Do physical laws require 2nd derivatives

Question

Is it a happy coincidence that second degree differential equations approximate reality or a necessity? They describe how a system will evolve from one state to the next, but surely the ultimate laws of nature will not depend on two prior initial conditions. Surely reality can compute its next state from the current state alone.

Answer 1

Actually if you think of classical (and non-relativistic) physics, the canonical equations of motion of the Hamiltonian formalism $$ \dot p=-\frac{\partial H}{\partial q}\, ,\qquad \dot q=\frac{\partial H}{\partial p} $$ are first order. In addition, the Hamilton-Jacobi equation $$ H+\frac{\partial S}{\partial t}=0\, ,\qquad p=\frac{\partial S}{\partial q} $$ is also of first order.

It is true that in Lagrangian mechanics the equations of motions are of second order, but my personal appreciation is that the Hamiltonian formalism - because the equations of motions are expressed in a more symmetrical way amenable to symplectic transformations and because it leads to deeper results such as Liouville's theorem and other associated invariants - is more fundamental.