Why only positions and velocities

[maze]

Why is the state of a physical system completely determined by only positions and velocities, rather than (possibly) other derivatives?

 

[Fredrik]

This can't be answered in the framework of classical mechanics, other than by pointing out that there's a theorem that guarantees that a differential equation of the form

$$\vec x''(t)=\vec f(\vec x'(t),\vec x(t),t)$$

has exactly one solution for each initial condition, i.e. for each pair of equations of the form

$$\vec x(t_0)=\vec x_0$$
$$\vec x'(t_0)=\vec v_0$$

We're just "lucky" that the functions that describe the acceleration caused by gravitational or electromagnetic interactions have that simple form.

I believe that the reason for it can be traced back to the fact (more of a conjecture really) that any theory of interacting matter must have a low energy approximation in the form of a quantum field theory in order to be consistent with special relativity. The QFTs can contain higher-order derivatives of the fields, which (I'm guessing) imply that the best possible classical equation of motion is a more complicated differential equation. But the terms in the Lagrangian that contain those higher order terms suffer from a condition called non-renormalizability, and that makes them negligible in the low energy limit.

 

[Ninjakannon]

Check out this YouTube video, I watched it only yesterday and I think it'll answer your question. It's 50 minutes, but well worth it!