From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: FEYNMAN 9 LEAST ACTION
Organization: Whole Earth 'Lectronic Link
Date: Thu, 15 Oct 1992 04:03:29 GMT


Feynman 9

"Newton's laws provide the mechanics; the principle of least action ensures
grace."

GENIUS p.60 "None of which mattered to Fynman when he encountered
Lagrange's method .... All he knew was that he did not like it.... to the
rest of the class [at MIT] the Lagrange formulation seemed elegant and
useful. It let them disregard many of the forces acting in a problem and
cut straight through to an answer. It served especially well in freeing
them from the right-angle coordinate geometry of the classical reference
frame required by Newton's equations. Any eference frame would do for the
Lagrangian technique. Feynman refused to employ it. He said he would not
feel he understood the real physics of a system until he had painstakingly
isolated and calculated all the forces. The problems got harder and harder
as the class advanced through classical mechanics. Balls rolled down
inclines, spun in paraboloids - Feynman would resort to ingenious
computational tricks ...

Feynman has first come on the principle of least action in Far Rockaway
after a bored hour of high-school physics when his teacher, Abram Bader
took him aside. Bader drew a curve ... the roughly parabolic shape a ball
would take if someone threw it up to a friend at a second-floor window. If
the time for the journey can vary, there are infinitely many such paths...
But if you know how long the journey took, the ball can have taken only one
path. ..the ball's energy: its kinetic energy, the energy of its motion,
and its potential energy, the energy it possesses by virtue of its presence
high in a gravitational field. ... Feynman was used to adding those
energies together. ... friction aside ... the total of kinetic and
potential energy never changes. The total energy is conserved.

Bader asked Feynman to consider a less intuitive quantity than the sum of
these energies: their difference.  Subtracting the potential energy from
the kinetic energy was as easy as adding them. It was just a matter of
changing signs. But understanding the physical meaning was harder.  Far
from being conserved, this quantity - the action, Bader said - changed
constantly. Bader had Feynman calculate it for the ball's entire flight to
the window. And he pointed out what seemed to Feynman a miracle. At any
particular moment the action might rise or fall, but when the ball arrived
at its destination, the path it had followed would always be the path for
which the total action was least. For any other path Feynman might try,,, -
a straight line from the ground to the window, a higher-arcing trajectory,
or a trajectory that deviated however slightly from the fated path [note
use of word 'fated' -JS] - he would find a greater average difference
between kinetic and potential energy."

Would that Feynman were still alive to check Gleick before publication. The
conclusion is correct but there is an important error of definition of
action. The action is the integral of the difference between the kinetic
and potential energies of the ball along the path chosen.  One must specify
the past place and time at which the ball starts and the future place and
time at which the ball arrives. The classical path that the macroscopic
ball takes is that path for which the path-dependent integral of the
difference between kinetic and potential energies is a minimum - for fixed
past and future end points in both space and time.  The differential in the
integral is that of time dt = ds/v where ds is the length of arc
differential and v is the instantaneous speed along that arc differential.
For example, if the path is given by the parametric equations
x = x(t) horizontal and y = y(t) vertical (1),
then
ds = (dx^2 + dy^2)^1/2 = vdt  (2)
where
v(t) = [(dx/dt)^2 + (dy/dt)^2]^1/2  (3)

The Lagrangian L of the ball of mass m is

L = Kinetic energy - Potential energy = mv(t)^2/2 - mgy(t)  (4)

assuming a constant gravitational field - a good approximation over changes
in elevation small compared to radius of earth. We are also assuming speeds
v small compared to that of light c so that Einstein corrections of special
relativity not important.

The nonlocal action A which is a global "functional" of the entire nonlocal
path [itself a local function of time] is then the path-dependent integral

A = Definite integral from t = t(i) to t = t(f) {Ldt} (5).

We have a space of local path functions given by equations (1) together
with the past and future constraints that x(t(i)), y(t(i)), x(t(f), y(t(f))
are fixed for all possible variations in the local function forms x(t) and
y(t) in between the past and future end points.  The actual motion taken by
the ball in physical space and time is a single point in the space of
possible functions outside of space and time. It is important to remember
that the fixed events in past and future co-create the intermediate
[present] motion in between.

A non trivial question is: are we free to choose the end points of the
motion arbitrarily? Can we always find a classical path of least action
connecting any two past and future endpoints for any potential?

Newton's local differential equations of motion (i.e. F = ma plus the force
law for conservative system at least with a velocity-independent potential)
are precisely the Lagrangian equations that minimize the action A.

At this classical level - the nonlocal nature of the least action principle
is not really important because we can replace with a local description in
which we start at the initial time t(i) and specify x(t(i)), y(t(i)) and
dx(t(i))/dt, dy(t(i)/dt - in principle exactly.  But this is precisely what
we are not allowed to do at the quantum level because of the Heisenberg
uncertainty principle which says, for example, that x(t(i)) and dx(t(i)/dt
cannot be specified together with arbitrarily small precision in each
quantity.

Not only that, but Einstein, Podolsky and Rosen, in 1935, showed that a
real nonlocal action-at-a-distance (apart from classical long-range forces
of gravity or electromagnetism) is required in order to prevent
Heisenberg's uncertainty principle from being violated. { ref. Pagels,
COSMIC CODE] It cannot be violated if quantum mechanics is a "complete"
theory of physical reality.

Back to Gleick's book on Feynman: "It is almost impossible for a physicist
to talk about the principle of least action without inadvertently imputing
some kind of volition to the projectile. The ball seems to choose its path.
It seems to know [with precognition -JS] all the possibilities in advance.
The natural philosophers started encountering similar minimum principles
throughout science.  Lagrange himself offered a program for computing
planetary orbits. The behavior of billiard balls crashing against each
other seemed to minimize the action. So did weights swung on a lever. So,
in a different way, did light rays bent by water or glass. Fermat, in
plucking his principle of least time from a pristine mathematical
landscape, had found the same law of nature.

Where Newton's methods left scientists with a feeling of comprehension,
minimun principles left a sense of mystery. 'This is not quite the way one
thinks in dynamics,' the physicist David Park has noted. One likes to think
that a ball or a planet makes its way instant by instant, not that it
follows a preordained path.  From the Lagrangian point of view the forces
that pull and shape a ball's arc into a gentle parabola serve a higher law.
Maupertius wrote, 'It is not in the little details ... that we must look
for the Supreme Being, but in phenomena whose universality suffers no
exception...'  Newton's laws provide the mechanics: the principle of least
action ensures grace.

... Feynman, having long since overcome his aversion to the principle of
least action, found the answer in quantum mechanics ... the question ..How
does the ball know which path to choose?" p.61