*KAM theory consequence: the motions are quasi-periodic. This is absolutely fine and fits well with how we understand the solar system. *But it can't be fine.

All Hamiltonian systems which are not integrable are chaotic.

Since the solar system is not integrable, and experiences unpredictable small perturbations, it cannot lie permanently on a KAM torus, and is thus chaotic.

KAM theory is valid for "sufficiently" small perturbations.

In reality, the perturbations in the solar system are far too large to apply KAM theory.

So, the mathematicians have to rely on computing Lyapunov exponents, in order to try to predict any region of instability/chaos.

Jack Wisdom (MIT): It is not possible to exclude the possibility that the orbit of the Earth will suddenly exhibit similar wild excursions in eccentricity.

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.

Lyapunov exponents and symplectic integration.

Let d(t) be the distance between two solutions, with d(0) being their initial separation. Then d(t) increases approximately as d(0)e

^{λt} in a chaotic system, where λ is the Lyapunov exponent. The inverse of the Lyapunov exponent, 1/λ, is called the Lyapunov time, and measures how long it takes two nearby solutions to diverge by a factor of e.

Sussman and Wisdom's 1992 integration of the entire solar system displayed a disturbing dependence on the timestep of the integration (measurement of the Lyapunov time).

Thus, different researchers who draw their initial conditions from the same ephemeris at different times can find vastly different Lyapunov timescales.

Wayne Hayes, UC Irvine

To show the importance and the dependence on the sensitivity of the initial conditions of the set of differential equations, an error as small as 15 meters in measuring the position of the Earth today would make it impossible to predict where the Earth would be in its orbit in just over 100 million years' time.

“The word ‘chaotic’ summarizes many fundamental concepts characterizing

a dynamical system such as complex predictability and stability. But above

all, it acts as a warming of the difficulties which are likely to arise when trying to

obtain a reliable picture of its past and future evolution. As an example, a

commonly accepted definition states that a system is ‘unstable’ if the trajectories of

two points that initially are arbitrarily close . . . diverge quickly in time. This has

strong implications, as small uncertainties in initial conditions . . . might [also] be

consistent with completely different future trajectories: The conclusion is that we

can exactly reproduce the motion of a chaotic system only if WE KNOW, WITH

ABSOLUTE PRECISION, THE INITIAL CONDITIONS – A STATEMENT

THAT, IN PRACTICE, CAN NEVER BE TRUE."

Alessandra Celletti, Ettore Perozzi, Celestial Mechanics: The Waltz of the Planets

Let us take a closer look the chaotic dynamics of planetary formation; thus, a clear indication that the initial conditions cannot be predicted with accuracy (as we have seen, a mere 15 meters difference in the data will have catastrophic consequences upon the calculations).

OFFICIAL SCIENCE INFORMATION

Four stages of planetary formation

Initial stage: condensation and growth of grains in the hot nebular disk

Early stage: growth of grains to kilometer-sized planetesimals

Middle stage: agglomeration of planetesimals

Late stage: protoplanets

For the crucial stages, the initial and early stages, prediction becomes practically impossible.

As if this wasn't enough, we have absolute proof that in the age of modern man planet Earth underwent sudden pole shifts (heliocentrical version), thus making null and void any integration of the solar system/Lyapunov exponents calculations which do not take into account such variations of the system's parameters:

http://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1635693#msg1635693http://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1546053#msg1546053Let me show you what sensitive dependence on initial conditions means, using one of the most famous examples: the Lorenz attractor butterfly effect.

In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario.

Here is the set of Lorenz equations:

Now, the set of differential equations which describe the planetary orbits is much more complicated than this.

NOTHING can be said about the RE heliocentrical system beyond a time scale of 300 YEARS.

Dr. Robert W. Bass

Ph.D. (Mathematics) Johns Hopkins University, 1955 [Wintner, Hartman]

A. Wintner, world's leading authority on celestial mechanics

Post-Doctoral Fellow Princeton University, 1955-56 [under S. Lefschetz]

Rhodes Scholar

Professor, Physics & Astronomy, Brigham Young University

"In a resonant, orbitally unstable or "wild" motion, the eccentricities of one or more of the terrestrial planets can increase in a century or two until a near collision occurs. Subsequently the Principle of Least Interaction Action predicts that the planets will rapidly "relax" into a configuration very near to a (presumably orbitally stable) resonant, Bode's-Law type of configuration. Near such a configuration, small, non-gravitational effects such as tidal friction can in a few centuries accumulate effectively to a discontinuous "jump" from the actual phase-space path to a nearby, truly orbitally stable, path. Subsequently, observations and theory would agree that the solar system is in a quasi-periodic motion stable in the sense of Laplace and orbitally stable. Also, numerical integrations backward in time would show that no near collision had ever occurred. Yet in actual fact this deduction would be false."

"I arrived independently at the preceding scenario before learning that dynamical astronomer, E. W. Brown, president of the American Astronomical Society, had already outlined the same possibility in 1931."

Dr. Robert Bass, Stability of the Solar System:

https://web.archive.org/web/20120916174745/http://www.innoventek.com:80/Bass1974PenseeAllegedProofsOfStabilityOfSolarSystemR.pdfDr. E.W. Brown

Fellowship, Royal Society

President of the American Mathematical Society

Professor of Mathematics, Yale University

President of the American Astronomical Society

What this means is that the interval of assured reliability for Newton's equations of gravitational motion is at most three hundred years.

Dr. W.M. Smart

Regius Professor of Astronomy at Glasgow University

President of the Royal Astronomical Society from 1949 to 1951

Within this 300 year time interval, we again have the huge problem of the sensitive dependence on initial conditions.