where we have used the mapping of particle positions in Eq. ( ) and Hamilton's equations, ṙi = ∂H FG ∂p i and ṘI = ∂H CG ∂P I , applied to the FG and CG Hamiltonians in Eqs. ( ) and ( ), respectively, to relate the FG and CG linear momenta and velocities. The mapping to the angular momentum of CG particle I is