SSC and the
Mathisson-Pirani SSC. We provide the frequencies of the circular equatorial
orbits for the pole-dipole-(spin induced) quadrupole approximation of the body
for both SSCs. In the process we develop an explicit four-velocity
four-momentum relation for a pole-dipole-quadrupole body under the
Mathisson-Pirani SSC.

General Relativity and Quantum Cosmology