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After the separation of the center-of-mass motion, a new privileged class ofcanonical Darboux bases is proposed for the non-relativistic N-body problem byexploiting a geometrical and group theoretical approach to the definition of{\it body frame} for deformable bodies. This basis is adapted to the rotationgroup SO(3), whose canonical realization is associated with a symmetryHamiltonian {\it left action}. The analysis of the SO(3) coadjoint orbitscontained in the N-body phase space implies the existence of a {\it spin frame}for the N-body system. Then, the existence of appropriate non-symmetryHamiltonian {\it right actions} for non-rigid systems leads to the constructionof a N-dependent discrete number of {\it dynamical body frames} for the N-bodysystem, hence to the associated notions of {\it dynamical} and {\it measurable}orientation and shape variables, angular velocity, rotational and vibrationalconfigurations. For N=3 the dynamical body frame turns out to be unique and ourapproach reproduces the {\it xxzz gauge} of the gauge theory associated withthe {\it orientation-shape} SO(3) principal bundle approach of Littlejohn andReinsch. For $N \geq 4$ our description is different, since the dynamical bodyframes turn out to be {\it momentum dependent}. The resulting Darboux bases for$N\geq 4$ are connected to the coupling of the {\it spins} of particle clustersrather than the coupling of the {\it centers of mass} (based on Jacobi relativenormal coordinates). One of the advantages of the spin coupling is that, unlikethe center-of-mass coupling, it admits a relativistic generalization.Comment: revtex file, 73 pages, 1 figur

Dynamical body frames, orientation-shape variables and canonical spin bases for the nonrelativistic N-body problem