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.

  • $\begingroup$ Related physics.stackexchange.com/q/18588/2451 and links therein. $\endgroup$ – Qmechanic May 30 '18 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 '18 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. $\endgroup$ – sammy gerbil May 30 '18 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 '18 at 19:51

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