  The classical motion of spinning particles can be described without employing
Grassmann variables or Clifford algebras, but simply by generalizing the usual
spinless theory. We only assume the invariance with respect to the Poincare'
group; and only requiring the conservation of the linear and angular momenta we
derive the zitterbewegung: namely the decomposition of the 4-velocity in the
newtonian constant term p/m and in a non-newtonian time-oscillating spacelike
term. Consequently, free classical particles do not obey, in general, the
Principle of Inertia. Superluminal motions are also allowed, without violating
Special Relativity, provided that the energy-momentum moves along the worldline
of the center-of-mass. Moreover, a non-linear, non-constant relation holds
between the time durations measured in different reference frames. Newtonian
Mechanics is re-obtained as a particular case of the present theory: namely for
spinless systems with no zitterbewegung. Introducing a Lagrangian containing
also derivatives of the 4-velocity we get a new equation of the motion,
actually a generalization of the Newton Law a=F/m. Requiring the rotational
symmetry and the reparametrization invariance we derive the classical spin
vector and the conserved scalar Hamiltonian, respectively. We derive also the
classical Dirac spin and analyze the general solution of the Eulero-Lagrange
equation for Dirac particles. The interesting case of spinning systems with
zero intrinsic angular momentum is also studied.

Quantum Physics