Potential theory

The potential theory or theory of irrotational vector fields deals with the mathematical and physical foundations of conservative ( irrotational ) force fields.

Important applications are some effective scalar fields in nature, in particular the gravitational or gravitational field and electric and magnetic fields. In fluid dynamics (aerodynamics and hydrodynamics ) can flow fields described as potential field, as many processes in atomic physics and the modeling of the exact figure of the earth.

The beginnings of the theory go to the Italian mathematician and astronomer Joseph -Louis Lagrange, the Englishman George Green and finally Carl Friedrich Gauss back, the case already had applications for the geoid determination in mind.

Key elements of theory building are the potential and its local discharges for which between the interior of a body (with its charge or mass distribution ) and the source-free outer space is to be distinguished (see Laplace equation).

Vector and scalar field

The potential theory based on the fact that at any conservative vector field is a scalar potential field exists, so that at each point of the vector field by the gradient of the potential field according to

Is given. The same can be prepared by formation of the divergence of the source and drain of the field to determine (for example, the electric charges in the electric field, the masses in the gravitational field ).

The potential theory now deals with how to at a given size, eg, the source field, calculate the corresponding other sizes. Depending on the particular question you speaking of various "problems".

Poisson problem

Of the potential of the Poisson equation

With as the Laplace operator. If the source field is given, the potential can be determined by integration: Since a single point source of strength at the point of the potential

Generated, it follows by summing or integration as a whole

Dirichlet problem

Frequently in physics can be the source fields not directly measure, probably, however their potential field on a specific geographic area. One such case is the study of the Earth's interior by geodetic or geophysical methods:

You can not drill into the deep interior of the earth, there to determine the density - you can, however, measure their effect on the Earth's surface in the form of gravity or the deflection of the vertical.

In such a case is determined on a part of the space, the field source itself, however, unknown. It is clearly only under certain constraints and can ia multiple solutions (see also reverse problem of potential theory ). An elegant mathematical solution of the Dirichlet problem is possible with the help of the Green's functions.

Potential of a simple layer

One difficulty in practical calculations in the potential theory is often the large amount of data to be processed, for example, for harmonic spherical function developments for the determination of the gravity field and geoid. To compute, for example, perturbations of satellites 50,000 mass functions of the earth, the Neumann method requires approximately 100,000 records and the inversion of huge matrices ( equations).

For this problem of satellite geodesy the Bonn Geodetic Karl Rudolf Koch has worked out in the 1970s under the name " potential of a simple layer " a so-called robust, highly effective method of calculation, in which the interference potential not by harmonic functions, rather than surface coverage on the Earth's surface is shown. These fictitious thin layers replace the unknown in detail sources or mass distribution in the deeper interior of the earth and in the earth's crust. The discontinuous at the model boundaries in principle calculation method proved to be immensely in practice and was able to reduce the computation time of the mainframe computer to a fraction.