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

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 body frame for deformable bodies. This basis is adapted to the rotation group SO(3), whose canonical realization is associated with a symmetry Hamiltonian left action. The analysis of the SO(3) coadjoint orbits contained in the N-body phase space implies the existence of a spin frame for the N-body system. Then, the existence of appropriate non-symmetry Hamiltonian right actions for nonrigid systems leads to the construction of a N-dependent discrete number of 1 dynamical body frames for the N-body system, hence to the associated notions of dynamical and 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 xxzz gauge of the gauge theory associated with the orientation-shape SO(3) principal bundle approach of Littlejohn and Reinsch. For N ≥ 4 our description is different, since the dynamical body frames turn out to be momentum dependent. The resulting Darboux bases for N ≥ 4 are connected to the coupling of the spins of particle clusters rather than the coupling of the 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.

Relativistic quantum mechanics and relativistic entanglement in the rest-frame instant form of dynamics

“…[7] and in II, they induce potential-dependent terms in the internal Lorentz boosts, so that the solution η 12 = η 12 [ ρ 12 , π 12 ] of the conditions K (int) ≈ 0 eliminating the internal 3-center of mass are potential dependent. 20 They are the non-relativistic energy levels of the the relative HamiltonianĤ rel resulting from the nonrelativistic limit ofM in Eq.(4.5). Let us remark that different relativistic theories (potential either inside or outside the square roots) can have the same non-relativistic potential as a limit.…”

The Chrono-Geometrical Structure of Special and General Relativity: A Re-Visitation of Canonical Geometrodynamics

“…The discovery of the rest-frame instant form made possible to develop a coherent formalism for all the aspects of relativistic kinematics both for N particle systems and continuous bodies and fields [15,16] generalizing all known non-relativistic results [5]: i) the classification of the intrinsic notions of collective variables (canonical non-covariant center of mass; covariant non-canonical Fokker-Pryce center of inertia; non-covariant noncanonical Møller center of energy); ii) canonical bases of center-of-mass and relative variables; iii) canonical spin bases and dynamical body-frames for the rotational kinematics of deformable systems; iv) multipolar expansions for isolated and open systems; v) the relativistic theory of orbits (while the potentials appearing in the energy generator of the Poincare' group determine the relative motion, the determination of the actual orbits in the given inertial frame is influenced by the potentials appearing in the Lorentz boosts: the vanishing of the boosts is the natural gauge fixing to the rest-frame conditions and selects the covariant Fokker-Pryce center of inertia); vi) the Møller radius (a classical unit of length identifying the region of non-covariance of the canonical center of mass of a spinning system around the covariant Fokker-Pryce center of inertia; it is an effect induced by the Lorentz signature of the 4-metric; it could be used as a physical ultraviolet cutoff in quantization).…”

Generalized Eulerian Coordinates for Relativistic Fluids: Hamiltonian Rest-Frame Instant Form, Relative Variables, Rotational Kinematics

“…[13,14] we can construct the following quantities 8) and, for n = u 1 , u 2 r 1 n = r n ·χ, r 2 n = r n ·N ×χ, r 3 n = r n ·N,…”

“…Then, following the methods of Refs. [13][14][15], we shall study the problem of the center-of-mass and relative variables, the separation of relative variables in orientational and vibrational ones by means of the introduction of dynamical body frames and Dixon's multipoles [16,17] of the fluid.…”