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I just began reading the Landau and Lifshitz book on classical mechanics. It states on the first page of Chapter 1 that:

Mathematically, this means that, if all the coordinates $q$ and velocities $\dot{q}$ are given at some instant of time, the accelerations $\ddot{q}$ at that instant are uniquely defined.

For given positions and velocities of a system of particles at a given instant, can't each particle have an arbitrary acceleration? Also, aren't accelerations determined from forces?

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  • $\begingroup$ Can you be more precise about where it says this, and maybe provide the quote along with context? $\endgroup$ – Mike May 5 '15 at 12:55
  • $\begingroup$ Oh, bottom of page 1, I guess. $\endgroup$ – Mike May 5 '15 at 12:59
  • $\begingroup$ @Mike Yup, that's right $\endgroup$ – IanDsouza May 5 '15 at 13:04
  • $\begingroup$ Your last sentence is the key here - given positions, velocities, and the equations for the forces in the system, the accelerations will be uniquely defined. No, the particles can not have arbitrary accelerations - they must have the accelerations as defined by the classical mechanics at play in the situation. $\endgroup$ – Jon Custer May 5 '15 at 13:44
  • $\begingroup$ Related: physics.stackexchange.com/q/18588/2451 , physics.stackexchange.com/q/4102/2451 and links therein. $\endgroup$ – Qmechanic May 5 '15 at 16:18
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The explanation comes from earlier in that paragraph:

If all the co-ordinates and velocities are simultaneously specified, it is known from experience that the state of the system is completely determined and that its subsequent motion can, in principle, be calculated.

This is just saying the familiar thing that if you know the laws of physics for the system in question, you have to specify the (generalized) positions and velocities at one instant, and then you can predict the motion for all time -- including the acceleration.

So the text isn't entirely clear, but it's definitely not saying that knowing just the generalized coordinates and velocities is enough; it's also implicitly saying that you need an equation of motion.

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