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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.

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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.

However, your guess is quite clever. While physically these are quite different concepts, the Euler–Lagrange equations also have a geometric interpretation. More specifically, one can formulate Classical Mechanics in terms of Differential Geometry. More specifically, in terms of Symplectic Geometry. This might explain the hint you got from the coordinate covariance of the equations. This approach is discussed, for example, on the book by V. I. Arnold or on the book by José and Saletan.

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.

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  • $\begingroup$ Many thanks for the answer. First, ok here I am ignoring any force or interactions so the Lagrangian is just the kinetic term. But your argument is fair enough. However, if we can derive the second law of Newton (N2) from Newtonian limit of geodesic equations, then we can recast them in form of EL equations (just recasting) and THEN we demand that this equation is true for any coordinate system which are functions of the Euclidean ones AND time. So, we can obtain EL equations from geodesic equations but we again should demand the covariance principle. Do you agree with this? $\endgroup$ Nov 13, 2021 at 7:31
  • $\begingroup$ @vahidhosseinzadeh Notice that in this situation we would be deriving EL for a particular sort of interaction (Newton's gravity), so it would not be general. The geodesic equation does not account, for example, for electromagnetic effects, so the general EL equations can't follow from it. Furthermore, the covariance of EL is far different from that of the geodesic equation: the geodesic equation is covariant with respect to changes of spacetime coordinates, while EL is covariant with respect to coordinate changes on "configuration space", which is the space formed by generalized coordinates $\endgroup$ Nov 13, 2021 at 8:15
  • $\begingroup$ @vahidhosseinzadeh For one particular case (Newton's gravity), the EL equations can be obtained from the geodesic equation. But this is not a derivation of the EL nor do their covariance have anything to do with general relativistic covariance. They just share the a similar mathematical structure, but that is no more profound than the fact that both the electric field and wavefunctions in QM are vectors in some vector space. They are both lie in vector spaces, but this relate them to each other in any profound way $\endgroup$ Nov 13, 2021 at 8:18

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