In classical mechanics, if you take a snapshot and get the momentary positions and velocities of all particles in a system, you can derive all past and future paths of the particles. It doesn't seem obvious why the position and its first derivative are enough and no further derivatives are needed.
For some reason the accelerations (forces) can be expressed by formulas that only mention the position and velocity of particles. For example, the gravitational force only requires knowing positions but some electromagnetic things need velocities as well. Why doesn't anything need the second derivative (acceleration)?
Does this say something about the universe or rather about our way of analysis?
Could we come up with a theory that only requires a snapshot of the positions? Could we devise a set of concepts and formulas where the second derivative is also required for prediction and instead of forces we'd be talking about stuff that induces third derivatives of motion?
Does modern physics (e.g. relativity) have something to say about this curious thing?