From non-relativistic to relativistic action

For a clean derivation, it is easier to start with the relativistic action $$S_0~=~-E_0Deltatau,qquad E_0~:=~m_0c^2.tag{1}$$ Eq. (1) itself is severely restricted by (i) dimensional arguments, (ii) relativistic invariance, i.e. it must be a function of proper time $Deltatau$, rest/invariant mass $m_0$, and the speed of light $c$; and (iii) it must reproduce the geodesic equation.

We can not take the non-relativistic limit $ctoinfty$ directly because of the rest energy (1) becomes infinite. However we can add a constant to the Lagrangian without affecting the EOM. So consider instead the equivalent action (without manifest relativistic symmetry) $$begin{align} underbrace{S}_{text{non-rel.}}~=~&int_{t_i}^{t_f}! dt~L~=~ underbrace{S_0}_{text{rel.}} + underbrace{E_0Delta t}_{text{const.}}cr ~stackrel{(1)}{=}~& E_0(Delta t-Deltatau) ,end{align} tag{2}$$ up to boundary terms.

After this shift, we can now consistently take the non-relativistic limit $ctoinfty$ and derive the well-known formula for the non-relativistic kinetic energy $$begin{align} L~stackrel{(2)}{=}~E_0(1-gamma^{-1})~=~&frac{1}{2}m_0{bf v}^2+{cal O}(c^{-2}), cr gamma ~:=~&frac{1}{sqrt{1-frac{{bf v}^2}{c^2}}}. end{align}tag{3}$$