Physical Trajectories Are Smooth, with Velocities at Least As Continuous As Brownian Motion

Olga Kosheleva, The University of Texas at El Paso
Vladik Kreinovich, The University of Texas at El Paso

Technical Report: UTEP-CS-22-87

Published in Mathematical Structures and Modeling, 2022, Vol. 63.


The fact that the kinetic energy of a particle cannot exceed its overall energy implies that the velocity -- i.e. the derivative of the trajectory -- should be bounded. This means, in effect, that all the trajectories are differentiable (smooth). However, at first glance, there seems to be no direct requirement that the velocities continuously depend on time. In this paper, we show that the properties of electromagnetic field necessitate that the velocities are continuous functions of time -- moreover, that they are at least as continuous as the Brownian motion.