Non-Newtonian Mechanics

Abstract

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.