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We consider the motion of relativistic particles described by an action that is linear in the torsion (second curvature) of the particle path. The Euler-Lagrange equations and the dynamical constants of the motion associated with the Poincaré group, the mass and the spin of the particle, are expressed in a simple way in terms of the curvatures of the embedded worldline. The moduli spaces of solutions are completely exhibited in 4-dimensional background spaces and in the 5-dimensional case wedoi:10.1142/s0217751x04018026 fatcat:sd26keuqw5eaddffezqcwyixvm