I'm not sure that "static" is the appropriate antonym to "dynamic" here.
If we set aside the fact that the term "dynamic" was apparently chosen here basically because it sounded cool, and for no good reason, and are trying to retroactively redefine it in a way where the word has meaning, then:
With recursion the computation order is predefined and deterministic.
With dynamic programming the computation order is more flexible since the core of the technique is just memoizing prior work. Your DP solution could be organized in recursive top-down fashion, or more optimally in bottom-up fashion if the algorithm is strictly recursive by nature, or neither of the above.
i.e. With dynamic programming in general you are just reusing results if/when they exist, else calculating them - it a more "go with the flow" dynamic approach than recursion where the order of calls is strictly prescribed.
Again, if the term linear programming makes sense for linear optimization problems, then the term dynamic makes sense for time dependent ones.
Also: the core of dynamic programming is breaking down a global optimization problem into many interacting local pieces. The Bellmann equation, which is the core mathematical result of dynamic programming, features neither recursion nor memoization.
OK, although that seems to be a different (maybe more thoughtful!) use of the term "dynamic programming".
It reminds also of "blackboard-based" AI systems where a bunch of cooperating agents with different capabilities jointly work on a problem by posing sub-problems and sharing them on a common blackboard where other agents can grab them and post a solution back on the board.
Look at the Wikipedia article on dynamic programming. They are very much related. My impression is that the deep part of the math of dynamic programming, the decomposition into interrelated subproblems, is trivial in the CS text book examples. Then all that's left are implementation aspects.
Why do you think that what I wrote implies they are static?
I didn't say that recursion and caching are the opposite of dynamic, I said they are essentially orthogonal concepts to it.
Generally speaking, dynamical systems are systems that have some sort of dependence on time [1]. In dynamic programming we have an optimization/decision problem in which time plays an essential role. The optimality is measured with respect to the whole time evolution. Bellmann showed that you can break this intertemporal optimization down into individual time steps.
So the key observation of dynamic programming is how to turn an "all times at once" problem into a "one time at a time" problem.
The reason that Hamilton's name shows up next to Bellmann here is that physicists and mathematicians (Lagrange, Hamilton, Jacobi) figured out that you can do the inverse: You can write the dynamical equations of physics as an intertemporal (all times at once) optimization problem. This has been one of the most influential ideas in theoretical physics _ever_, especially after Noether's theorems leveraged this structure to connect symmetries and conserved quantities. In many fields of theoretical physics you no longer write down differential equations to model a dynamical system, you write down a "Lagrangian", and mean that your system follows those trajectories that minimize the integral of the Lagrangian over all time. So the central ideas of dynamic programming are extremely tightly related to physics and dynamical systems.
Edit: After reading a bit more I think I understand why people focus on this. The algorithmic examples with shortest paths sort of show the point. To me the core point of Dynamic Programming is before you start thinking about recursion or memoization. It's when you establish that you have optimal substructures:
what is dynamical is that the subproblems depend on each other, in the way that what happens at time t+1 depends on what happens at time t, or, more to the point, what is the optimal decision at time t depends on what the optimal decision at time t+1 is. If your subproblems are ordered by time, then of course you don't need recursion to solve them, you just iterate over time steps, hence my confusion.
