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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.

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  • $\begingroup$ Related physics.stackexchange.com/q/18588/2451 and links therein. $\endgroup$
    – Qmechanic
    May 30, 2018 at 10:39
  • $\begingroup$ In newtons model the only accelerations are caused collisions and gravity both of which can be calculated at a given instant with the gravitational constant of the universe and the positions of all the particles. simply knowing the initial acceleration is unneeded. besides acceleration changes over time anyway so knowing the initial acceleration is kinda pointless. $\endgroup$
    – Ummdustry
    May 30, 2018 at 11:07
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  • $\begingroup$ Good question but it has been asked and answered many times on this site. I think you need to justify why your question is different, or why the other answers are not satisfactory. $\endgroup$ May 30, 2018 at 14:14
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    $\begingroup$ I think what is tricky about this is that personal experience tells us the exact opposite. It tells us that you must know accelerations. You must know if the car is braking or not. However, those personal experiences are based around not having perfect information. If you knew the position of the drivers' foot, you could determine if they were braking, and if you knew the precise hydraulics of the brake system, you could even determine how hard. Newton is considering a situation with perfect position/velocity information for all particles. $\endgroup$
    – Cort Ammon
    May 30, 2018 at 19:51

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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>

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