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[tex]m \frac{d^2 x^{\mu}}{d {\tau}^2} = \frac{e}{c}F^{\mu \nu} \frac{d x_{\nu}}{d \tau} + \frac{g}{c}G^{\mu \nu} \frac{d x_{\nu}}{d \tau}[/tex]

Where G is the dual field tensor

Could anyone help me with writing the Lagrangian that gives rise to this equation?

Here is what I have tried till now.

1. Solve the original Lagrangian, get the equation of motion. Add the additional term, of G, and work backwards. However, I have realised that the required relation between A (potential) and a (dual potential, if I can call it that) is non-linear and non-local.

[tex] G^{ \mu \nu } = \partial_{\mu} a_{\nu} - \partial_{\nu} a_{\mu} [/tex]

2. So i tried hit and trial, wrote down, using covariance, symmetry, and dimensions as my guides, several possible Lagrangians, and got the corresponding equation of motion, hoping to give the one that i want.

However, I have been unlucky so far. Have been working on it for several hours.

Can such a Lagrangian be written down explicitly in terms of A and a??