The Ubiquitous Lorentz Force

Abstract

Minimizing the Action integral of a Lagrangian provides the Euler-Lagrange equation of motion in the elegant machinery of Lagrangian Mechanics. However two relations define the divergence of current and energy-momentum, and provide an alternative motivation for the Euler-Lagrange equation without invoking the considerable machinery required for the principle of least Action. The derivation of these two relations proceeds with only local differential operations on the Lagrangian, and without a global integration defining an Action. The two relations connect local continuity equations (as a vanishing divergence) for current and energy-momentum to Lorentz force, symmetry, and the Euler-Lagrange equation. The Euler-Lagrange equation is common to both relations so providing sufficient motivation for acceptance as equations of motion. The essential relationships between the central concepts of energy-momentum, force, current, symmetry and equation of motion provide pedagogically interesting clarity. The student will see that how these concepts relate to each other as well as the definition of each concept in isolation.