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When you begin learning physics, you start with equations of motion applied to various physics systems. In classical mechanics course you learn, that exists Lagrangian/action of a system, which gives you, after applying Euler-Lagrange equations, the solution for equations of motion - we can say that Lagrangian is something higher than EoM, because it consists all physics of a system inside, and even gives us information about conserved values, during to symmetries. Lagrangian formalism is so elegant and simple, that we try to apply it almost everywhere, mainly in field theory and quantum mechanics.

My question is - is there a mathematical object which is more fundamental to Lagrangian in a same way like Lagrangian is more fundamental to equations of motion?

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    $\begingroup$ in what sense is the Lagrangian superior to the equations of motion? $\endgroup$ – AccidentalFourierTransform Apr 20 '16 at 12:31
  • $\begingroup$ @AccidentalFourierTransform I'm guessing that's supposed to mean "more general", perhaps $\endgroup$ – David Z Apr 20 '16 at 12:34
  • $\begingroup$ Have you seen the Hamiltonian approach? $\endgroup$ – Viktor Apr 20 '16 at 12:36
  • $\begingroup$ @DavidZ or "simpler" maybe? [but it's not always true that the Lagrangian is simpler than the eom, so I'm not sure...] or even "more fundamental"? $\endgroup$ – AccidentalFourierTransform Apr 20 '16 at 12:37
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    $\begingroup$ @CheshireCat In that case: the EoM are unique and the Lagrangian is not. To me, this means that the EoM are superior to the Lagrangian, and not the other way around! $\endgroup$ – AccidentalFourierTransform Apr 20 '16 at 12:42
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Going from action to EOMs is simple: it is just (functional) differentiation. Going the other way from EOMs to the action is hard: It is (functional) integration, and sometimes impossible!

OP is now essentially asking:

Can we integrate one more time?

Well, not the action itself. But if we replace the EOMs and the Lagrangian $L$ with their dynamical (as opposed to kinematic) counterparts, namely the (generalized) forces $Q_i$ and the (generalized) potential $U$, respectively, then it is sometimes possible (typically in SUSY theories) to integrate one more time in a certain sense: The result is known as a prepotential.

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  • $\begingroup$ Thank you mate, you gave me some nice sources to read :) $\endgroup$ – Cheshire Cat Apr 20 '16 at 22:30
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You can "derive" the Lagrangian formulation from Shannon entropy arguing Liouville's theorem in reverse.

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    $\begingroup$ can you provide a source for this? $\endgroup$ – anon01 Apr 21 '16 at 3:10
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    $\begingroup$ The second answer of this works physics.stackexchange.com/q/47581 You can also argue unitarity and Feynman's path integral ansatz requires the Lagrangian to have its form in order for classical mechanics to be recovered. $\endgroup$ – user114978 Apr 21 '16 at 3:18

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