The Lagrangian and the energy-momentum tensors in the perturbation theory of classical electrodynamics

Abstract

The theory of infinitesimal transformations is incorporated in the variational principle for the solution of perturbation problems in classical electrodynamics. Every variable quantity is transformed to the unperturbed coordinate system and expanded into a power series of a small perturbation parameter. The Lagrangian of any given order of magnitude in the unperturbed coordinate system characterizes a closed physical system. This is proved by showing that the total energy-momentum tensors of the first and second orders are symmetric. The proof for the higher orders is obtained by deduction. Sturrock's asymmetric tensor is recognized as being the result of using an incomplete Lagrangian. Small-amplitude solutions can also be obtained by first applying variation to the total Lagrangian function and then transforming and expanding the results in different orders of magnitude. This method, as exemplified by the proof of the symmetry of the second-order energy-momentum tensor, is in many cases more convenient.