Classical Electrodynamics · Problem 12.1

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · relativistic-lagrangian-hamiltonian

Problem

(a) Show that the Lorentz invariant Lagrangian (in the sense of Section 12.1BB) L=−mUαUα2−qc UαAαL = -\frac{mU_\alpha U^\alpha}{2} - \frac{q}{c}\, U_\alpha A^\alpha gives the correct relativistic equations of motion for a particle of mass mm and charge qq interacting with an external field described by the 4-vector potential Aα(x)A^\alpha(x). (b) Define the canonical momenta and write out the effective Hamiltonian in both covariant and space-time form. The effective Hamiltonian is a Lorentz invariant. What is its value?

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