>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>
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.
>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.
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.
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.
David Bowman
dbowman@gtc.georgetown.ky.us