SECTION FIVE

MATHEMATICAL CONSTRUCTIONS OF SINGULARITY

The general idea of mathematical physics is that the mathematics can be used to study physical processes. For example, the motion of a system of bodies can be represented by integrating t he differential equations describing the forces acting upon the bodies. This research has gone one subtle step further, by suggesting that the mathematics is more than a model for some processes but the process itself; e.g. in the case of the Fourier Series describing the incremental steps of allowable energy in the general 3BP. ( ONE )

Celestial Mechanics has an expression for this idea called "integrals of motion." They are laws that apply in restricted situations, in a small neighborhood of an optimal trajectory or solution. For example, consider the Jacobi Integral (total energy is constant) for the circular coplanar 3BP. It is a "parity check" matrix for non coplanar, non circular problems; just like the steps decided upon by the variable step integrator for the 3BP computer model. ( TWO )

As an introduction to this purely mathematical section, it is interesting to consider the numbers that we know must exist in nature. The ratio of a circle versus its radius comes first by most accounts - the number 1 for radius, of a circle with circumference equal to pi. Strictly from the mathematical perspective, the fact that pi is an unknown and unknowable number (infinite number of digits, never repeating) implies that circles are not possible in nature - e.g. circular orbits - and perhaps they are mathematically equivalent to a point singularity, when posed in the right context (e.g. in the complex plane).

This, in turn, implies that nature acts to avoid circles - or at least to circumvent them whenever possible. A concept which leads inevitably to motion on the unit circle, which is appropriately represented by another unknown and unknowable number, "i" or , the imaginary number. Where uniform motion on the unit circle - nature's way of representing a higher order singularity - is represented by - which makes "e" a third fundamental value of nature, also infinitely unrepeating.

Taking this logic one step further, and you surmise that a spherical orbit is also a singularity condition. That is, a thin spherical shell (or hoop in 2D) of uniform density reduces to a point mass - i.e. a singularity. Nature does its utmost to avoid motion in these type of systems, doing so by sinusoidal increments (sin/cos waves being the projection of a point rotating on a circle - if that circle moves, it forms a 3D helix). Theoretically, every motion of every body is a variation of the unit point/circle/sphere, in concentric domains of ever increasing extent until ultimately only life itself maintains the dynamical stability of the fragile whole.

On the level of life, this stability can be studied, modeled, and understood using the mathematics. This section looks at a few important concepts in mathematics from the perspective of the dynamics of the 3BP - which is itself, in the final reckoning, a singularity of nature.

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© 2004 WH Clark