Single-Particle Electrodynamics - Assassination Science

equations, one then finds, for either Lagrangian (2.77) or (2.78), the equation
of motion
d τ (mU) = 0,
and thus, by the definition of the mechanical momentum p,
d τ p = 0.
We can likewise obtain the equation of motion for an electric charge, by
simply adding a term to the free-particle Lagrangian:
L = 1 2 mU 2 + q(U ·A).
The canonical momentum four-vector for the electric charge is then directly
obtained:
b α ≡ ∂ U αL = mU α + qA α ;
in other words, we again find the minimal coupling result:
b = p + qA. (2.79)
The Euler–Lagrange equation for b then yields
d τ b − q∂(U ·A) = 0,
which, on using (2.79), yields
d τ p = −qd τ A + q∂(U ·A).
We now need to use the relativistic convective derivative,
d τ A ≡ (U ·∂)A,
to find
d τ p = −q(U ·∂)A + q∂(U ·A)
≡ q(∂∧A)·U.
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