Single-Particle Electrodynamics - Assassination Science

exercise will be of use in guiding us in future sections. We know that, for
the purpose of computing a charged point particle’s equations of motion, the
electric charge interaction Lagrangian is
L int = qU α A α . (5.3)
We can turn this Lagrangian into a Lagrangian density by converting the
point particle’s electromagnetic current, qU α , into a current density, J α (x),
via
∫
J α (x) ≡ dτ qU α (τ) δ (4) [x − z(τ)] . (5.4)
Then (5.3) is equivalent to the Lagrangian density
L int = J α A α , (5.5)
and so the complete Lagrangian density, as far as the electromagnetic field
is concerned, is simply the sum of (5.1) and (5.5):
L = L free + L int = 1 4 F αβ F αβ + J α A α . (5.6)
5.3.2 The Maxwell equation
A straightforward application of the field Euler–Lagrange equations to the
Lagrangian density (5.6) yields the familiar (inhomogeneous) Maxwell equation,
equation (B.7) of Section B.2.7:
∂ µ F µν = J ν . (5.7)
5.3.3 Solution of the Maxwell equation
One’s next task is to determine the solution of (5.7) for some given electric
current distribution J ν (x). While it is possible to obtain this information
directly from (5.7), it is simpler to employ again the four-potential A µ , purely
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