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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.

The principle is that a metric can be chosen on the extended space such that the free geodesic equation of motion of the extended coordinate imply that it's canonical momentum is a constant of motion which can be interpreted as the electric charge on the original configuration space. In other words, when the solution of the equations of motion of the extra coordinate are substituted in the equation of motion, one gets the standard equation of motion of a particle on the original configuration space with the electric charge equal to (the constant) canonical momentum of the extended coordinate.

This approach offers an explanation of the quantization of the electric charge semiclassically, because since the extra dimension is circular the corresponding momenta should be quantizaed.

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