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We have recently covered the Lagrangian in our lectures, whereby it was shown that all equations of motion ($x(t)$) satisfying the Euler-Lagrange equation with Lagrangian $L=T-V$, where $T=\frac{1}{2}mv^2$ and $F=-\frac{\partial V}{\partial x}$, must also satisfy $F=m\ddot{x}$.

However, to me, it wasn't intuitive at all why this must have been the case: why, if we minimise (or maximise) some arbitrary quantity $L$, which didn't even have to be conserved, we for some reason get Newton's Second Law, which also implies that $L$ is in fact energy and has to be conserved.

Is there any intuitive way of understanding why ALL possible types of motion MUST minimise this arbitrary quantity $S$ and why, if we do minimise it, we get that $L$ is conserved and happens to represent the total energy of an object?

Qmechanic
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Max
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    As a matter of fact, L is NOT conserved and does NOT represent the total energy of an object. I think you are confusing Lagrangian and Hamiltonian formalism there. – Akerai Nov 17 '19 at 16:58
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    If you are asking why the stationary solution of the Lagrangian gives the right equations of motion that is not easily understood without going further into Quantum Field Theory, but in short it is simply a very good approximation. In QFT path integral formulation the Lagrangian appears as a phase of an exponential that is being integrated over to find transition amplitudes. The part of the solution with stationary phase adds the most to the probability and can be approximated as the only solution in classical case. – Akerai Nov 17 '19 at 16:59
  • Possible duplicates: https://physics.stackexchange.com/q/15899/2451 , https://physics.stackexchange.com/q/9/2451 , https://physics.stackexchange.com/q/78138/2451 and links therein. – Qmechanic Nov 17 '19 at 17:10
  • @Akerai Finally someone agrees - thank you! The Stationary Action Principle is a result of interference of quantum waves representing matter. Here is my answer stating this fact: https://physics.stackexchange.com/q/489241/ – safesphere Nov 17 '19 at 17:53
  • L isn’t conserved T+V is conserved – Eli Nov 17 '19 at 19:49
  • @Akerai Okay this is really silly... I spent an entire day effectively correcting a sign... I was indeed thinking of the Hamiltonian, you're right. And, from your explanation, it appears that the Lagrangian really is nothing more than a solution to the equation where the LHS of the Euler-Lagrange equation equals F-ma, or some other maximisation equation from QM. That is, there isn't much physical meaning to the Lagrangian apart from that it solves some equations. – Max Nov 17 '19 at 21:59
  • Baffled by comments claiming that this has something to do with quantum mechanics. This is a purely classical thing. –  Nov 17 '19 at 23:59
  • The question as written had two parts. One was a reasonable question about the intuition behind minimizing the action. The second was a trivial confusion, which was cleared up in comments. Since the first one is a duplicate, I'm voting to close. –  Nov 18 '19 at 00:04

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