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)del((x)over dot)(x)over dot = 0 of a linear connection (L)del defined on a convenient pull-back vector bundle. By using a geometric averaging method, an associated averaged Lorentz connection <(L)del > 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)del > remain nearby close to the auto-parallel curves of (L)del. Applications of this result in beam dynamics and plasma physics are briefly described. (C) 2014 Elsevier B.V. All rights reserved.