Fiber averaged dynamics associated with the Lorentz force equation

It is shown that the Lorentz force equation is equivalent to the auto-parallel condition L∇ẋẋ=0 of a linear connection L∇ defined on a convenient pull-back vector bundle. By using a geometric averaging method, an associated averaged Lorentz connection 〈L∇〉 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 〈L∇〉 remain nearby close to the auto-parallel curves of L∇. Applications of this result in beam dynamics and plasma physics are briefly described.