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Persistent URL http://purl.org/net/epubs/work/53143
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Record Id 53143
Title Averaged dynamics associated with the Lorentz force equation
Abstract An averaged version of the Lorentz force differential equation is discussed. Once the geometric setting has been introduced, we show that the Lorentz force equation is equivalent to the auto-parallel equation $< ^L\nabla>_{\dot{x}}\dot{x}=0$ of a projective linear connection $^L\nabla$ on the pull-back vector bundle $\pi^*{\bf TM}$. Using a geometric averaging procedure, we obtain the associated {\it averaged connection} $< ^L\nabla>$ and we consider its auto-parallel equation $< ^L\nabla>_{\dot{\tilde{x}}}\dot{\tilde{x}}=0$. We prove that in the ultra-relativistic limit and for narrow one-particle probability distribution functions whose support is invariant under the flow of the Lorentz's force equation, the auto-parallel curves of the averaged connection $< ^L\nabla>$ remain close to the auto-parallel curves of $^L\nabla$. We discuss some applications of these result. The relation of the present work with other geometric approaches to the classical electrodynamics of point particles is briefly discussed. Finally, in relation with the geometric formulation of the Lorentz force, we introduce the notion of {\it almost-connection}, which is a non-trivial generalization of the notion of connection.
Organisation CI , STFC
Keywords Physics
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Language English (EN)
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Preprint 2009. http://arxiv.org/abs/0905.2060 2009