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's worth pointing out that Hamilton didn't develop his ideas to explain mechanics, but to explain optics. It was a decade or so before he realized that the very same mathematics that could explain the angles of rays of light in a telescope could also explain stuff like momentum in a mechanical system. Of course, this recognition kicked off the long chain of events that led to quantum mechanics.

I wonder if the successor to Euler-Lagrange is already out there somewhere, but not yet applied to an area of physics where it will be revolutionary.