Question - A particle of mass in and charge q moves in the field of a vector potential A(r, t) = -(E0ct + B0z)y^. (Note that we are using the cgs Gaussian units of your text book.) Here E0 denotes a constant electric field amplitude and B0 denotes a constant magnetic field amplitude. The initial particle position is r(0) = 0 and the initial particle velocity is r·(0) = 0.
(a) Determine the Lagrangian L(x, y, z, x·, y·, z·, t) which describes the particle's motion.
(b) Write the Euler-Lagrange equations for this system.
(c) Find the particle trajectories x(t), y(t), z(t) by solving the equations and imposing the initial conditions.
(d) Determine the Hamiltonian H (x, y, z, px, py, pz, t) for this system.
(e) Convince yourself that the canonical equations of motion are consistent with results of the Lagrangian analysis. (It is not necessary to repeat your solution.)