Dynamical Body Frames, Orientation-Shape Variables and Canonical Spin Bases for the Non-Relativistic N-Body Problem

TL;DR

This paper introduces a new class of canonical bases for the non-relativistic N-body problem, utilizing geometric and group theoretical methods to define body frames, orientation, and shape variables, with implications for spin coupling and relativistic generalization.

Contribution

It develops a novel geometrical and group-theoretical framework for defining dynamical body frames and orientation-shape variables in the N-body problem, extending to momentum-dependent frames for N≥4.

Findings

For N=3, the dynamical body frame is unique and reproduces the xxzz gauge.

For N≥4, the dynamical body frames are momentum dependent.

The approach connects spin coupling with shape variables, enabling relativistic extensions.

Abstract

After the separation of the center-of-mass motion, a new privileged class of canonical Darboux bases is proposed for the non-relativistic N-body problem by exploiting a geometrical and group theoretical approach to the definition of {\it body frame} for deformable bodies. This basis is adapted to the rotation group SO(3), whose canonical realization is associated with a symmetry Hamiltonian {\it left action}. The analysis of the SO(3) coadjoint orbits contained 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-symmetry Hamiltonian {\it right actions} for non-rigid systems leads to the construction of a N-dependent discrete number of {\it dynamical body frames} for the N-body system, hence to the associated notions of {\it dynamical} and {\it measurable} orientation and shape variables, angular velocity, rotational and vibrational configurations. For N=3 the dynamical body frame turns out to be unique and our approach reproduces the {\it xxzz gauge} of the gauge theory associated with the {\it orientation-shape} SO(3) principal bundle approach of Littlejohn and Reinsch. For $N \geq 4$ our description is different, since the dynamical body frames 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 clusters rather than the coupling of the {\it centers of mass} (based on Jacobi relative normal coordinates). One of the advantages of the spin coupling is that, unlike the center-of-mass coupling, it admits a relativistic generalization.