A lagrangian that doesn't single out time as a special coordinate will parameterize the coordinats by some arbitrary coordinate, s. Then .

The action is then: . The must be lorentz invariant.

For a free particle, you get: :

So . Integration by parts gives: so This is a second order differential equation for . Using you can get: which is a simpler equation to solve.

source psfile jl@crush.caltech.edu index charged_particle_lagrangian