After some years of physics studies I'm accustomed to the Hamiltonian principle but I sometimes still wonder why physicists tacitly assume that the eq.s of motion of any physical theory (no matter if quantized or not, relativistic or not, strings etc.) can be obtained as Euler-Lagrange equations of some variational problem which severely restricts the possible eq.s of motion. Did I overlook something obvious? Even Ramond (in

*Field Theory - A Modern Primer*) says

How do we know that maybe important new physics don't lie beyond the realm of Euler-Lagrange eq.s?It is a most beautiful and awe-inspiring fact that all fundamental laws of Classical Physics can be understood in terms of one mathematical construct called the Action.