KAM Theory

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ScholarPedia.org
http://www.scholarpedia.org/article/Kolmogorov-Arnold-Moser_theory#Lower_dimensional_tori

Attempt to illustrate category:toroidal-poloidal harmonics against a toy theory of Reality could evolve from this page or another. As a round peg doesn't fit in a square hole, integer harmonics cannot express the tangential equivalence of co-acceleration of central waveform gradients within a torus knot of a waveform particle.


Kolmogorov-Arnold-Moser (KAM) theory deals with persistence, under perturbation, of quasi-periodic motions in Hamiltonian dynamical systems.” —Scholarpedia.org


Reasons of interest in KAM Theory

Toro_d1.gif
Fig. 1.  Illustration of 3D helical motion on a torus surface, and the 2D rectangular transformation of the helical path as parallel lines. From Scholarpedia.org, contributed by the curtor of Scholarpedia, Dr. Luigi Chierchia; Dipartimento di Matematica, Universita' di Roma Tre

If a torus surface is mapped to a rectangle, or 3D torus coordinates transformed to a rectangular surface, by matrix algebra, then straight lines crossing the rectangular transformation may represent persistent, repeated crossing paths at certain integer ratios of the height and width.


The persistent path on the torus, translated in reverse from the rectangle to the torus, is a helix twisting on the torus surface around the center hole of the torus ring (Fig. 1, left).


Certain slopes of the straight and parallel crossing-paths that cut the rectangular transform (See Fig. 1, right) will connect with the starting point, to create repeating crossings on the same path, a harmonic condition. This resonance occurs when the slope is an interger ratio of the height and width of the rectangular crossing.


The top and bottom of the rectangular transform represent a line around the outer diameter of the torus. The center line, horizontally, of the rectangle represents travel through the hole in the torus.


To transform the center surface to the rectangle on the same retilinear mapping, the transformation must stretch the topology of the torus center hole surface around the hole diameter to the same distance as the circumference of the outer diamter.


The rectangular graph therefore is a representation, geometrically, of a compressed torus topology that varies on a vertical gradient. The gradient of compression/stretch is scaled horizonatlly, so that a gradient change at the rectangle center line is rapid, with less gradient of change at the top and bottom. The gradient change units grow closer at the top and bottom, and more separated at the middle of the rectangular transform.


The parallels of this topological surface expanded into orthogonal kinematical (across time) relationships separated by 90° in phase is exactly the classical model in topological consideration, first envisioned by James Clerk Maxwell, a mathematical dynamicists.


Why for?

The point and purpose of this amateur adventure is to discover a method of explaining geometrically how a 3D dynamic requires a topological gradient in the reactive qualities of the orthogonal forces of electromagnetism. The rectangular crossing-paths of Fig. 1 in electromagnetics represent phase relationships between orthogonal angular momenta.


Furthermore

There IS interaction “between the lines” of a rectangular mapping. This represents the forces used commonly in Z-pinch machines, or the mutual attraction of like charges in motion through magnetic induction produced by their motion.


Considering the effect of the entire waveform of a resonant system upon each point within the line with the induced Z-pinch against each portion of the dynamic path of Fig. 1, the effect would be to modulate the compression-gradient of the rectangular transformation in a sinusoidal fashion, where the top and bottom edges are translated as sinusoidal curves in phase (as they meet in the middle of the torus hole in phase). This statement is in reference to the self-entangled waveform of an electron, for instance, and the Z-pinch effect being responsible for holding the EM oscillations upon the aether particles, vacuuons""[1]


The interlocking of harmonic and integer ratio waveform in a three-dimensional relationship as a wave-particle of EM Theory is proportional in the coupling to the inertial quality of the waveform, for reasons expanded in a developing toy theory of a vector boson -slash- time glass explaination of the hyperbolic function of the space-compression of a troidal EM harmonic.


The energy content absorbed is 'pushed' into the resonance as a phrase-sensitive chamber accepting a structured pulse that is harmonically absorbed. The pushed energy is that energy content of an inertial mass increase thereafter exhibited by the resonant system.


To be continued… hopefully —while the work continues designing the power-supply needed for extremely sharp rise-times for optimizing the coupling with the Aharonov-Bohm Effect upon the bosonic strain distributed within a matter lattice of a paramagnetic insulator (Fe3O4 or such).


DonEMitchell 13:30, 26 November 2010 (MST)


See also

“This fractal is the byproduct of various efforts to find an answer to the question whether our Solar System is stable or not. It is named after Russian mathematician A.N. Kolmogorov who developed a theory predicting the form and stability of the orbits of the planets. The theory was independently confirmed by Mr. Kolmogorov's student, V.I. Arnold and the German mathematician J. Moser (hence the name: Kolmogorov, Arnold, and Moser).



  1. Crandal, They All Told the Truth