They are actually, literally equivalent to each other. Differential equations aren't a "purely random walk," they are a statement of what is about to happen based on what is happening. You have three formalisms: Newtonian (differential equations), Lagrangian (action minimization), and Hamiltonian (differential equations, but derived from a scalar field kind of like in Lagrangian mechanics). They are all different ways of writing down the same thing, and their advantages and disadvantages are related to situational convenience.
Thank you for the response, I find this very interesting.
So is it true to say that the production of a particular solution to a differential equation is a statement about how a system will behave based on its initial conditions, and that the statement captures within it the principle of action minimization by virtue of the fact that it is a derivation of information from natural laws?
"Action is minimized" and "F = ma" are the same in the sense that x+5=0 and x+6=1 are the same. In both cases you can go from one to the other with a sequence of deductions. There's no way to tell which one is the "natural law" because you can take either one as a fact and derive the other as a consequence.
