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[Phys-l] zeroth order and first order



Hi Folks --

I've seen the following confusion arise in several different
situations, so I reckon it's worth discussing.

In the case of a car tire, we have
-- a zeroth-order term representing the tension in the bead, plus
-- a first-order term representing the pull of the sidewall.

In the case of an airplane wing, we have
-- a zeroth-order term representing ambient atmospheric pressure, plus
-- a first-order term representing gauge pressure (suction or overpressure).


Now it is possible to have a holy war between the big-endians and the
little-endians.
A) The big-endians argue that the zeroth-order term is infinitely
important, and it is beyond meaningless to discuss the first-order
term except as a perturbation on the zeroth-order term.
B) The little-endians argue that for a wide class of practical
purposes, including calculations of the net upward force, the
zeroth-order term drops out, so that for such purposes the
calculation can be correctly expressed just in terms of the
first-order term, without mentioning the zeroth-order term at
all.


I hope the folks on this list are grown-up enough to see that each side
is /mostly/ right. In the case of the tire:
A) It is infinitely important to have a lot of tension in the bead;
otherwise the system would explode. Literally.
B) On the other hand, if you are only interested in the net upward force,
*after* you have ascertained that there is "enough" tension in the bead,
the bead-tension is of no *further* importance. By symmetry, it drops
out of the calculation.
C) On the third hand, if you want a microscopic understanding of the
mechanism, you might need to refer back to the bead tension, because
in a figure such as
http://www.av8n.com/physics/img48/tire.png
the lower sidewall cannot push up on the rim; it can only have a
*reduced* downward force, so microscopically we need the bead tension
in order to explain the mechanism of the upward force.

The same thing happens for an airplane wing:
A) The ambient pressure is infinitely important. If we didn't have any
ambient pressure, we wouldn't have airplanes at all.
B) On the other hand, if you are only interested in the net upward force
on the wing, *after* you have ascertained that there is "enough"
ambient pressure to ensure that no shock forms, then the ambient pressure
is of no *further* importance. The calculation can be done entirely in
terms of gauge pressure (suction and overpressure).
C) Yeah, I know that the air has no microscopic mechanism to suck upwards
on the top of the wing. But the end result is the same: A reduction in
downward force has the same effect as an increase in upward force.

The same thing happens in a gazillion other settings, including anywhere
there is a gauge invariance.

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

Pedagogical remarks:

1) This is the sort of thing that drives students crazy. The experts
understand the mechanism and know the zeroth order term is going to drop
out, so the first-order term is the starting point and ending point of
their discussion. Meanwhile the students want to know about the mechanism,
and until they understand the mechanism they cannot possibly know what
drops out and what doesn't, so they have no idea where the experts are
coming from.

2) I am *not* suggesting that we dumb-down all discussions to the point
where we carry around the zeroth-order term in calculations where it
isn't needed. At some point we need to teach students how to make the
separation between zeroth-order and first-order terms.
-- We need to realize they weren't born knowing this, but
-- they need to learn it at some point.


IMHO this is a big part of what physics is. When I hear somebody say
"by symmetry, the zeroth-order term drops out" my ears perk up and I
say to myself, "Ho, ho, who have we here? He talks like a physicist."