4. [30 pts]
(a) The Lagrangian for a particle with charge q and mass m in an electromagnetic field described by electric scalar potential φ(r(t), t) and magnetic vector potential A(r(t), t) is

Obtain the equation of motion of the charged particle from Hamilton’s principle.

(b) The Lagrangian density of an electromagnetic field with a charge density ρ(r, t) and current density J(r, t) is given by

The Lagrangian of an electromagnetic field is given by a volume integral of the Lagrangian density: