(a) Write the action for special relativity as

S = α ∫ dτ = α ∫ Ldτ

where α is a constant and we use coordinate time t to parameterize the spatial coordinates. (Note that the constant α doesn’t change the fact that extremizing S will give us the equation of motion for an inertial observer.)

Expand L for small speeds v and match to the non-relativistic particle Lagrangian to determine the factor α.