WHY WAS THE IRRATIONAL NUMBER CONCEPT SO BADLY NEEDED BY THE HELIOCENTRISTS?
And now we get to the crux of the matter: when Newton invented his results on the three body problem, he did not realize that the differential equations approach, built upon the irrational number concept, cannot possibly describe mathematically the heliocentric planetary system because of the existence of chaotical orbits, the quadratic homoclinic tangency problem.
The first great mathematician who discovered with amazement and shock that Newton's differential equations include chaotical orbits was Henri Poincare.
It is only at the highest level of academic circles specialized in bifurcation theory (thus, well-hidden from public view) where we find the truth about the original H. Poincare quotes, which do show that a differential equation (initial value d.e.) approach to celestial mechanics IS IMPOSSIBLE.
As Poincare experimented, he was relieved to discover that in most of
the situations, the possible orbits varied only slightly from the initial
2-body orbit, and were still stable, but what occurred during further
experimentation was a shock. Poincare discovered that even in some of the
smallest approximations some orbits behaved in an erratic unstable manner. His
calculations showed that even a minute gravitational pull from a third body
might cause a planet to wobble and fly out of orbit all together.
Here is Poincare describing his findings:
While Poincare did not succeed in giving a complete solution, his work was so impressive that he was awarded the prize anyway. The distinguished Weierstrass, who was one of the judges, said, 'this work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics.' A lively account of this event is given in Newton's Clock: Chaos in the Solar System. To show how visionary Poincare was, it is perhaps best if he described the Hallmark of Chaos - sensitive dependence on initial conditions - in his own words:
'If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. but even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.' - in a 1903 essay 'Science and Method'
That is why the conspirators had to invent a very complicated new theory, called chaos theory, with the help of G.D. Birkhoff and N. Levinson; their work was the inspiration for S. Smale's horseshoe map, a very clever way to describe Poincare's original findings as "workable" and "manageable". The formidable implications are, of course, that chaotical motion of the planets predicted by the differential equation approach of the London Royal Society is a thing that could happen ANYTIME, and not just some millions of years in the future, not to mention the sensitive dependence on initial conditions phenomenon.
Even measuring initial conditions of the system to an arbitrarily high, but finite accuracy, we will not be able to describe the system dynamics "at any time in the past or future". To predict the future of a chaotic system for arbitrarily long times, one would need to know the initial conditions with infinite accuracy, and this is by no means possible.
http://web.archive.org/web/20090108031631/http://essay.studyarea.com/old_essay/science/chaos_theory_explained.htmhttp://ptrow.com/articles/ChaosandSolarSystem5.htmSmale Horseshoe concept:
http://www.its.caltech.edu/~mcc/Chaos_Course/Lesson23/Predicting.pdfhttp://en.wikipedia.org/wiki/Horseshoe_mapKAM theory:
http://www.math.rug.nl/~broer/pdf/kolmo100.pdfVelikovsky stability theory:
http://www.ralph-abraham.org/interviews/abraham-ebert.htmlButterfly effect:
http://en.wikipedia.org/wiki/Lorenz_attractor (this is the only time you will see wikipedia type articles in my messages)
http://en.wikipedia.org/wiki/Butterfly_effectE. Lorenz did not realize that a system of three nonlinear differential equations could not approximate at all such a complicated natural phenomenon; there is no butterfly effect, the weather in Asia will not change due to the movement of a butterfly's wings in North America (sensitive dependence on initial conditions).
http://homepages.ulb.ac.be/~gaspard/G.Acad.00.pdfHomoclinic orbits:
http://arxiv.org/PS_cache/nlin/pdf/0702/0702044v2.pdfPoincare chaos:
http://web.archive.org/web/20061208155727/http://pims.math.ca/pi/current/page25-29.pdfDynamics and Bifurcations, J. Hale and H. Kocak (pages 248, 477, 486-490)
Introduction to Applied Nonlinear Dynamical Systems and Chaos, S. Wiggins (pages 286, 384, 420-443, 550, 612), both edited by Springer-Verlag; the information in these pages actually show the mathematical and physical implications of chaos theory.
The Duffing oscillator (prototype for nonlinear oscillations), the driven Morse oscillator, Poincare's three body problem equations, the librational motion of a satellite equations, the Ginzburg-Landau equation (nonlinear Schrodinger eq.) which reduces to the Duffing oscillator, all will have parameter values for which the stable/unstable manifolds of a saddle point will come into contact tangentially - homoclinic tangency.
Differential equations can be used on a very limited base (classical mechanics, quality-control, electronics/electrical engr., thermodynamics, and even here with certain assumptions/simplifications) and not at all in order to describe/predict biological processes and cosmological theories, where the aether theory comes into play to explain all the details.
Moreover, the system parameters will be varying functions of time, not to mention that the coefficients of the forcing/damping functions will not be "sufficiently small" in actual practice.
The assumptions actually made in describing various phenomena in several branches of physics are very well described in the classic Mathematics applied to deterministic problems in natural sciences by C.C. Lin and L. Segel (chapters 1, 4, 6, 8 ); page 43 exemplifies the extraordinary philosophical implications of the differential equation approach in modern physics:
http://www.ec-securehost.com/SIAM/CL01.htmlNow we know that Pythagoras never existed actually, as there were no ancient Greece/Rome/Egypt in our radical new chronology, and that the conspirators invented the irrational number concept in order to deceive the public regarding the Pythagorean comma (instead of a circle of fifths, we would have a spiral of fifths); they also invented, through J.S. Bach, the equal temperament scale in order to hide the real scale they used to produce levitation of large blocks of stone.
