Why do people say that Hamilton's principle is all of classical mechanics? How to get Newton's third law?

Question

From the principle of least (or stationary) action, we get that a classical system will evolve according to Euler-Lagrange equations: $$\\frac{d}{dt}\\bigg (\\frac{\\partial L}{\\partial \\dot{q_i}}\\bigg) = \\frac{\\partial L}{\\partial q_i} .$$ I have often read and heard from physicists that this differential equation encapsulates all of classical mechanics. A glorious reformation of Newton's laws that are more general, compact and much more efficient. I get that if you plug in the value of the Lagrangian, you re-obtain Newton's second law. But Newtonian mechanics is based on 3 laws, is it not? The law of inertia is a special consequence of the second law, so we don't need that, but what about the third law, namely that forces acts in pairs; action equals minus reaction? My question is, can we obtain Newton's third law from this form of Euler-Lagrange equation? I understand that Newton's third law for an isolated $2$-body system follows from total momentum conservation, but what about a system with $N\\geq 3$ particles? If not why do people say that it's all of classical mechanics in a nutshell?

Answer 1

Euler-Lagrange equations (EL equation) describes the principle of motion for particles in a given potential function V.

Newton's third law is the result of the specific property of potential V, which is independent of EL equation. When V=V(|r_i-r_j|), Newton's third law will naturally come out and can be derived mathematically.