where we have applied Hamilton's equation, ω I = ∂H CG ∂L I , for the CG angular velocity ω I to Eq. ( ) and used the fact that the mapping of particle orientations defined by Eq. (10) requires the mapped FG angular velocity to equal the CG angular velocity. This identifies I I I FG,I ω I as the total angular momentum of the FG particles with respect to their center-ofmass, i∈ζ I ∆r i × p i , given the definition of the CG orientation Ω I in terms of the principal axes of inertia of the group of FG particles that are mapped onto the CG particle. The mapping M M M LI (p n , r n ) for the angular momentum depends on the FG positions due to the dependence of the angular momentum on FG particle positions with respect to the centerof-mass.