Charged particle in an electrodynamic field Show that the velocity dependent potential
U=eϕ(r,t)−er˙⋅A(r,t)
represents the Lorentz force F=eE+ev×B that acts on a charge e moving with velocity v in the general electrodynamic field {E(r,t),B(r,t)}. Here {ϕ,A} are the electrodynamic potentials that generate the field {E,B} by the formulae
E=−gradϕ−∂t∂A,B=curlA.
Show that the potentials ϕ=0, A=tzi generate a field {E,B} that satisfies all four Maxwell equations in free space. A particle of mass m and charge e moves in this field. Find the Lagrangian of the particle in terms of Cartesian coordinates. Show that x and y are cyclic coordinates and find the conserved momenta px, py.