This question is a little strange, so stay with me. I understand that some physical theories apparently cannot be defined in terms of Lagrangians (although these generally are not fundamental theories), whereas some can. But this got me thinking about a related but different question - what are the necessary things we must posit to exist before we can even conceive of defining a Lagrangian theory?

To give an analogy, any theory within a Newtonian framework must take as primitive concepts a Euclidean space parameterised by time as an external variable, with some notion of mass. Whether a given theory obeys or fails to obey Newton's laws is only meaningful once we have these basic concepts, this basic ontology.

So what are the fundamental concepts and things to exist that are necessary for Lagrangian theories to be definable, even if a given theory may not have such a formulation?

  • $\begingroup$ Lagrangian theory can be derived from Newtonian theory, once you have defined energy. $\endgroup$ – Peter Diehr Jun 15 '18 at 11:51
  • $\begingroup$ But it is much more general, Lagrangian formulations exist for QM, QFT, GR, etc. Especially in the case of the standard model it seems to be a primitive notion from which other things are defined. As I stated explicitly, I only brought up Newtonian mechanics as an analogy. $\endgroup$ – user6873235 Jun 15 '18 at 12:57
  • $\begingroup$ The lageangian is an expression of the least action principle where energy and time are Fourier conjugates. Thus energy is a function that defines the system evolution in time. $\endgroup$ – safesphere Jun 15 '18 at 13:34
  • $\begingroup$ You need a real variable called time and a set of real valued quantities whose values are differentiable functions of time. $\endgroup$ – WillO Jun 15 '18 at 16:16
  • $\begingroup$ One thing a lot of people overlook is that Lagrangian theories are fundamentally variational; that is, they assume the existence of something (the action) that nature automatically minimizes. Not many know that you can actually formulate constraints on physical systems at least mathematically that cause the variational/Lagrangian picture to fail! $\endgroup$ – aghostinthefigures Jun 16 '18 at 0:55

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