X 0 1 2 3 4 5 6 7 8
Section Three
F and G Forces
11. Relativity vs CM
X 0 1 2 3 4 5 6 7 8 |
|
Section Three |
| F and G Forces |
| ___ |
| 11. Relativity vs CM |
| 12. Kepler's 2nd |
| 13. Complex Plane |
| 14. Szebehely Eqn. |
| 15. Transitioning |
| 16. Shock Wave |
| 17. Quaternions |
| 18. Envelope Curve |
TAB 17
QUATERNION MODEL FOR ECCENTRICITY ACCUMULATIONThis speculative paper extrapolates the idea that hysteresis is reminiscent of accumulation phenomena (a fundamental concept in solid state physics theory, to be covered later) because a dynamical system behaves differently when it is increasing than when it is decreasing.
Solar system data shows a repeating pattern, like in atomic orbitals (e.g. 1s, 2s, 3s,…) - fractal theory at work.
Within each level, orientation of e-increments presumably changes. Many variations are possible. A "Wankel Engine" is likely a stable 3BP triangle of equilateral points up to perhaps a twelve cylinder rotary aircraft engine (developed later). Following is a periodic orbit of the 3BP that is characteristic of the rotating triangular shaped pistons in the Wankel Engine.
A Periodic Orbit in the Three Body Problem
A mathematical shorthand representing each transition from circle-to-low-eccentricity-ellipse is helpful, showing how transformations along each coordinate axis affect various aspects of the ellipse - e.g. the orbital parameters. Following is an illustration suggesting how the three coordinate axes might be interpreted.
The Fluxion Coordinate System
The general idea is to develop a Cartesian type coordinate system so that phasors along each of the orthogonal axes cause a unique change in the orbit and so that, all together, they encompass all of the nine orbital elements.
Another requirement for this "Fluxion Coordinate System" (Fluxion is the name given to elements of the Calculus by Newton, and seems an apt word to use in this particular application) is for there to be some kind of nesting mechanism, allowing sequential representation of fractal levels. The following figure hints of this kind of nesting in the orbital elements of the planets, which suggests that the scheme shown above might work.
Orbital Elements versus the Fluxion Coordinate System
The next requirement for the Fluxion system is that it must exhibit the kind of structure that underlies the geodesy of Earth as revealed by fourier series modeling. Sturm-Lioville theory says that the spherical harmonics are eigenfunctions (which are mutually perpendicular or orthogonal functions) that are an independent basis for the gravitational potential model. Since these terms also correlate to the eccentricity equation ( 1 ) and each term is an eigenfunction; they are all orthogonal, so the form a basis for modeling the Earth's gravitational field. Consider the simple case of the zonal harmonics, and how the represent graphically the higher order terms of the eccentricity equation.
Zonal Harmonics and phasors in Fluxion Space
The general idea is to orient these phasors so that they are aligned parallel to one of three principal axes (essentially the organization of a gravity ellipsoid, if one could be derived analytically). This notion is posed by MacCullogh's Method (Vallado, 488)
where the first term is the gravity potential, the second is the two body potential, and A, B, and C are the moment of intertia about the three principal axes; and 
is the polar moment of intertia. This is called MacCullogh's Forumla.
The sectoral harmonics are not much harder to conceptualize because they are symmetric about the polar axis, as indicated in the next figure. Here the phasors are seen to form the boundary lines between the different zones, which is exactly the mathematical significance of the Lagrange polynomials.
Sectoral harmonics and their Fluxion Equivalent
The tesseral harmonics are more complex, and are essentially a combination of the above two methods, displacing a phasor and rotating it about the polar axis, to form the lines between the tesseral segments.