# Relation between electric charge and gauge parameter of the moduli space of monopoles

I am studying about the moduli space of a 2 monopole system from Harvey's notes, and Manton's paper. In both of these, (Harvey section 6.2), after constructing the Lagrangian for a two dyons system, the author replaces the electric charge $e$ with $\dot{\chi}$. $$e \to \dot{\chi}$$ to obtain the the equation of motion is geodesic form, from which the Taub-NUT metric can be seen. Why is this identification made? Why does the rate of change of the gauge parameter, give the electric charge?

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A theory describing a charged particle on a configuration space $M$ can be obtained from the reduction of a theory of a particle moving on an extended configuration space $M \times S^1$. (To be precise symplectic reduction). This is the simplest example of the Kaluza-Klein approach. Please see: Marsden and Ratiu: Introduction to mechanics and symmetry section 7.6: (page 196) treating the case of a nonrelativistic charged particle.