is the world lagrangian?

My day started with a long and very fun conversation with Monica Pate (NYU) about conservation laws in classical physics. As we all know, conservation laws are related to symmetries; each symmetry of the laws of physics creates a conservation law. Or does it? Well, it's a theorem! But it's a theorem when the laws of physics are lagrangian (or hamiltonian). That is, every symmetry of a hamiltonian system is associated with a conservation law in that system. So I asked: How do we know if or whether the world is lagrangian or hamiltonian? How could we know that? My best guess is that we know it because of these very conservation laws! The situation is complex.