Lagrangian to the Euler-Lagrange equation

[Trying2Learn]

TL;DR

Can one "reason" out thge Euler-Lagrange equation in dynamics

Hello all,

I understand the formation of the Lagrangian is: Kinetic Energy minus the potential energy.
(I realize one cannot prove this: it is a "principle" and it provides a verifiable equation of motion.

Moving on...

One inserts the Lagrangian into the form of the "Action" and minimizes it.

One then extracts the Euler-Lagrange equation for a dynamical system: d(L/dq) = d( dL/dq-dot)/dt

So...

Has anyone ever attempted to "reason out" or "justify" THIS FINAL form to be used in dynamics?

I know one can begin with it, and demonstrate it leads to Newton's F=ma; but has anyone tried to justify this
Euler-Lagrange equation in and of itself? Is it "reasonable?"

 

[BvU]

I seem to remember it has something to do with the principle of d'Alembert