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In the 1st page of Landau's "Mechanics", he used "known from experience" and "in principle" to state that a $N$-degree of freedom particle system is completely determined by $2N$ variables ($N$ generalized positions and $N$ generalized velocities) as its initial condition:

Mechanics, p1

Is there a theoretical justification for the statement? i.e., why higher order derivatives ($>=2$) do not matter if $q$ and $q'$ are specified? Or other combinations of $2N$ variables (e.g., $N$ positions, $N/2$ velocities, $N/2$ accelerations) can also uniquely determine the system?

bruin
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    This is just the Newton laws. Without external forces, an object moves linearly indefinitely. Which means that the second derivative is zero. To change that a force needs to be applied. Or conversely, everything that changes this state is a force. The natural place to detect that force is in the second derivative. Thus is born $m\ddot x=F(t,x,\dot x)$. A second order system has exactly the claimed properties. – Lutz Lehmann Oct 15 '19 at 11:51
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    The second derivative, i.e. the accelerations, are detrmined by the forces in play. The only forces in the n-body problem are the gravity by the bodies themselves, which is completely determined by their locations. The third derivative is similarly determined by change in location, i.e. the velocities. Once you know the locations and velocities, all higher derivatives follow. – Jaap Scherphuis Oct 15 '19 at 11:57
  • @LutzL @ Jaap Thanks for your answers. Now, if I understand correctly, there is an implicit assumption that the forces involved in the study of the system are known or at least can be determined (somehow.) So the 2nd sentence above means that: $\text{some-assumptions-of-the-system}+q+\dot{q}\implies F \implies \ddot{q}$. Right? – bruin Oct 16 '19 at 01:15
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    Yes. Also the experience that in all naturally occurring processes this is sufficient. One could of course construct something using an acceleration sensor to control some actuator, then the acceleration is input and the equation describing it would have higher order. – Lutz Lehmann Oct 16 '19 at 06:01

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