Higher than Lagrangian/action?

Question

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?

Answer 1

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.

Answer 2

You can "derive" the Lagrangian formulation from Shannon entropy arguing Liouville's theorem in reverse.