[Physltest] [Phys-L] Re: accelerating charge

[John Denker <jsd@AV8N.COM>]

I wrote:

> > I suspect whenever you have two formulas that differ by an
> > integration by parts, you can create some kind of mystery
> > by pushing the boundary terms into the far past and future.

Carl Mungan wrote:
> Interesting speculation. Can you think of some other examples?

Consider an aerodynamic force such as lift or drag.  Over a wide
range of Reynolds numbers, it will scale like velocity squared.
Specifically,
         F = .5 rho S v . v                                [1]
where rho is the density of the air, and S is some area.
The impulse is therefore
         I = integral .5 rho S v . v dt                    [2]
which can be integrated by parts to obtain
         I = .5 rho S v . x - integral .5 rho S a . x dt   [3]
and if we drop the boundary term we get
         I' = - integral .5 rho S a . x dt                 [4]
which can be interpreted in terms of a pseudo-force
         F' = - .5 rho S a . x                             [5]

There are various ways of obtaining [4] and [5].  For starters,
if you're doing dimensional analysis, you can derive [4] just
as easily as [2];  there is no dimensional reason to prefer
one over the other.  This isn't a particularly deep observation;
it just reminds us that there's more to physics than dimensional
analysis.

Perhaps more interestingly, you can contrive scenarios where
the velocity is always perpendicular to the position vector,
so v . x vanishes, so the boundary term vanishes.  Picture
circular motion (including nonuniform as well as uniform
circular motion).  In such a scenario, [4] and [5] are
rigorously equivalent to [2] and [1].

In such a scenario, you have no basis in mathematics or in
prior experiment for deciding whether I or I' is the relevant
quantity.  If you want to generalize the concepts and formulas
so they can be applied to non-circular motion, you may need
to do the experiment to find out which version holds up.

Note that (I) is the impulse i.e. momentum, so it is subject
to a conservation law.

Everything down to this point has been strictly true AFAIK;
now I will speak more figuratively.

The foregoing serves, loosely, as a parable for the radiation
of an accelerated charge.  The quantity (I) maps onto the
radiated energy, which is subject to a conservation law.
There are cases where the boundary terms are strictly or at
least approximately ignorable ... and other cases where they
are not.

The EM case is particularly tricky, because the self-energy
of a charged particle is infinite, and this gives you a place
to "hide" almost any constant boundary-term contribution.

In the EM case, the scenario of circular motion is again
special.  There are conservation laws that hold in such a
scenario that do not hold in general.  If all your experience
is restricted to such a scenario, there is no basis in math
or prior experiment for deciding which laws will generalize
to other scenarios and which will not.  You have to do
experiments in the new scenarios.

On the other side of the same coin, the specialized case of
hyperbolic motion -- i.e. uniform rectilinear acceleration --
can produce confusion because it has a new and different
set of 'extra' conservation laws, which may or may not
have deep physical significance.

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

When I was a kid, the four-color-map question and Fermat's
last "theorem" were wonderful examples of unsolved problems
where the question was simple enough for a kid to understand.

Both of those have since bitten the dust, and the Poincaré
conjecture might be on its last legs.

At least we've still got the Goldbach conjecture(s).
   http://mathworld.wolfram.com/GoldbachConjecture.html

Some higher-level questions may be found at
   http://www.claymath.org/millennium/

For questions with more physics flavor, I recommend
   http://www.physicstoday.org/pt/vol-54/iss-2/p11.html
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