Also if the 3 vector gives the lorentz force law what exactly does the o'th component represent in this case?

As a follow up, in calculating the 0'th component would the time derivative act on $\gamma$ i.e does $$\frac{\partial ( \gamma \phi )}{\partial c t} = \gamma \frac{\partial \phi}{\partial ct}$$ ?

Thank you very much! Helped a lot! My issue was that, as you mentioned in your second point, I was breaking $\frac{\partial}{\partial x^{\mu}}$ up into separate components and I was getting lost in the details and It didnt click that its the 3 vector part that gives the lorentz force law

Awesome thanks! I didn't think about getting rm from the total energy. I'll try applying it to what I've done and see if I can match your graph output.

of course sorry, I wasn't thinking. Thank you!

@Eli Why did you set $\dot \sigma =0$ ? The question states that the surface is moving. and what is $J$ in the second equation?