PiMath.de Planetary Systems of the Earth 1
Classic Systems
  Copyright © Klaus Piontzik  
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2.11 - General attempt

With the construction of the spatial oscillation structure a mathematical and physical model is available that allows to explain the structures of the earth on a basis of waves.

The question is:

What is a general approach for an oscillation structure?
The answer is found in the Laplace equation.

Pierre-Simon (Marquis de) Laplace (28.03.1749 bis 05.03.1827) was a French mathematician, physicist, and astronomer. He worked in the fields of probability theory and differential equations.
Laplace was always more physicists as a mathematician. He used the mathe-matics as a tool. Today, the mathematical procedures which Laplace developed and applied, become more important than his actual astronomical work.
The most important mathematical tools are the Laplace operator, Laplace's equation, Laplace's formula, as well as the Laplace transformation.

The Laplace operator Nabla is a mathematical operator that is a general mathematical provision (calculation way). The Laplace operator is a differential vector operator within the multidimensional analysis.

The Laplace operator occurs in Laplace's equation, for example. Twice continuously differentiable solutions of this equation are called harmonic functions.

Laplace equation:
Equation 2.11.01 Laplace-Equation
  to be read: Nabla f is zero


Expressed in cartesian coordinates (x, y, z):

Equation 2.11.02 Laplace-Equation cartesian

Applies in only one dimension:

Equation 2.11.03 Laplace-Equation one dimensional


This is the equation for a harmonic oscillator, such as a pendulum or a spring without friction.

The Laplace's equation represents a mathematical formula to describe oscillation phenomena in space.

The common approach to a solution with a central configuration is to transform the Cartesian coordinates (x, y, z) in spherical coordinates (lambda, phi, r).

Then, the entire function is decomposed into two part functions. Where a function the radial part contains and the other function the part of the angle.

The general approach to a solution function of Laplace's equation in spherical coordinates looks like this:


Equation 2.11.04 Lösungsfunktion der Laplace-Gleichung

Both part functions R and Y can be solved in each case individually.

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 Planetare Systeme 1

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Der Autor - Klaus Piontzik