Dynamical Body Frames, Orientation-Shape Variables and Canonical Spin...
Luca Lusanna
Bases for the Non-Relativistic N-Body Problem
Author(s): David Alba, Luca Lusanna and Massimo Pauri
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.
Paper: hep-th/0011014
Dated: Thu, 2 Nov 2000 10:44:59 GMT (72kb)
Comments: revtex file, 73 pages, 1 figure
Report-no: FI-TH-0020
URL: http://arXiv.org/abs/hep-th/0011014