Lagrangian of Charged Particle Evaluated On-Shell

I am trying to calculate the Lagrangian of a charged particle in background gauge field evaluaed on-shell.

Let $A^{\mu}(x)$ be a gauge field. The action of a charged particle in this background gauge field is

$$S[A]=\int_{-\infty}^{+\infty}d\tau\left\{m\left(\frac{dx^{\mu}}{d\tau}\right)^{2}+e\frac{dx^{\mu}}{d\tau}A_{\mu}(x)\right\}$$

The equation of motion is

$$mg_{\alpha\beta}\frac{d^{2}x^{\beta}}{d\tau^{2}}=eF_{\alpha\beta}\frac{dx^{\beta}}{d\tau}$$

Now I want to plug the solution to the above equation of motion back into the action.

Is there any way to evaluate the on-shell action as a functional of $A^{\mu}$ without solving the equation?

• Have you tried integrating by parts and solving the equation of motion for $\frac{d^2x^\beta}{d\tau^2}$ and then inserting that in the action that you just integrated by parts. This should at the very least simplify the action a lot (0?). – Michael Aug 12 '18 at 1:58
• @Michael Could you be specific? Which term do you integrate by part? – Libertarian Monarchist Bot Aug 12 '18 at 9:51