About philosophy behind mathematics of Lagrange mechanics and Newtonian mechanics

Question

I wanted to ask how does Euler-Lagrange mechanics describe the Newtonian mechanics? By which i mean , since this results are mathematical and not something we found using experiment,afaik , it should be the case , i think , that at some fundamental level F=ma is equivalent to some equation in Lagrange equations so that both can describe the same thing , otherwise i think it would be too coincidental. Thanks

Answer 1

It works because of the least action principle, which is the most fundamental principle of modern physics. But I doubt if there is an explanation of why it works in the real world apart from it works because it works (Wigner was discussing this a lot).

In short, it means that every system has an action which it tries to minimize. The mathematical minimization of this variational problem for classical mechanical system, results in Euler-Lagrange equations. If you select cartesian coordinates to describe your system, Euler-Lagrange equations give Newtonian equations. Thus, Newtonian mechanics is just a specific case of Euler-Lagrange mechanics and its results/predictions can be confirmed in experiments.