From: "JACK L. URETSKY (C) 1996; HEP DIV., ARGONNE NATIONAL LAB, ARGONNE, IL 60439" <JLU@hep.anl.gov>
Date: Wed, 15 Jan 1997 14:02:46 -0600 (CST)
Hi all-
One of the points to be made in an elementary physics course is
that we deal with approximations to reality. This is because most problems
are too difficult for us humans to solve. The art of physics is largely
the art of finding viable approximations.
Nature is inherently non-linear. There are, however, linear
approximations that describe many phenomena with sufficient accuracy for
many practical purposes. An example is the set of Maxwell's equations in
a vacuum. These equations are linear and, as we know, useful for many
purposes.
Superposition is a consequence of linearity. The criterion for
superposition to work is that the phenomenon in question be subject to
linear equations. Superposition is then a trivial consequence.
One must, however, be careful, as John has reminded us. Linear
partial differential equations are generally subject to boundary conditions.
The superposition principle need not work when one is dealing with solutions
for two different sets of boundary conditions.
The interesting question, I suggest, is not to ask for examples
of situations where superposition does not work (it hardly ever does), but
to ask for the circumstances and reasons where it does appear to work.
Regards,
Jack