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?