Noether's theorem and the work-energy theorem for a charged particle in an electromagnetic field

Abstract

Noether's theorem is based on two fundamental ideas. The first is the extremum of the action and the second is the invariance of the action under infinitesimal continuous transformations in space and time. The first gives Hamilton's principle of least action, which results in the Euler–Lagrange equations. The second gives the Rund–Trautman identity for the generators of infinitesimal transformations in space and time. We apply these ideas to a charged particle in an external electromagnetic field. A solution of the Rund–Trautman identity for the generators is obtained by solving generalized Killing equations. The Euler–Lagrange equations and the Rund–Trautman identity are combined to give Noether's theorem for a conserved quantity. When we use the Lagrangian and the generators of infinitesimal transformations for a charged particle in an external electromagnetic field, we obtain the work-energy theorem.