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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.

trynalearn
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  • Related https://physics.stackexchange.com/q/18588/2451 and links therein. – Qmechanic May 30 '18 at 10:39
  • 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. – Ummdustry May 30 '18 at 11:07
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    Possible duplicate of Newton's principle of determinacy or Why are position and velocity enough for prediction and acceleration is unnecessary? or Why are coordinates and velocities sufficient to completely determine the state and determine the subsequent motion of a mechanical system? – sammy gerbil May 30 '18 at 12:21
  • 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. – sammy gerbil May 30 '18 at 14:14
  • @sammy gerbil I have looked at all of these links and IMO none of them offer an intuitive/physical explanation of why acceleration is not required. All of the answers are mathematical. I have a similar question: https://physics.stackexchange.com/q/399647/45664. – user45664 May 30 '18 at 18:18
  • @sammy gerbil I feel like asking the question one more time while asking for an intuitive/physical answer and specifically excluding math-only answers, and also include all of the existing links. Would this be immediately declared a duplicate? – user45664 May 30 '18 at 18:25
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    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. – Cort Ammon May 30 '18 at 19:51
  • It's also tricky because the intuition needs to convince you of a negative statement. It's not like saying "you need to know velocity in addition to position," which can be proven with flash photography. All you need is one example to make that work. To prove the negative you must exhaust all other possibilities, which is an endless task. – Cort Ammon May 30 '18 at 19:55
  • Your other question asked for an intuitive answer to a mathematical question, but you selected the more mathematical of the 2 answers as best answer. Why are you being more demanding about this question? There are many answers to this question which are far less mathematical than the one you selected. ... Instead of posting a new question, which may only attract the same answers, you can post a bounty to an existing open question. The process allows you to explain why current answers are unsatisfactory and what kind of answer you want. – sammy gerbil May 31 '18 at 10:44
  • The statement is not true in general. It is only true for a limited class of Newtonian systems. One can easily construct physical systems which do not behave according to this principle, they are then simply not Newtonian systems. – FlatterMann Nov 06 '22 at 15:38

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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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