  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.
