Hamiltonian mechanics electromagnetic field

  1. 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)δtA(x,t) , B = ∇xA , (x,t) elements of R3xRt

    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
     
  2. jcsd
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