What is the Lagrangian for Electromagnetic Fields?

[lavster]

Homework Statement

In my notes i have the following two equations written with no explanation where thehy came from... can someone help please!?
$$L=(u, x )= -mc\sqrt(u^{\beta} u_\beta)-\frac{q}{c}A^{\alpha}u_\alpha, L(v,r, t) = -mc^2(1-\frac{v^2}{c^2})-\phi +\frac{q}{c}vA$$
L is lagrangian, m mass c speed of light u is speed, A is potential four vector made up of scalar (phi) and vector potential A

Homework Equations

$$L = \frac{1}{2}m u_{\alpha} u^{\alpha}-\frac{1}{c}A^{\alpha}j_{\alpha}-\frac{1}{16\pi}F^{\alpha\beta}F_{\alpha\beta}$$
F is electromagnetic field tensor

The Attempt at a Solution

I have no idea :(

 

[hunt_mat]

Have you tried putting them through the Euler-Lagrange equations and seeing what you
get? It's the case that you usually have to guess at lagrangian, so there is no explanation from where they came from.