Non-Newtonian mechanics

The classical motion of spinning particles can be described without recourse to particular models or special formalisms, and 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 four-velocity in the usual Newtonian constant term p(mu)/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 nonlinear, nonconstant relation holds between the time durations measured in different reference frames. Newtonian mechanics is reobtained as a particular case of the present theory. namely for spinless systems with no zitterbewegung. Then we analyze the strict analogy between the classical zitterbewegung equation and the quantum Gordon-decomposition of the Dirac current. It is possible a variational formulation of the theory through a Lagrangian containing also derivatives of the four-velocity: we get an equation of the motion, actually a generalization of the Newton law a = F/m, where non-Newtonian zitterbewegung-terms appear. 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 (a x v)/4m and analyze the general solution of the Eulero-Lagrange equation oscillating with the Compton frequency omega = 2m. The interesting case of spinning systems with zero intrinsic angular momentum is also studied.