Hamiltonian mechanics electromagnetic field

Liquidxlax

1. The problem statement, all variables and given/known data

Let (V(x,t) , A(x,t)) be a 4-vector potential that constructs the electromagnetic field (in gaussian Units) by

E(x,t) = -∇V(x,t) - (1/c)δ_tA(x,t) , B = ∇xA , (x,t) elements of R³xR^t

Consider the lagrangian

L=.5mv² - eV(x,t) + (ev/c)(dot)A(x,t)

a) compute and interpret the Euler-lagrange equatinons of motion for this system
b) determine the hamiltonian
c) determine hamilton's equations of motion. Are they gauge invariant?

2. Relevant equations



3. The attempt at a solution

3 simple questions about this, and hopefully not too stupid of questions

when i apply the euler-lagrange equations do i take the curl of A or the gradient? if gradient what does that mean to take the ∇A?

is there only one equation of motion because i fail to see the other possibility if there is one.

Otherwise i can solve the rest

thanks