This includes some numerical examples, with diagrams.
The first example involves motion in two dimensions, i.e. circular
Keplerian orbits. This is interesting, because the multi-dimensional
behavior is richer than the one-dimensional behavior, in ways that
would not have been easy to guess.
This is important, because Liouville’s theorem is exceedingly
fundamental. If you could build a device that violated the Liouville
theorem, it could almost certainly be used, quite directly, to:
1. Violate the second law of thermodynamics.
2. Violate the Heisenberg uncertainty principle.
3. Violate the brightness theorem (in optics).
4. Violate the fluctuation/dissipation theorem (in physics, electronics, etc.)
5. .... and so forth ....
I mention those items without explaining them, because explaining them
would essentially require explaining all of physics.
Also note that ideas of “conservation of phase space” crop up in
connection with symplectic integrators, which are a category of
numerical methods, useful for computing the solutions to differential
equations. Symplectic integrators usually (but not quite always)
have advantages over non-symplectic integrators, especially when
the equation being integrated comes from a non-dissipative physical
system. If you’re going to use such methods, it’s nice to understand
just what is being conserved, and why.
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I threw this together just now, so there are likely to be lots of
bugs. Comments and suggestions are welcome.
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Pedagogical and philosophical remark:
In camp "A", there are lots of folks for whom the equations are paramount,
and other things like pictures are irrelevant, annoying, or worse.
Meanwhile, in camp "B", there are lots of folks who don't really
understand something unless they can visualize it. The pictures guide,
motivate, and clarify the calculations ... and vice versa.
The track record shows that both groups are about equally successful at
doing physics.
However, I would argue that approach "B" has pedagogical advantages, for
the simple reason that if you present the equations and the pictures,
students in camp "B" will appreciate it. Students in camp "A" can ignore
the pictures that were presented more easily than students in camp "B"
can construct pictures that weren't presented.
I mention this because I spent a couple hours googling for diagrams
illustrating Liouville's theorem, without success. So I put up my
own diagrams.
For as long as I can remember, even back when I was a sorcerer's
apprentice, I've been able to visualize these things in my head.
That's fine as far as it goes, but when explaining things to other
people, it helps to have tangible diagrams to show them.