Fiber averaged dynamics associated with the Lorentz force equation

TL;DR

This paper demonstrates that the Lorentz force equation can be reformulated as an auto-parallel condition of a specific linear connection, and introduces an averaged connection that approximates particle trajectories in ultra-relativistic regimes, with applications in beam and plasma physics.

Contribution

It introduces an averaged Lorentz connection and shows its auto-parallel curves approximate true trajectories in certain physical regimes, providing a geometric framework for particle dynamics.

Findings

Auto-parallel curves of the averaged connection stay close to true trajectories in ultra-relativistic limit.

The geometric averaging method simplifies analysis of particle motion.

Applications in beam dynamics and plasma physics are discussed.

Abstract

It is shown that the Lorentz force equation is equivalent to the auto-parallel condition $\,^L\nabla_{\dot{{x}}}\dot{{x}}=0$ of a linear connection $^L\nabla$ defined on a convenient pull-back vector bundle. By using a geometric averaging method, an associated {\it averaged Lorentz connection} $\langle\,^L\nabla\rangle$ and the corresponding auto-parallel equation are obtained. After this, it is shown that in the ultra-relativistic limit and for narrow one-particle probability distribution functions, the auto-parallel curves of $\langle\,^L\nabla\rangle$ remain {\it nearby} close to the auto-parallel curves of $^L\nabla$. Applications of this result in beam dynamics and plasma physics are briefly described.