Demystifying the Lagrangians of special relativity

TL;DR

This paper clarifies the Lagrangian formulation of special relativity, deriving key results like Lorentz invariance and mass-energy equivalence, making the subject more accessible and contextually grounded.

Contribution

It systematically derives and explains the Lagrangian formalism in special relativity, including transformations of electromagnetic potentials and fields, connecting mechanics and field theory.

Findings

Derivation of spacetime interval invariance

Development of relativistic particle Lagrangian

Invariance of electromagnetic Lagrangians and equations of motion

Abstract

Special relativity beyond its basic treatment can be inaccessible, in particular because introductory physics courses typically view special relativity as decontextualized from the rest of physics. We seek to place special relativity back in its physics context, and to make the subject approachable. The Lagrangian formulation of special relativity follows logically by combining the Lagrangian approach to mechanics and the postulates of special relativity. In this paper, we derive and explicate some of the most important results of how the Lagrangian formalism and Lagrangians themselves behave in the context of special relativity. We derive two foundations of special relativity: the invariance of any spacetime interval, and the Lorentz transformation. We then develop the Lagrangian formulation of relativistic particle dynamics, including the transformation law of the electromagnetic potentials, the Lagrangian of a relativistic free particle, and Einstein's mass-energy equivalence law ($E=mc^2$). We include a discussion of relativistic field Lagrangians and their transformation properties, showing that the Lagrangians and the equations of motion for the electric and magnetic fields are indeed invariant under Lorentz transformations.