Fiber averaged dynamics associated with the Lorentz force equation

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It is shown that the Lorentz force equation is equivalent to the auto-parallel condition ˡ∇xx=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.