On the Possible Trajectories of Spinning Particles. I. Free Particles

TL;DR

This paper derives equations of motion for spinning particles using the Frenet-Serret frame, analyzing their trajectories in free and external fields, and classifies possible motions for massive and massless particles with spin.

Contribution

It provides a comprehensive set of equations for spinning particles in arbitrary fields and characterizes their free motion, including polarization and trajectory types.

Findings

Massive particles always have longitudinal polarization.

Possible free motions include oscillatory, circular, and rosette trajectories.

Massless particles can fluctuate or move along complex paths with arbitrary spin directions.

Abstract

By means of the method of moving Frenet-Serret frame the set of equations of motion is derived for spinning particle in an arbitrary external field, which is determined by potential depending from both position and the state of movement, as well as by two pseudo-vectors one of which is easily associated with external magnetic field, and another still remains undetermined. The equations give a possibility to describe the motion of both massive and massless particles with spin. All solutions of the equations of motion in the absence of external fields were found, and besides, we give more precise definition of a free object. It turns out that the massive particles always possess a longitudinal polarization. There are possible transversal motions of the following types: 1) oscillatory motion with proper frequency, 2) circular motion, and 3) complicated motion along rosette trajectories round the center of inertia with the velocity, varying in finite limits. Free massless particles can either fluctuate or move along complicated paths around fixed centers of balance, when the spin of the particles can have any direction.