Electromagnetic minimal coupling

Electromagnetic minimal coupling replaces the free kinetic momentum in an action or Hamiltonian by a combination involving the electromagnetic potential. For $L=-m\sqrt{{\dot{x}}^a{\dot{x}}_a}-q{A}_a{\dot{x}}^a$, the canonical covector is $P_a=-m{\dot{x}}_a/\sqrt{{\dot{x}}^2}-q{A}_a$, so $P_a+q{A}_a$ recovers the potential-independent covector. Under $A_a\mapsto A_a+{\partial }_a\chi$, the action changes by an endpoint term. The fixed-endpoint equations are the relativistic Lorentz force equation and are gauge invariant.