  We theoretically investigate the rigid body dynamics of an optically
levitated nanodumbbell under parametric feedback cooling and provide a
simplified model for describing the motion. Differing from previous studies,
the spin of the nanoparticle about its symmetry axis is considered
non-negligible. Simulations reveal that standard parametric feedback cooling
can extract energy from two of the five rotational degrees of freedom when the
nanoparticle is levitated using a linearly polarized laser beam. The dynamics
after feedback cooling are characterized by a normal mode describing precession
about the laser polarization axis together with spin about the nanoparticle's
symmetry axis. Cooling the remaining mode requires an asymmetry in the two
librational frequencies associated with motion about the polarization axis as
well as information about the two frequencies of rotation about the
polarization axis. Introducing an asymmetric potential allows full cooling of
the librational coordinates if the frequencies of both are used in the feedback
modulation and is an avenue for entering the librational quantum regime. The
asymmetry in the potential needs to be large enough for practical cooling times
as the cooling rate of the system depends non-linearly on the degree of
asymmetry, a condition that is easily achieved experimentally.
