Newton's standard theory of gravitation is reformulated as a gauge theory of the extended Galilei Group. The Action principle is obtained by matching the gauge technique and a suitable limiting procedure from the ADM-De Witt action of general relativity coupled to a relativistic mass-point.

In the Wigner-covariant rest-frame instant form of dynamics it is possible to develop a relativistic kinematics for the N-body problem which solves all the problems raised till now on this topic. The Wigner hyperplanes, orthogonal to the total timelike 4-momentum of any N-body configuration, define the intrinsic rest frame and realize the separation of the center-of-mass motion. The point chosen as origin of each Wigner hyperplane can be made to coincide with the covariant non-canonical Fokker-Pryce center of inertia. This is distinct from the canonical pseudo-vector describing the decoupled motion of the center of mass (having the same Euclidean covariance as the quantum 1 Newton-Wigner 3-position operator) and the non-canonical pseudo-vector for the Møller center of energy. These are the only external notions of relativistic center of mass, definable only in terms of the external Poincaré group realization. Inside the Wigner hyperplane, an internal unfaithful realization of the Poincaré group is defined while the analogous three concepts of center of mass weakly coincide due to the first class constraints defining the rest frame (vanishing of the internal 3-momentum). This unique internal center of mass is consequently a gauge variable which, through a gauge fixing, can be localized atthe origin of the Wigner hyperplane. An adapted canonical basis of relative variables is found by means of the classical counterpart of the Gartenhaus-Schwartz transformation. The invariant mass of the N-body configuration is the Hamiltonian for the relative motions. In this framework we can introduce the same dynamical body frames, orientation-shape variables, spin frame and canonical spin bases for the rotational kinematics developed for the non-relativistic N-body problem.

In this paper we study the perturbations of the charged, dilaton black hole, described by the solution of the low energy limit of the superstring action found by Garfinkle, Horowitz and Strominger. We compute the complex frequencies of the quasi-normal modes of this black hole, and compare the results with those obtained for a Reissner-Nordström and a Schwarzschild black hole. The most remarkable feature which emerges from this study is that the presence of the dilaton breaks the isospectrality of axial and polar perturbations, which characterizes both Schwarzschild and Reissner-Nordström black holes.

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.