Sometimes physics is hard work ... but sometimes it's nice to do some
physics just for fun. Here are a handful of stories...
Recently there was some discussion of dropping metal squares and
comparing them to cardboard squares. That motivated me to do some
informal experiments:
-- I dropped a square in the broadside "pancake" orientation
-- I dropped a square in some other orientations.
-- I dropped a circular disk in various orientations.
-- I experimented with spin-stabilized disks, like Frisbees.
-- I dropped some rectangles in various orientations.
-- I dropped some airplane-shaped cutouts in various orientations.
-- I dropped a long thin slat in the /tumbling/ mode.
-- I experimented with adding pennies to the cardboard, to shift
the center of mass, thereby changing the stability.
-- I wondered about the definition of "stability". Most of these
things are unstable by any definition ... yet under suitable
conditions they can be sufficiently "close" to stable that the
instability doesn't matter. This is an important concept.
I don't really have the vocabulary to think about or talk about it.
-- et cetera....
The point here is that I wasn't answering a multiple-guess quiz question.
Indeed, nobody had asked a question about this at all. I was not
testing a hypothesis. Originally, the only goal was to see what
would happen. Somewhere during the process I /invented/ a pseudo-goal,
namely to see what shape would fall the quickest and what shape would
fall the slowest. Later I cooked up a new pseudo-goal, namely to see
which shapes could fall both slowly and quickly, depending on initial
conditions.
Even so, the only real goal was to see what would happen. I reckon
all this serves a purpose, but it's hard to identify any short-term
concrete purpose. There may be some longer-term purpose, insofar as
all available data tells me that wondering about stuff tends to increase
smartness. Maybe some day some subset of these experiments will be
useful as a classroom demonstration, but that's not why I did it.
Mostly I did it because it was interesting.
=============================
Recently I started thinking about conservation of charge. I have no
idea what provoked this line of thought.
In line with my deeply-ingrained instinct to look at the connections
between ideas, I asked myself
a) How is conservation of charge related to the Maxwell equations?
b) I had learned about (a) in three-dimensional notation ... but
electromagnetism is much simpler and more informative in four
dimensions, and the idea of conservation is simpler and more
informative in four dimensions, so one might hope that (a) would
look nicer using four-dimensional bivectors ... and one would not
be disappointed. Expressing (a) is ugly in 3D, but in 4D it
comes out as
∇•(∇•F) = 0 [1]
where F is the electromagnetic field bivector. Equation [1] is
amazingly simple. Proving it is trivial, because
∇•(∇•anything) is zero, [2]
for *ANY* bivector (or higher-grade object, just not a vector or
scalar). This is looking more like a fundamental mathematical
identity and less like a special property of the EM field.
c) Continuing that line of thought, we can write
∇∧∇∧(*F) = 0 [3]
where (*...) is the Hodge dual. This is even more elegant, because
∇∧∇∧(anything) is zero, [3]
without restriction, for any scalar, vector, bivector, or anything
else. This should be obvious from the antisymmetry of the wedge
operator and the symmetry of mixed second derivatives. It also
has a profound topological interpretation, namely "the boundary of
a boundary is zero". Reference: Misner, Thorn, Wheeler _Gravitation_.
There must be some corresponding topological interpretation for
equation [3], but I can't immediately think of what it is.
d) Comparing (b) and (c), one has to wonder under what conditions
∇•X is equal to *∇∧*X. In other words, given that for any vector
V, V∧X always raises the grade of X by one (unless the product
vanishes) maybe we can redefine the dot product so that V•X always
lowers the grade of X by one (UtPV). This would make some results --
such as equation [1] -- simpler and more elegant. However, overall
it would be a bad idea, because it would quickly run afoul of the
requirement that
|V•X| should be equal to |X•V| [5]
and I care about equation [5] a lot more than I care about marginal
polish on equation [1]. There exist non-symmetrical "left
contraction" operators that can be used if need be, but for the
moment that doesn't seem worth the trouble.
e) Then I started thinking about how conservation of charge is
related to gauge invariance.
*) There are probably lots of angles I haven't thought of.
Again, the point of the story is that I did not /have/ to think about
any of this, because there was no concrete, short-term goal. Nobody
asked me a question about this. I was not testing a hypothesis.
There was no deadline. It was just something interesting to think
about.
I didn't think about it all at once. I wondered about it for ten
minutes one day and ten minutes the next, and so on.
On the other side of the same coin, there is another reason why I did
not have to think about any of this. I could probably have googled
the right answers on most of these topics ... or I could have just
flipped to the relevant page in MTW. However, that would have defeated
the purpose. I didn't value the answer nearly so much as I valued the
process of wondering about it. Googling makes you more knowledgeable,
whereas wondering about stuff makes you smarter.
=====================
Back when I was in high school, I spent many many hours playing with
waves in a ripple tank. I had no idea that it would actually be
useful. In retrospect, it's obvious why knowing about waves is
useful, but I didn't know that back then. It was just interesting.
=====================
Also in high school, I spend many hours with an oscilloscope and a
couple of wave synthesizers, making Lissajous patterns. I had no
idea what that could possibly be good for. It was just interesting.
However, on several occasions since then I have faced a real-world
problem where a Lissajous display was exactly the right solution.
Also BTW there is a book _Theory of Sound_ by some guy named Rayleigh
that describes yet more clever things that can be done this way,
things I hadn't thought of.
======================
In a previous message I mentioned wondering about the connections
between geometrical optics, Feistel ciphers, and symplectic
integrators ... which you might think are wildly different, but
turn out to overlap in an interesting way.
=================================================
The reason I mention all this in this forum is that I worry that some
parts of the education system are too fixated on short-term goals, to
the detriment of long-term goals ... and to the detriment of having fun.
As you might imagine, I can be VERY focused on short-term goals
when I want to be ... but sometimes its nice to decompress and
work on something that's not an emergency.
To say the same thing another way, the last time you wrote down your
class goals -- or personal goals -- was "wondering about stuff" listed
as one of the goals?
It's super-hard to teach this sort of thing. One suggestion is to
have the students play a game of Six Degrees of Kelvin Baryon ...
i.e. to start with two random physics and find some chain of
connected ideas that lead from one to the other.