= Electromagnetic minimal coupling
Electromagnetic minimal coupling replaces the free kinetic momentum in an action or Hamiltonian by a combination involving the electromagnetic potential. For $L=-m\sqrt{\dot x^a\dot x_a}-qA_a\dot x^a$, the canonical covector is $P_a=-m\dot x_a/\sqrt{\dot x^2}-qA_a$, so $P_a+qA_a$ 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.
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