TAB 13

THE COMPLEX PLANE

The "real" plane analysis of gravity in the Two Body Problem showed there were three separate functions that satisfy the 2BP - r, f, and g. Some interesting mathematical support was offered, but in the end it was inconclusive. Now consider the same problem in a new coordinate system, the complex plane.

The Two Body Problem has an elegant solution in the complex plane.

 where is clockwise rotation

(1)

Notice how elegantly this representation breaks out the individual terms for the coriolis force, centrifugal force, and gravitational force. In Cartesian coordinates the only force was due to gravity. This suggests two actual forces exist in additional to the usual gravitational force.

This analysis is for a circular orbit. Now consider the same analysis for an ellipse of small eccentricity. Consider a representation of an orbit of small eccentricity as follows:

("higher order terms") (2)

where and (the semi parameter).

Thus

(3)

These two equations are a Taylor Series expansion in sine/cosine about a unit circle in rotating coordinates. Substituting these into equation 1 you get a function in terms of R and its derivatives,

(4)

That is,

(5)

which are both simple functions of t only while both a and p are slow variables, that is and The total force is a function of three terms again - and . Also observe that, like the f and g functions, a series representation is possible of these forces in the complex plane using trigonometric series for sine and cosine.

In equation 4, the first term is motion on the unit circle (for the normalized or regularized dynamical system) and the last two terms are a vector in the complex plane with both vectors rotating at the same frequency (i.e. an octurnion).

Distorted Space-Time in the Deleted Neighborhood

Posed in terms of the forces, it might fit better to the f- and g-force hypothesis and also with the series representation of gravity - being comprised of a fundamental frequency (the unit circle) plus a perturbation. In this context, the orbital motion is centered at the center of the ellipse (not at one focus), which is how the 1/r force works. This is just how the symmetric planet was proposed to act.

It is appropriate to elaborate further on the complex/symmetric plane analogy (in order of decreasing strength of correspondence)

both represent orbital motion as perturbations from the unit circle

derivatives of all order exist for the equations of motion

both are coordinate systems rotating at a constant rate, normalized to the unit circle

contour integrals have the same action as zero velocity curves

the complex integral and the Jacobi integral are equivalent

both have a reflection principle

This strongly suggests that the symmetric plane of the solar system is, in fact, the well known complex plane and that the difference to the invariant plane is due only to the mathematical representation of the origin - barycenter versus geometric center, new versus old. Barycenter is a point singularity, while the symmetric plane avoids this as a coordinate system origin by simulating a "deleted neighborhood" there, a small 3D sphere.

In other words, in order to solve the unrestricted general 3BP it is first necessary to transform the coordinates into a symmetric plane.

Forces on a particle in a Rotating Coordinate System

Analysis

The above illustration shows the forces acting on a bug crawling along a radial line on a rotating turntable.

• Corriolis Forces act only when a particle is moving in a rotating coordinate system and always act perpendicular to the direction of motion. On Earth, these forces are what cause the circulation of air around high/low pressure systems.
• Transverse Forces are perpendicular to the radius vector, but occur only if the rotating coordinate system is accelerating (which is not the case in the Three Body Problem scenario, which assumes a constant rate of rotation). Later it will be shown that this is the displacement force causing tesseral harmonics.
• Centrifugal Forces are do to rotation about any axis and act outward along the radius vector.

Planet's Halo Orbit vs. its Axis of Rotation

 

The above illustration is a dynamical model of the L6 Lagrange Point that defines the planet's motion versus the symmetric plane as a figure-8 loop (the fan blade) while a slight "imbalance" in the motor shaft (the difference between the invariant and symmetric planes) causes the planet's angular momentum vector to describe a small cone about the nominal axis of rotation. This dynamical imbalance happens because the axis of rotation of the fan is not exactly a principal axis of the system. This is how planet's rotation brings the whole system back into a dynamic equilibrium.

You will recall, per the uppermost illustration, how this deleted neighborhood is related to the f- and g-forces (12) where the action of a second 1/r force is superimposed upon the regular inverse squared gravity field. The resultant action, considering the center of the unit ball to be the L6 Lagrange Point, is exactly the geometry of a halo orbit around its Lagrange Point in a rotating coordinate system. In non-rotating coordinates, the halo orbit is just a regular elliptical orbit. This geometry is further delineated in the geometric study of the hyperbolic fly by or gravity assisted fly by of a planet or large mass.

Planet Fly By Geometry

 

A generalized solution substitutes an ellipse for the circle (unit ball), and this ellipse is centered at the end of the unit vector as shown (i.e. a circle at some angle to the plane of the page). This is the classical 1/r force system characteristic of all halo orbits. Having no body at the focus, for halo orbits still, the 1/r force is the only possible way to quantify an elliptical orbit about (and whose center is) the Lagrange Point.

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© 2004 WH Clark