Newton's Principle of Determinacy (intuitive explanation)

Question

I was reading Arnold's Mathematical Methods of Classical Mechanics.

In it he speaks of "Newton's Principle of Determinacy". He says for a mechanical system (collection of point masses in 3D Euclidean Space), it's future is uniquely determined by providing all the positions and velocities of the points.

He adds that we can imagine a world in which we would also need to know the acceleration, but from experience we can see that in our world this is not the case.

It is not clear to me that you don't need to know the accelerations of each particle too.

I am told this has something to do with the fact that the equation for motion is a 2nd order ODE, and so from a mathematical point of view, it can be seen that positions and velocities give all information.

Yet I am wondering if someone can explain why we don't need to know accelerations from a intuitively physical point of view based on our personal experiences, as Arnold alluded to.

Answer 1

According to Arnold, the statement that says ``we don't need acceleration to predict the future'' is simply an experimental fact. Furthermore, it's said later in Arnold's book that positions and velocities determine the acceleration [see section D (Newton's equations), pp 8].

There is a function $\mathbf{F}:R^N \times R^N \times R \to R^N$ such that $$\mathbf{\ddot{x}} = \mathbf{F}(\mathbf{x,\dot{x},}t).$$ Newton used equation above as the basis of mechanics (Newton's equation).

Now, suppose there are corrections to the equation above including terms of $\dddot{x}, \ddddot{x},...$ and so on. It is possible the show that differential equations with these ``correction terms'' have problems with causality and locality. See, e.g., the discussion in the link: <physics.stackexchange.com/q/18588/2451>