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.