Averaged Dynamics Associated with the Lorentz Force Equation

Details Averaged Dynamics Associated with the Lorentz Force Equation Averaged Dynamics Associated with the Lorentz Force Equation

2009-05-13

arxiv.org/abs/0905.2060v10

Physics

Mathematical Physics

37 pages, one section deleted; correction of minor errors

Scientific paper

We introduce a geometric framework for the dynamics of point charged particles described by the Lorentz force equation. We prove that the Lorentz force equation is equivalent to the auto-parallel equation $ ^L\nabla_{\dot{x}}\dot{x}=0$ of a linear connection $^L\nabla$ on the pull-back vector bundle $\pi^*{\bf TM}$. Using a geometric averaging procedure, we obtain an associated {\it averaged connection} $< ^L\nabla>$ and the associated auto-parallel equation $< ^L\nabla>_{\dot{\tilde{x}}}\dot{\tilde{x}}=0$. In the ultra-relativistic limit and for narrow one-particle probability distribution functions whose supports are invariant under the flow of the Lorentz force equation, the auto-parallel curves of $< ^L\nabla>$ remain close to the auto-parallel curves of $^L\nabla$. We show some applications of this result in beam dynamics and plasma physics.

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