paul@phs.UUCP (Paul C. Dolber) (06/30/85)
A rather off-the-wall question for net.physics, I suppose, but no more so than some of what I've seen here lately, and definitely a physics question. From George Owen's "The Universe of the Mind" (Johns Hopkins Press, Baltimore, 1971): "Heron of Alexandria made a major contribution to the theory of reflection by observing that when light is emitted from a point A and is reflected from a plane surface to a point B, the path corresponding to equal angles of incidence and reflection is the shortest path. He assumed that the ideal path, i.e., the shortest, represented the physical situation, and in this assumption he was quite correct... This approach to the question of reflection has much greater significance than the result shown above. The implication is that the laws of nature obey some ideal principle -- in this case, that the time for a ray to proceed via a reflection from A to B is a minimum, although in Heron's age it was not realized that the velocity of light is finite. When one incorporates the finite velocity of light and the fact that the velocity of propagation along both segments is the same, the result of Heron's construction implies that the transit time of a light ray along the reflected path is a minimum. Recognition of this fact led Fermat to the Least Time principle, in deriving the law of refraction of light rays" [pp 56-7]. "Attempting to derive Snell's law of refraction, Fermat, like Heron of Alexandria in his analysis of reflection, suggested that a light ray in passing from a point A to a point B in a medium of variable refractive index (or in a medium wherein the velocity of light varied) would make the passage in the least possible time" [p 108]. While I don't know that George really meant that part about the laws of nature obeying some ideal principle, the utility of the Least Time principle does strike me as a very odd thing indeed. Unfortunately, I don't recall what else he said about it in the course he taught for which this book was used. What I would like to know is: Is there some known physical reason why light *must* follow the least time path? or can one only conclude that it's an accident? or the result of some cosmic design? Please don't get more detailed in your replies than is absolutely necessary; the course was known (until the year I took it) as "Physics for Poets," which ought to give you a good idea of the depth of my understanding of physical principles. Regards, Paul Dolber (...duke!phs!paul).
cjh@petsd.UUCP (Chris Henrich) (07/02/85)
[] In article <1033@phs.UUCP> paul@phs.UUCP (Paul C. Dolber) writes: >...(an example of how the path of a light ray gets to its end >point in "least time")... > What I would like to know is: Is there >some known physical reason why light *must* follow the least time >path? or can one only conclude that it's an accident? or the result >of some cosmic design? There is a mathematical reason, if not exactly a physical reason. The best description of how light propagates, for most purposes, is a "wave" model: something which is defined at each point in space and varies continuously as a function of position and time; the "wave equation" relates the change at one point to the values at neighboring points. (That's a hand-waving definition of what a partial differential equation does.) The wave equations and other differential equations of physics often imply "variational principles:" that a wave propagates (or a particle moves) along a line which happens to be the solution of the following kind of problem: find the curve which minimizes a certain function depending on all the points that it passes through. The best-written book about this connection between physical principles and variational problems known to me is "The Variational Principles of Mechanics" by Cornelius Lanczos. It is not light reading, and presumes a knowledge of calculus. Does the existence of variational principles in physics imply something about God? This question goes beyond the subject matter of science or mathematics; these are about experiments, hypotheses, equations, theorems, etc. I believe that the answer is "Yes," but this is not a thing that I try to prove by mathematics. Regards, Chris -- Full-Name: Christopher J. Henrich UUCP: ..!(cornell | ariel | ukc | houxz)!vax135!petsd!cjh US Mail: MS 313; Perkin-Elmer; 106 Apple St; Tinton Falls, NJ 07724 Phone: (201) 758-7288
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (07/03/85)
> 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.
rimey@ucbvax.ARPA (Ken Rimey) (07/04/85)
>> What I would like to know is: Is there >>some known physical reason why light *must* follow the least time >>path? > There is a mathematical reason, if not exactly a >physical reason. The best description of how light >propagates, for most purposes, is a "wave" model ... Here's a try at a physical reason: Let's say that the ray of light doesn't only follow the path of least time, but follows all possible paths. There is really a collection of rays, but those that don't approximately follow the least-time path cancel each other. Rays that follow paths close to that of the least-time ray get to the destination slightly later, but only slightly. These rays therefore arrive roughly in phase, and interfere constructively. For each ray that follows a bizarre, unphysical path, there are rays that follow nearby paths that arrive later, and rays that arrive sooner. These arrive with an even mix of phases, and interfere destructively. The result is that only the rays *close* to the least-time ray are seen at the destination. These ideas can be expressed mathematically, but only with difficulty. Does anybody buy it? Ken Rimey
mikes@AMES-NAS.ARPA (07/09/85)
From: mikes@AMES-NAS.ARPA (Peter Mikes)