# Newtonian limit of geodesic equation and Euler-Lagrange equations

As far as I know the Euler-Lagrange (EL) equations $$\frac{\partial L}{\partial q^m}-\frac{d }{dt}\frac{\partial L}{\partial \dot{q}^m}=0$$ are covariant time dependent coordinate transformations,

$$q’^m(q^n,t)$$

So for example we can use the EL equations in a rotating coordinate. On the other hand, we can use geodesic equations

$$\frac{d^2x^\mu}{d\tau ^2}+\Gamma_{\alpha\beta}^\mu \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau}=0$$

and this also gives us a covariant equation of motion in terms of a spacetime metric. Now it seems that the EL equations are the Newtonian limit $$c \rightarrow \infty$$. I am wondering if my conclusion is correct and if there is any quick proof for that.

The Euler–Lagrange equations are more general than the geodesic equation, in fact. They do not apply exclusively to Newtonian Mechanics, but rather to any theory to which you can write a Lagrangian (possibly with some modifications). That includes, as a particular case, the motion of particles on curved spacetime. You'll notice that the Lagrangian $$L = \frac{1}{2}g_{\mu\nu} \frac{\textrm{d} x^\mu}{\textrm{d}\tau}\frac{\textrm{d} x^\nu}{\textrm{d}\tau}$$ leads to the geodesic equation as the Euler–Lagrange equations. In this situation, the coordinates are taken to be whichever spacetime coordinates you prefer (i.e., time is taken to be one of the dependent coordinates), and the independent parameter is taken to be either proper time or an affine parameter. For more details, see, e.g., Bob Wald's General Relativity, particularly pp. 43–45. He writes down the Lagrangian I wrote on Eq. (3.3.14).
Notice also that the Euler–Lagrange equations will often describe interactions with other pieces of a system or other fields, such as electromagnetic effects. These are not taken into account in the geodesic equation. A $$c \to +\infty$$ limit on the geodesic equation for a weak field would lead you to the theory of a particle falling in Newtonian gravity. This is shown, e.g., on Wald's book, Section 4.4a. In particular, see p.77.
In summary, the Euler–Lagrange equations are not the non-relativistic limit of the geodesic equation. The Euler–Lagrange equations are in fact far more general, provided the correct Lagrangian is specified. In the $$c \to + \infty$$ limit for weak gravity, the Newtonian motion for a particle on a gravitational field will be obtained.