Subject: Re: Optical Rays In Heterogeneous Media
Date: Sat, 27 Dec 1997 21:32:07 -0600
From: jim@smithwicks.sanga-mtl.com (james dolan)
Newsgroups: sci.math

h-obrown@worldnet.att.net writes:

-It might seem from all these reflective cases that we could
-just say the governing principle is that the reflected ray
-makes an equal angle with the incident ray relative to the
-surface normal.  However, we can set up specific circumstances
-to demonstrate that even the "equal angles" rule doesn't
-always apply, and we can do this based on the understanding
-of the interference and "sum over all paths" described by
-Feynman.


it's interesting that before mentioning that the stationarity rule
doesn't always apply, you changed its name to "the 'equal angles`
rule" to spare it the indignity of being pointed out as sometimes
failing (as must anything that's essentially "classical").

it's strange the way everyone says that stationarity principles are
more fundamental in classical mechanics (and optics) than minimum
principles are when it's really the other way around.  if stationarity
principles were more fundamental, then the sign (plus or minus) of the
action would be entirely arbitrary, whereas in reality it's not
arbitrary at all.  it's determined in a uniform way by thermodynamic
considerations for all physical systems that have ever been
encountered by human beings.

a way to formulate a minimum principle applying to all of classical
mechanics (and optics) without exception is as follows:

in a classical mechanical (or optical) system where we insist (taking
it either as part of the definition of what it means to be in the
classical regime, or as an extra condition if you prefer) that all the
data is smooth, the physically possible paths are precisely those
which over sufficiently short time intervals absolutely minimize the
action between the endpoint configurations.

if anyone knows of any exceptions to the above generalization i'd be
interested to hear of them.  (though of course the inevitable presence
here of weasel words like "in the classical regime" makes it a
somewhat fuzzy question as to what constitutes an exception to the
generalization as opposed to an exception to the conditions defining
the classical regime.)

real physical systems (such as light bouncing off of a mirror or a
ball bouncing off of a wall) which can be described to a good
approximation as classical systems with non-smooth data can be
described to an equally good approximation as classical systems with
smooth data.