And I don’t understand why the acceleration can be specified if we know the coordinates and velocity
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$\begingroup$ The paragraph is incorrect or very poorly worded. Knowing the location of a point, say $\vec{r}=0$ and knowing its velocity, say $\vec{v}=(1,0,0)$, what is the acceleration? $\endgroup$– npojoCommented Sep 16, 2018 at 9:35
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3 Answers
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The passage you posted is worded poorly, in my opinion. Here is what I think it means.
If you have some law of physics that says you have a force on a particle that only depends on its position and velocity at a given instant, you could write
$$ F = F(q, \dot q) $$ that is, the force $F$ is a function of $q$ and $\dot q$. From Newton's law, $$ F = m \ddot q $$ so $$ \ddot q = \frac{1}{m} F(q, \dot q). $$ In other words, if you know $q$ and $\dot q$, you know $\ddot q$.
answered Sep 16, 2018 at 6:47
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Try throwing a ball.
The only acceleration on it is due to the gravitational field, roughly $10 m/s^2$ and directed downwards.
However we see straight away that the velocity of the ball changes and so we can see that knowing position and velocity doesn't determine the acceleration.
Mathematically, this is just the observation that specifying the zero and first derivative of a function doesn't help determine its second derivative.
answered Sep 16, 2018 at 7:17
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Do we agree that you can write the velocity as the time derivative of x, and the acceleration as time derivative of velocity more commonly noted as $$\dot{x} = v$$ $$\ddot{x}=\dot{v} = a $$ ?
