French
mathematician Henri Poincare first blew the whistle on the "closed
system" thinking of Newtonian mechanics. According to classical
physics, Newtonian physics, a closed system is perfectly orderly and
predictable. A pendulum in a vacuum, free of friction and air resistance,
will conserve its energy. The pendulum will swing back and forth for
all eternity. It will not be subject to the dissipation of entropy,
which eats its way into systems by causing them to give up their energy
to the surrounding environment. Planets, like pendulums, cannot be disturbed
unless by outside chance, and they must be unvarying in their perambulations
around the sun.
But Poincare
asked a question about the stability of the solar system. Why he asked
this question, we do not know; but he did. And the reaction to his question
was the standard linear/faith brush-off: "Of course they are stable!
They've been stable for a long time. Heck, we can predict eclipses years
in advance!" It was a tenet of the scientific faith that knowing
the law of force and mass of the bodies, any good scientist could predict
the interactions with Newton's equations. The law of force, the inverse
square of the law of gravitation, was all wrapped up in a nice, neat
package.
But Poincare
had been doing some math on the side, and he knew that there was a small
difficulty here: for a system containing only two bodies, Newton's equations
work. For an ideal two-body system, the orbits are stable. The problem
arises when going from two to three bodies, such as including the Sun
in the equations, Newton's equations become unsolvable! For formal mathematical
reasons, the three-body equation cannot be worked out closer than an
"approximation:"
Well,
one would think that an "approximation" might be okay. We
can live with that. It's nothing to keep one awake at night, right?
Well, Poincare knew that the approximation method appeared to work for
the first few terms added, but when the number of terms increased, if
you add more and more bodies to the system, even including a few spare
asteroids and their very minute perturbations of the system, over long
periods of time, at some point the orbits shift and the solar system
begins to break apart under its own internal forces.
Mathematically,
this problem is nonlinear and nonintegrable. When you add a term to
a two body system it increases the nonlinear complexity, or feedback
of the system. Poincare did this, and was satisfied that a 3 body system
remained pretty stable. Small perturbations, but so what? With just
the Sun, the Earth and the Moon, we can sleep safely in our beds at
night. Right?
Wrong.
What happened next was a shock. Poincare discovered that with even the
very smallest perturbation, some orbits behaved in an erratic, even
chaotic way. His calculations showed that a minute gravitational pull
from a third body might cause a planet to wobble and weave drunkenly
in its orbit and even fly out of the solar system altogether!
One
will be struck by the complexity of this figure which I do not even
attempt to draw. Nothing more properly gives us an idea of complication
of the problem of three bodies and, in general, of all the problems
in dynamics where there is no uniform integral. [H. Poincare quoted
by M. Schroeder in Fractals,
Chaos, Power Laws]
Poincare
had discovered that chaos is the essence of the nonlinear system. He
revealed that even a completely deterministic system like our solar
system could do crazy things with the least provocation. The smallest
effects could be magnified through positive feedback and a simple system
can explode into shocking complexity.
This is
quite a different matter from the "negative feedback control mechanism"
that controls the "temperature" of our reality.
In nonlinear
dynamics and complex systems, it is even possible to locate potential
critical pressure points in such systems. At these critical points (which
probably relate to what Gurdjieff, in his "Law of Octaves"
called "semi-tones' or "stopinders", but we are dealing
with a gigantic "octave" this time), a small change can have
a tremendous impact.
It is
in this context that Ark and I see our "mission."
Because
it so happens that, at this particular point in history, in this particular
reality that so many of us seem to be sharing, there are these communicants
from somewhere, calling themselves the Cassiopaeans, who just "happened"
to arrive on the scene after a lifetime of searching for the way back
to God by yours truly.
Continue
to page 168
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