THE GENERAL TRAJECTORIES OF ACCELERATED PARTICLES IN SPECIAL RELATIVITY
A. SFARTI *
CS Department, UC Berkeley, 387 Soda Hall, Berkeley, United States.
*Author to whom correspondence should be addressed.
Accelerated motion in special relativity is a subject that tends to be treated under very restrictive conditions, the motion is considered to be “uniformly accelerated”, meaning motion under constant force and the treatment is in one dimension only. In the present paper we treat the general case, of non-uniform force and we extend the treatment to all three spatial dimensions. The paper is divided into three main sections: the first section is an overview of the existent solutions for the unidimensional trajectories, the second section deals with the general, three dimensional trajectories under constant force. The third section introduces the case of non-uniform force. The case of non-uniform force is further subdivided into two sub-cases: force that is explicitly time-dependent and, the more complicated case of velocity-dependent (aka Lorentz) force. All cases teach us how to deal with increasing levels of non-linearity in the equations of motion. In each case we will show how to find fully symbolic (closed) solutions for the trajectories. The last case, of the Lorentz force, is especially interesting because it is a real life case, taken from the particle accelerator applications as in the design of velocity selectors used for particle separation. What makes it even more interesting is the fact that the solution uses a physics approach at the point where the mathematical approach hits a dead end. The subject is of interest for particle physicists as well for graduate students and teachers.
Keywords: Accelerated motion, non-uniform force, Lorentz force, particle accelerators