Challenging Susskind's "rule" for classical mechanics

[Isaac0427]

Hi all! I was directed (on another thread) to professor Leonard Susskind's first lecture on classical mechanics. I learned a lot from it, however he introduced a "rule" for "acceptable" classical fields/equations that I have a little trouble with. He explains classical mechanics as if you know the position in space and time of an object, you can determine everywhere it has been, and everywhere it will go (I know my wording is a little off). He says that if you have a flowchart of the states of the particle, each state (in order to be "acceptable" in classical mechanics) may have no more than 1 arrow going to the state, and 1 arrow going away from the state. This, as I have seen it, generally works, however I believe I have found an exception that I don't think he talks about. In a field with a point that acts as a sink, the sequence may look like this (with the sink being r₀): r₃->r₂->r₁->r₀->r₀ and so on. If you made a flowchart, the state r₀ would have 2 arrows pointing towards it (see the lecture if you are confused with what I'm saying). Many fields with sinks, such as the flow of a fluid or gravity, are in classical mechanics, but they break this rule. Am I understanding this correct?
Thanks in advance!

 

[fzero]

I think that by "acceptable" Susskind is talking about closed, conservative systems, like a gas of point particles subject to Coulomb forces in a perfectly insulated box. Other types of systems, such as "dissipative" systems can have trajectories that converge since information is leaving the system. For instance, if we had a ball rolling in a frictionless bowl, we could follow the trajectory forwards and backwards in time using Newton's laws. However, if we introduce friction, we know that all trajectories will eventually end with the ball at rest at the bottom of the bowl. Once the ball is at rest, all information about the previous trajectory has been lost to heating the bowl/environment.

 

[jfizzix]

Though it might look very much like it should be, the rule is not actually broken.

Another element of classical mechanics is that it is reversible.
If you could have multiple initial states reach precisely the same final state, then it would also have to be that the same initial state could reach multiple final states. This would be non-deterministic, and not permissible in classical mechanics.

If you consider two different initial states, and propagate them in time, you might find that their final states get more and more similar (depending on what's going on) without truly being identical.

That being said, if you have multiple initial states very close together, their final states could be very different.
Such behavior is known as chaotic (small changes over long times give wildly different results, (see for example turbulent fluid dynamics and weather forecasting)).

 

[phinds]

Another element of classical mechanics is that it is reversible.
If you could have multiple initial states reach precisely the same final state, then it would also have to be that the same initial state could reach multiple final states.

I am curious about that statement. If you imagine a bowl with a ball at the Northern rim let go to roll to the bottom, with friction, eventually it will come to rest at the bottom. It would do exactly the same thing if it had started at the Southern rim. That covers different initial states ending up at the same final state. What I don't get is how one of those initial states could end up at any different final state.

 

[A.T.]

with friction ... That covers different initial states ending up at the same final state.

The final state isn't the same. The thermal movement caused by friction is different.

 

[anorlunda]

I think Susskind's rule closely associated with Liouville's theorum? The number of possible states must remain constant.

In the same classical course a few lectures later, he talks more specifically about Liouville's Theorum. If you think that you can disprove that theorum, have at it.

 

[phinds]

The final state isn't the same. The thermal movement caused by friction is different.

OK, I'm not seeing it, but I'll take your word for it. Is it because the friction causes slightly chaotic events?

 

[kith]

OK, I'm not seeing it, but I'll take your word for it. Is it because the friction causes slightly chaotic events?

I wouldn't put it that way. The final state of the ball may be exactly the same in both cases. What the reversibility requirement forbids is that the state of the whole system (ball+bowl) is the same in both cases. So the crucial thing is to include the state of the bowl.

Fixing macroscopic quantities -like the amount of heat in the bowl- isn't enough to fix the microscopic state to which the fundamental laws apply (as you probably know, entropy can be viewed as a measure of how many different microstates are compatible with a given macrostate).

So even if we couldn't distinguish between the two situations by performing actual measurements on the ball and the bowl, the reversibility of Newton's second law tells us that the underlying microstates -which are not directly accessible- have to be different.

 

[phinds]

I wouldn't put it that way. The final state of the ball may be exactly the same in both cases. What the reversibility requirement forbids is that the state of the whole system (ball+bowl) is the same in both cases. So the crucial thing is to include the state of the bowl.

Fixing macroscopic quantities -like the amount of heat in the bowl- isn't enough to fix the microscopic state to which the fundamental laws apply (as you probably know, entropy can be viewed as a measure of how many different microstates are compatible with a given macrostate).

So even if we couldn't distinguish between the two situations by performing actual measurements on the ball and the bowl, the reversibility of Newton's second law tells us that the underlying microstates -which are not directly accessible- have to be different.

That's what I was missing. Thanks for that clarification.

 

[A.T.]

OK, I'm not seeing it, but I'll take your word for it. Is it because the friction causes slightly chaotic events?

Yes. For the final state to be identical, all the velocities of all the atoms would have to be identical.

 

[phinds]

Yes. For the final state to be identical, all the velocities of all the atoms would have to be identical.

Got it. Thanks.