Uniqueness of Acceleration: Understanding Landau's Mechanics

[micromass]

This is one of the better trolls we have had in a long time I must say.

Dunno, that guy from yesterday with his numerology was pretty good.

 

[WannabeNewton]

Dunno, that guy from yesterday with his numerology was pretty good.

Yeah but he was too polite. What kind of troll is polite?

 

[berkeman]

This [STRIKE]is[/STRIKE] was one of the better trolls we have had in a long time I must say.

I corrected your post.

 

[Jano L.]

It is an experimental result.

You can invent a framework where the third derivative of the position depends on position, velocities and accelerations, where you have to know all three to get full knowledge about the system. According to all measurements done so far, we do not live in such a universe-

I think this was a nice answer. In other words, we use 2nd order equations with 2 initial conditions and not more because that is the simplest framework that works well in practice. We even call it "Newton's laws", but this does not prevent the possibility that some motions are out of scope of such theory, we just did not observe such motions.

Parenthetically. Even in the framework of 2nd order differential equations, position and velocity is not always sufficient. Consider motion of particle in the potential

$$
U(x) = -\frac{|x|^{3/2}}{3/2}
$$
with initial conditions
$$
x(0) = 0
$$
$$
\dot{x}(0) = 0
$$
and equation of motion
$$
m\ddot{x} = -\partial U/\partial x.
$$

This has infinity of solutions

$$
x(t) = 0,
$$
or
$$
x(t)=\left(\frac{1}{12m}\right)^2(t-t_0)^4 \theta(t-t_0),
$$
for any $t_0>0$. Which one is realized can be determined if we know the value of $d^4x/dt^4(t)$ in addition.

 

[lightarrow]

In Landau's Mechanics it states "If all co-ordinates and velocities are simultaneously specified, it is know 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 velocities $\dot{q}$ are given at some instant, the accelerations $\ddot{q}$ at that instant are uniquely defined."

My question is why is this so. I understand that from knowing the co-ordinates of a mechancial system the future evolution of a system is not uniquely determined. But how does the additional knowledge of the velocities uniquely determine the acceleration of the system and hence its future mechanical state?

It is sincerely very difficult to understand what you intended to ask in this thread, so I can try with the following.

We know from 2° Newton law that:

$$m\ddot{x} = F$$ where F is the force field. You seem to be asking why F depends on $${x}, \dot{x}, t$$ only, and not even on $$\ddot{x}$$for example. So let's assume a force field F' depended on the second derivative of the position, so that Newton's law would be:$$m\ddot{x} = F'({x},\dot{x},\ddot{x},t)$$But this would still be a second order differential equation on the unknown x(t), so its solution would depend again from 2 initial conditions only, for example, the usual:$$x(0),\dot{x}(0).$$