D. Hempel on Pythagoras' irrational numbers:
http://www.davidicke.com/forum/showthread.php?t=10283http://www.breakingopenthehead.com/forum/showpost.php?s=b7d281def62a68bb3f0971352e1ed848&p=30829&postcount=5Scientists used to, before the chaos theory, believe in the
theory of reductionism, many still do. Reductionism imagines nature as equally
capable of being assembled and disassembled. Reductionists think that when
everything is broken down a universal theory will become evident that will
explain all things. Reductionism implied the rather simple view of chaos
evident in Laplace's dream of a universal formula: Chaos was merely complexity
so great that in practice scientists couldn't track it, but in principle they
might one day be able to. When that day came there would be no chaos,
everything in existence would be perfectly predictable, no surprises, the
world would be safely mutable. The universe would be completely controlled by
Newton's laws.
Chaos touches all things in existence, and all sciences,
mathematics, physics, biology, anthropology, entomology, astronomy, even the
Ivory Tower science of Newtonian physics. In the last years of the 19th
century French mathematician, physicist and philosopher Henri Poincare
stumbled headlong into chaos with a realization that the reductionism method
may be illusory in nature. He was studying his chosen field at the time; a
field he called the mathematics of closed systems, the epitome of Newtonian
physics. A Closed system is one made up of just a few interacting bodies
sealed off from outside contamination. According to classical physics, such
systems are perfectly orderly and predictable. A simple 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. Classical scientists were
convinced that any randomness and chaos disturbing a system such as a pendulum
in a vacuum or the revolving planets could only come from outside chance
contingencies. Barring those, pendulum and planets must continue forever,
unvarying in their courses.
It was this comfortable picture of nature that
Poincare blew apart when he attempted to determine the stability of our solar
system. For a system containing only two bodies, such as the sun and earth or
earth and moon, Newton's equations can be solved exactly: The orbit of the
moon around the earth can be precisely determined. For any idealized two-body
system the orbits are stable. Thus if we neglect the dragging effects of the
tides on the moon's motion, we can assume that the moon will continue to wind
around the earth until the end of time. But we also have to ignore the effect
of the sun and other planets on this idealized two-body system. Poincare's
problem was that when an additional body was added to the situation, like the
influence of the sun, Newton's equations became unsolvable. What must be done
in this situation is use a series of approximations to close in on an answer.
In order to solve such an equation, physicists were forced to use a theory
called Perturbation. Which basically works in a third body by a series of
successive approximations. Each approximation is smaller than the one before
it, and by adding up a potentially infinite amount of these numbers,
theoretical physicists hoped to arrive a working equation. Poincare knew that
the approximation theory appeared to work well for the first couple of
approximations, but what about further down the line, what effect would the
infinity of smaller approximations have? The multi-bodied equation Poincare?
was attempting was essentially a Non-linear equation. As opposed to a
differential or linear equation. For science, a phenomenon is orderly if its
movements can be explained in the kind of cause-and-effect scheme represented
by a differential equation. Newton first introduced the differential idea
throughout his famous laws of motion, which related rates of change to various
forces. Quickly scientists came to rely on linear differential equations.
Phenomena as diverse as the flight of a cannonball, the growth of a plant, the
burning of coal, and the performance of a machine can be described by such
equations. In which small changes produce small effects and large effects are
obtained by summing up many small changes. A non-linear equation is quite
different. In a non-linear equation a small change in one variable can have a
disproportional, even catastrophic impact on other variables. Behaviors can
drastically change at any time. In linear equations the solution of one
equation allows the solver to generalize to other solutions; in non-linear
equations solutions tend to be consistently individual and unrelated to the
same equation with different variables. In Poincare's multi-bodied equation,
he added a term that added nonlinear complexity to the system (feedback) that
corresponded to the small effect produced by the movement of the third body in
the system. As he experimented, he was relieved to discover that in most of
the situations, the possible orbits varied only slightly from the initial
2-body orbit, and were still stable but what occurred during further
experimentation was a shock. Poincare discovered that even in some of the
smallest approximations some orbits behaved in an erratic unstable manner. His
calculations showed that even a minute gravitational pull from a third body
might cause a planet to wobble and fly out of orbit all together.
Poincare's discovery was not fully understood until 1953 by Russian physicist A. N. Kolmogorov. Initially
scientists believed that in theory they could break up a complicated system
into its components before experimentation because any changes in patterns
would be small and not effect an established construct such as an orbit.
Kolmogorov was not prepared to accept that the whole universe is a fraction of
a decimal point away from self-destruction. Unfortunately his research didn't
help. Kolmgorov concluded, from his own calculations, that the solar system
won't break up under its own motion provided that the influence of an
additional gravitational source was no bigger than a fly approximately 7000
miles away, and the cycles per planetary year did not occur in a simple
ratio like 1:2 1:3 or 2:3 and so on.
But, what happens when the planet's years form a simple ratio? Well, that would mean that with each orbit, the
disturbance is amplified due to a steady input of gravitational energy. It
creates a resonance feedback effect much like a normal microphone amplifier.
Say you lie an amplifiers input mic directly in front of its output speaker.
Any sound that enters the microphone will be played back through the speaker
louder, that playback will be picked up by the mic and amplified once again,
eventually the volume will reach its critical point and the speaker will blow
out. Well, if this were so, is there proof? Does this really happen in space?
Could this occur in our solar system? The answer is yes.
No other comments are needed.