The lagrangian formulation is:

Then set

To do this relativistically, make sure that S is a lorentz scalar. , so we want to be a lorentz scalar.

For a free particle, position does not matter, so so .

With a potential, you get:

For a free particle in EM field: possible lorentz invariants: . Which means so .

This is not a lorentz covariant formulation.

source psfile jl@crush.caltech.edu index continuous_system_lagrangian charged_particle_lagrangian electromagnetic_field covariant_lagrangian