Classical Electrodynamics · Problem 12.14

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · alternative-electromagnetic-lagrangian-density

Problem

An alternative Lagrangian density for the electromagnetic field is L=−18π ∂αAβ∂αAβ−1c JαAα\mathscr{L} = -\frac{1}{8\pi}\,\partial_\alpha A_\beta \partial^\alpha A^\beta - \frac{1}{c}\,J_\alpha A^\alpha (a) Derive the Euler--Lagrange equations of motion. Are they the Maxwell equations? Under what assumptions? (b) Show explicitly, and with what assumptions, that this Lagrangian density differs from (12.85) by a 4-divergence. Does this added 4-divergence affect the action or the equations of motion?

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