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We use Lagrangians and variational calculus for almost all of physics, from Newtonian mechanics to QFT. Is there any theorem in mathematics that guarantees that all possible dynamics of objects (say time evolution of fields, particles) can be translated into a problem of variational calculus provided we find the right Lagrangian?

In most cases (say EM and CM), we start with observable dynamics then condense them into a differential equation (Maxwell's equations and Newton's second law). Then we recast the DE into a Lagrangian. But the same is not exactly true of QFT is it?

In some sense, are there physical processes whose dynamics cannot be modelled using variational calculus?

Qmechanic
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Lost_Soul
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    There are some theories which are conjectured that are not described by Lagrangian approach. For examples (2,0) 6d superconformal theory. – reza-ebadi Jul 11 '22 at 10:04
  • Hysteresis (i.e. history dependence) and dissipation are "problematic" (dissipation implies that volume in the phase space is NOT conserved, in contrast with the Liouville theorem). However, "fundamental" theories where all the DOF are accounted for can typically be modelled in Lagrangian terms. See also the "inverse problem": https://physics.stackexchange.com/a/561253/226902 – Quillo Jul 11 '22 at 10:06
  • Idk if this is a trivial fact you have thought of, but the reason we use variational calculus is because we want to minimize a quantity. For example the EoMs arise as a consequence of the action minimization. Hence, your question might be restated as "is there a physical process, whose dynamics are not a consequence of some sort of action (or other quantity) minimization?" – schris38 Jul 11 '22 at 10:06
  • I do not know the exact question to your answer, I am just trying to contribute with soem food for thought... – schris38 Jul 11 '22 at 10:07
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    Possible duplicates: https://physics.stackexchange.com/q/3500/2451 , https://physics.stackexchange.com/q/163073/2451 and links therein. – Qmechanic Jul 11 '22 at 10:07

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