Uniquely Defined Accelerations

[LukasMont]

From Landau's book, here's the following extract:

"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. Mathematically this means that, if all the co-ordinates q and q˙ are given at some instant, the accelerations q¨ at that instant are uniquely defined".

I don't understand how that can be the case. Knowing the positions and velocities in a given moment allow me to calculate the new positions at a dt time afterwards, if accelerations are all zero. If not, I'll need to know the accelerations from some other source, like knowing the forces acting on the system, etc. I don't see how merely having positions and accelerations in a given moment gives you the accelerations in that moment as well.

 

[Haborix]

I think in this case Landau has a little more in the back of his mind. I would interpret "from experience" to mean something like all the classical forces we know depend on at most the first derivative of position. In that sense, any acceleration at a particular time is determined by the position or velocity at that time.

 

[LukasMont]

I think in this case Landau has a little more in the back of his mind. I would interpret "from experience" to mean something like all the classical forces we know depend on at most the first derivative of position. In that sense, any acceleration at a particular time is determined by the position or velocity at that time.

@Haborix, that was my interpretation as well...but I thought it would be better to bring the topic into discussion, since sometimes in Landau's books he makes this fast, "simple" conclusions which are actually deeper than his tone may convey. Thanks for your input!

 

[Haborix]

@Haborix, that was my interpretation as well...but I thought it would be better to bring the topic into discussion, since sometimes in Landau's books he makes this fast, "simple" conclusions which are actually deeper than his tone may convey. Thanks for your input!

That's definitely a good instinct to have when you go through Landau, but I think in this instant he's just getting the reader primed to see how these facts fall out naturally from extremizing the action.