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From: gwyn@brl-tgr.ARPA (Doug Gwyn )
Newsgroups: net.physics
Subject: Re: Least Time Principle
Message-ID: <11372@brl-tgr.ARPA>
Date: Wed, 3-Jul-85 11:43:32 EDT
Article-I.D.: brl-tgr.11372
Posted: Wed Jul  3 11:43:32 1985
Date-Received: Fri, 5-Jul-85 06:00:52 EDT
References: <1033@phs.UUCP>
Organization: Ballistic Research Lab
Lines: 31

> Is there some known physical reason why light *must* follow the least
> time path?

Many of the fundamental laws of physics can be expressed as "minimum"
principles, or rather, as variational principles.

One needs to be a bit subtle in formulating a "minimum" principle.
In the case of reflection, an even shorter path (or time) would be
obtained by a direct beam from transmitter and receiver, instead of
bouncing off the mirror.  In general relativity, the path of light
is a maximum (in 4 dimensions), not a minimum.

The general form of a variational principle is:

	The actual behavior of a system is such that some
	quantity computed from its behavior is "stationary"
	with respect to small variations in behavior from the
	actual behavior.

That is, the computed quantity (typically, an energy or path length)
is the same for all possible system behaviors that are "close" to the
one that actually happens.  If you know some calculus, this should
remind you of max or min of a function occurring at the point where
the derivative is zero.  (The variational principle amounts to a
strange sort of "variational derivative" being zero.)

Why it is possible to derive the fundamental equations of several
areas of physics from variational principles has not been
satisfactorily explained.  My own view is that a variational principle
is just a statement of structural stability, and the only physical
laws we can hope to find are those that are structurally stable.