How do we know that solving Euler-Lagrange equation gives us the correct equation of motion in any coordinate?

Question

As far as I understand it: If we defined a quantity called the Lagrangian as the difference between the kinetic energy and potential energy for a particle in one dimension in Cartesian coordinates, then solving Euler-Lagrange equation give us the law of motion of that particle (we can check that by solving Euler-Lagrange equation and comparing the results with Newton's second law we find them the same). All that happened in Cartesian coordinates. I understand all that.

But now we make the jumb: That if we do this process for any coordinate, i.e. we defined the Lagrangian in a general coordinate and solved Euler-Lagrange equation, that will also give us the law of motion in that coordinate with any dimension.

First question: How can we justify that this process will give us the law of motion in general coordinate if we can't check that as in the Cartesian coordinates case.

Second question: is asking for a justification is justified itself? Because from what I have read it looks like the answer is trivially yes we can do that. But I don't understand why. My feeling is that it actually needs justification and it's not self evident that we can do that.

Edit: l'm looking for a mathematical justification. Meaning if we accept that Newton's second law of motion is true in its current form in Cartesian coordinates, then solving Euler-Lagrange equation in general coordinates will give us the law of motion in general coordinates.

What I don't understand is that it seems such a basic question to ask, but I almost never seen it addressed in the lectures on classical mechanics. The answer is just assumed to be yes.

Answer 1

The answer depends on the precise context of OP's question:

1. Within the framework of Newtonian mechanics in Cartesian space $\mathbb{R}^3$, one may prove the Lagrangian formulation in terms of holonomic constraints and generalized coordinates. For a derivation, see e.g. Chapter 1 in H. Goldstein, Classical Mechanics.

2. In contrast, outside the framework of Newtonian mechanics in Cartesian space $\mathbb{R}^3$, we are on our own up to general guidelines, such as, e.g., locality. A geometric covariant action formulation in curved spacetime typically serve as a first principle for the physical system.

Concerning covariance under change of coordinates, see also e.g. this & this related Phys.SE posts.

Answer 2

It sounds like you're taking for granted that the system obeys Newton's equations and that this means that it will extremize the action when the Lagrangian is expressed in particle coordinates. Your question, then, is why the classical path should extremize the action in arbitrary coordinates.

The answer, I think, is simply that the notion of extremizing a path is coordinate-independent. We can find the action associated with any path regardless of the coordinate system. A valid change of coordinates will leave all actions unchanged. Thus, a path either extremizes the action or it doesn't, regardless of coordinate system.

However, you might also ask why classical paths should extremize the action when there is no relation to Newtonian equations of motion. In that case, I believe the extremization of the path is taken as a fundamental postulate and generally not derived from some other more reliable source.