Where Do I Start? Deriving Equations of Motion

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Equation: d/dt(dL/dv) - dL/dx = d/dt(mv / sqrt(1 - v^2/c^2)) - qE, a Lagrangian-style relativistic equation of motion with a charge q in field E.


After some arguments about the equation and some other shenanigans beyond this question, we find that
Euler-Lagrange equation: d/dt of (partial L / partial v) minus partial L / partial x equals zero.

With the equation equaling zero, we can derive equations of motion.
 
Mark44 said:
What's your question? Your title, "Where do I start" doesn't provide much guidance.

I'm just not even sure what my first step would be in this case.
 
To show some property of the Lagrangian, you first need the Lagrangian.
We don't know where to start if you do not share the full problem.