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Re: geometric algebra (was: Re: Flatland is not dark



It is clear from the mail of John Denker, but
I would like to add that with GA you maintain the
same mathematics since you are studying
rotations in newtonian mechanics
until Lorentz transformations in relativity.

Arnulfo Castellanos-Moreno


----- Original Message -----
From: "John Denker" <jsd@AV8N.COM>
To: <PHYS-L@lists.nau.edu>
Sent: Monday, April 19, 2004 5:57 PM
Subject: Re: geometric algebra (was: Re: Flatland is not dark


Larry Smith wrote:

Do you posit that this is true for _any_ physics application of the
cross product, i.e., that there is a "different and grander truth" in
GA? (Another re-wording of the question is: "Are there any cases
where the cross product is better than GA?")

Interesting questions.

Executive summary answer:
-- Sometimes GA is a huge win, yielding new insights.
-- Sometimes GA is a only a small win, yielding the same
results, perhaps slightly more cleanly and more easily.
-- I know of only one case where I have something partially
favorable to say about cross products.

Details:

*) I might as well start with the one case that is
arguably favorable to cross products. Suppose you want
to sketch the "magnetic field lines" popping out of one
end of a bar magnet and flowing around to the other end.
GA says you should draw tubes, not lines. There are
drawings of such tubes in the _Gravitation_ book by Misner,
Thorne, & Wheeler. And you can make a strong case that
the tubes are a better representation of the physics.
But the lines are oftentimes a good-enough approximation,
and they are a bunnnnch easier to draw.

So ... I'm not enough of a GA zealot to tell people to
entirely stop drawing magnetic field lines.

Maybe we can compromise as follows: Draw the line,
but don't mark a north-to-south arrowhead on it.
Instead mark a bivector-like circulation around it.
That way we can at least get rid of the right-hand rule,
which mis-represents the symmetry of electromagnetism.


The rest of the score is pretty lopsided:

*) Angular momentum, torque, and precession. GA is a
big win here, and the more naive the audience the bigger
the win. I've tried explaining cross products to
non-scientists, with disappointingly little success. In
contrast, I can explain precession in terms of bivectors
and a decent fraction of the audience seems to "get it".
http://www.av8n.com/how/htm/motion.html#fig-angular-precession

*) Electromagnetism, in particular magnetic fields:
Again GA is a win, in (at least) the three following
sub-areas:

-- It allows a reasonably graphic explanation of why
the magnetic field of a long straight wire is an
automatic consequence of Coulomb's law plus special
relativity.
http://www.av8n.com/physics/straight-wire.htm

-- It allows a nice answer to little Pierre's
question about the symmetry of a compass needle plus
wire plus battery.
http://www.av8n.com/physics/pierre-puzzle.htm

-- As mentioned at the start of this thread, it allows
us to see a grander truth a grander truth about
electromagnetism in dimensions other than 3+1.

*) Calculating area and volume: GA is at least as
good as other techniques such as the triple scalar
product. Maybe not spectacularly better, but no
worse:
http://www.av8n.com/physics/area-volume.htm

*) When it comes to representing rotations in
three dimensions, there are no easy solutions,
even for easy problems (except the truly trivial
ones). For easy problems, GA can do the job,
but I have not found it to be much easier than
non-GA techniques ... although this could be due
to vastly more experience with non-GA than GA.
But when it comes to hard problems, GA is a huge
win. Everybody who computes 3D rotations these
days -- e.g. for an autopilot or for a
flight-simulator game -- uses a four-variable
representation, i.e. quaternions, which is a
particular subalgebra of geometric algebra.
If you try to do this using the non-GA approach,
e.g. with Euler angles, you will be hounded by
the mathematical equivalent of "gimbal lock".
http://www.av8n.com/physics/rotations.htm

*) When it comes to representing the Lorentz
group, i.e. special relativity, i.e. rotations
plus boosts, then GA is a huge win. It becomes
clear that a boost in the X direction is profoundly
analogous to a "rotation" in the XT plane. The
Wigner rotation (and the Thomas precession, which
is a special case thereof) can be seen as natural
consequences of the fact that boosts don't commute
with rotations, just like rotations don't commute
with rotations around other axes. I've been
meaning to write up a rant about this but haven't
gotten around to it.

*) I haven't looked into it, but I imagine that in
classical inviscid fluid dynamics, the circulation
can be handled by bivectors the same way a magnetic
field is, since the equations are so similar.

=================

So ..... That's the end of my list. Can anybody think
of any other applications where someone might be tempted
to use cross products?