  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.
