At 02:19 AM 3/18/97 EST, David Bowman wrote:
^^^^^^^^^
Does your mom know your up this late playing with your computer ;-).
>Brian Harper (while wearing a theistic scientist/ID hat) wrote:
>
>>The principle of least action is clearly a teleological principle,
>>is non-mechanistic ["Anyone desiring to regard the principle of
>>least action as mechanical would today have to apologize for doing
>>so." -- Max Planck] and history shows beyond any doubt that the
>>principle derived from a strong conviction that natural law is
>>closely related to a higher will and more specifically that natural
>>laws should mirror Divine attributes such as beauty, elegance and
>>efficiency.
>>
>>Now, it may be argued that any mechanics problem which can be
>>solved by the principle of least action can also be solved by
>>direct application of the non-teleological, mechanistic, purposeless,
>>Laws of Sir Isaac. This is not terribly significant since one
>>often finds that natural phenomena can be described by several
>>different theories. The principle of least action is far greater
>>than Newton's Laws, though, for the following reasons:
>> <SNIP>
>
>DB>Brian then gives a partial list of reasons why the principle of least action
>is theoretically superior to the Newtonian formulation of classical
>mechanics. There are more reasons than Brian listed.
>
>5. The least action/Lagrangian/Hamiltonian formulation allows a direct
>connection between the symmetry principles of mechanics and the laws of
>nature. For instance, Noether's Theorem is easily proved in such a
>formulation. This theorem relates each underlying continuous symmetry of the
>laws of nature to the dynamically conserved quantities. (I.e. translational
>invariance in time implies energy conservation; translational invariance in
>space implies momentum conservation; isotropy in space implies angular
>momentum conservation; etc. -- just to name a few.)
>
>6. This formulation explains why the various quantities of mechanics have the
>mathematical formulas they do (i.e. why momentum is usually m*v, why kinetic
>energy is m*v^2/2, etc. in nonrelativistic versions, and why these quantities
>have the relativistic formulas that they do in the relativistic formulation)
>as direct consequences of the symmetry group under which the action is
>invariant.
>
>7. This formulation naturally allows extensions of mechanics to deeper
>formulations of natural law. For instance, even the dynamical equations of
>*general relativity* for spacetime curvature follow from an application of the
>least action principle (to the Hilbert Action). Also the least action/
>Lagrangian/Hamiltonian formulation allows the most straightforward
>generalization of classical physics to its corresponding quantum mechanical
>version.
>
Thanks for these additions. An interesting aspect of all this for
me is that I've used these two approaches to solving mechanics
problems all my life without ever really thinking about them.
I use a particular method for a particular problem based on utility.
So, it is really fascinating to see all the metaphysical implications
of these various concepts.
>>BH>It is true that the teleological approach has fallen into disrepute
>>since Galileo. Considering the great success of the principle of
>>least action, perhaps its time for a change.
>
>DB>Now for the rub where I put on my Devil's advocate hat.
>
>Feynman did for teleology in mechanics what Darwin did for teleology in
>biology. Feynman's PhD thesis showed how the principle of least action in
>classical mechanics is an *automatic consequence* of the deeper quantum
>mechanical principle of superposition of complex amplitudes. In the quantum
>formulation all conceivable trajectories democratically contribute on an
>equal footing to the complex probability amplitude for a given process. In
>the small (DeBroglie) wavelength limit of the quantum formulation where the
>accumulated action for some process is much larger than Planck's quantum of
>action, then the classical trajectory (of minimal action) *automatically*
>emerges as the overwhelmingly most likely path due to a process of
>constructive interference among all the DeBroglie waves, while all other non-
>extremal (classically) disallowed paths have their contribution to the
>process cancel out via destructive interference. Thus the minimal action
>trajectory in classical mechanics is the necessary zero-wavelength limit of
>quantum mechanics. In short, quantum mechanics explains how nature can be so
>smart to as to always find the minimum action trajectory for the system.
>Feynman showed that classical mechancics, as summed up by the least action
>principle, is related to the wave nature of quantum mechanics in exactly the
>same way as geometric optics, as summed up by Fermat's principle, is related
>to the wave nature of electromagnetism contained in Maxwell's equations. Both
>geometric optics and classical mechanics are expressible in terms of an
>apparently teleological extremum principle whose source is found in the
>zero wavelength limit of a wave theory which has no such apparently
>teleological implications. Rather than this being the best of all possible
>worlds, we find out that this is really a democratic (i.e. equal-weighted)
>superposition of all possible worlds.
>
Unfortunately, quantum mechanics is one of those things I know very little
about. I just finished reading an excellent biography of Feynman (_Genius_,
by Gleick) in which the above was discussed, in a more watered down way
of course. Your description above helped me understand a little better what
was going on.
Apparently, the principle of least action played an important role
in Feynman's career. He was first introduced to the principle by one
of his high school teachers in a very interesting way. I'm thinking of
using this myself in my lectures. First, Feynman was asked to solve
a fairly simple problem by classical methods, namely computing the
trajectory of a ball thrown by some kid on the street to his friend in
third story window. After this, his teacher defined for him the "action"
and he was asked to compute the action for the path he just determined.
He was then asked to consider several paths adjacent to the one he just
calculated and to compute the action for each of these paths. I think
this approach to introducing the idea can be quite dramatic for someone
who has never heard of the principle and actually thinks about what they
are doing. It seems almost as if the ball is conscious, somehow weighing
all the paths ahead of time and selecting the path that minimizes the action.
Apparently this had an impact on Feynman since he refused to solve any
problem by the principle of least action as an undergrad at MIT. Too
mysterious. Reading between the lines, perhaps these experiences
motivated Feynman's thesis work that you discuss above.
Be that as it may (and speaking as a layman) I was somewhat surprised to
hear your assessment "Feynman did for teleology in mechanics what Darwin
did for teleology in biology" even though the author of the biography
mentioned above said something very similar. It seems to me that Feynman
has merely replaced one mystery by another mystery that is perhaps even
greater. To be sure, Feynman provided an explanation for the principle
of least action. But this explanation does not seem to me in any way
similar to Darwin's explanation of biological design.
Let's turn to a more general point. One of the points of my devil's
advocate position was that the teleological approach was successful
where we are regarding success in a rather pragmatic way. The proof
is in the pudding, and the pudding turned out to be very tasty even
though some may not like the oven it was cooked in. In other words,
even if we conclude that mechanics is not teleological we still must
admit that the teleological approach reaped a great benefit to
science. How long would it have taken to discover the principle
of least action without the teleological approach?
>DB>Personally, I think it takes a much greater intelligent designing genius to
>create a world that obeys the laws of quantum physics which has the teleology
>of classical physics automatically fall out of it, than to create a merely
>classical world to begin with. If such reasoning is applied to the
>biological world (which I'm not 100% sure I want to do) we could say that it
>takes a more intelligent designer to set up a system of variation and selection
>to automatically create via evolution the exquisitely designed and adapted
>organisms that we see today than one who just separately directly creates
>discontinuously each of the biological innovations.
>
I agree here totally. In fact, considerations such as these played an
important role in my switch from PC to EC.
Brian Harper
Associate Professor
Applied Mechanics
The Ohio State University
"Because there's no primordial soup;
we all know that, right?" -- Leo Buss