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It makes a lot of sense to me to imagine a cannonball flying through space as not so much experiencing a macroscopic non-conservative drag force, but as pushing a bunch of air molecules and giving away its momentum and energy to them, thereby losing speed. I'd also imagine that if one took a snapshot of the cannonball in mid-flight, replicated the system exactly down to each molecule (assuming this is possible), and then reversed all of the velocities and angular velocities, and pressed 'play', the cannonball would end up flying back to where it was launched. Is this right?

Drag, therefore, is simply an emergent phenomenon that is a result of our simplification of the entire system, isn't it? And is really the result of kinematic principles and classical mechanics.

In thinking about this I noticed that if we think of drag on mechanical terms, conservation of mechanical energy is held -- and in the result, time-reversal symmetry (right?). So conservation of mechanical energy is therefore "linked" to time-reversal symmetry -- would this be related to how mechanical energy is the conserved quantity associated with time translation symmetry, as well?

What about something like a sugar cube dissolving in tea? If I dropped a sugar cube into tea, and then watched it dissolve, froze that system and recreated it while swapping all velocities of all of the molecules, would the sugar cube re-gather and then pop out of the cup?

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This is called the "Lorschmidt paradox", and it is discussed ad-nauseum in the late 19th century physics literature. It is very interesting, but very old. – Ron Maimon Jan 11 '12 at 15:16

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Great question!

First of all, not all systems exhibit time reversal symmetry, the most common examples are systems with an external magnetic field. But this is not the real answer. The question is - if we assume that the microscopic laws (cannon ball-air interaction and sugar-water interaction in your examples) obey time reversal symmetry, how come the macroscopic universe clearly does not obey this symmetry?

The answer comes when we are dealing with a large number of particles. If your canon ball was to collide with a single air molecule (or a dozen), the system would be perfectly time reversal symmetric. However, when dealing with a macroscopic number ($\sim10^{24}$) of particles, one cannot solve the equations of motion for all the particles and thermodynamic considerations should be used. Namely - the entropy must increase with the system's evolution. Therefore, the macroscopic laws are not time-reversal symmetric, while the microscopic laws are.

This is, as you correctly stated, a result of our simplification of the system, and of our inability to track the motion of individual molecules. For example, because the cannon-ball system is time invariant, energy is conserved. However, we describe the energy transfer from the cannon ball to the air (due to drag) as "heat", and not as kinetic energy of individual molecules. Actually, we say that energy that was once stored in degrees of freedom which are of interest (the coordinates of the ball) was transferred to degrees of freedom which are we are not able to track (coordinates of air molecules). Nevertheless, it is guaranteed to be conserved, due to the microscopic time translational symmetry.

This is thoroughly discussed here and here, and in many thermodynamics/stat mech textbooks.

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I'm not saying that one would try to solve the equations of motion -- I'm suggesting one reconstructs the exact physical conditions, and have the physics play out themselves. – Justin L. Jan 11 '12 at 8:00
Strictly speaking, in classical mechanics the answer is that the dynamics are fully reversible. However, In the real world you are bounded, even theoretically, by the uncertainty principle, not to mention the outrageous impossibleness of reconstructing the the system with reversed velocities. Also, many-body dynamics are generically chaotic, and infinitesimal deviations of initial conditions will result in a significantly different evolution. – yohBS Jan 11 '12 at 11:53
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@YohBS: while you are right that systems with an external magnetic field break time-reversal symmetry, if you time-reverse the magnetic field source, you get correct reversed behavior. Only a classical Kaon field (if you could make a macroscopic one), would violates T intrinsically measurably directly. Internally generated magnetic fields do not break the T symmetry. It is good to say this clearly somewhere. The surprise here is that external static fields other than magnetic fields do not break time reversal symmetry, because they aren't sourced by motion. – Ron Maimon Jan 11 '12 at 17:13
@Ron Maimon. You are, of course, right about that, but this is a minor point in my answer. I edited it now. – yohBS Jan 12 '12 at 23:00

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