Potential theory

The potential theory or the theory of the vortex-free vector fields deals with the mathematical-physical basics of conservative (vortex-free) force fields .

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

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

Central elements of the theoretical building are the potential and its local derivatives, in which a distinction must be made between the interior of a body (with its charge or mass distribution ) and the source-free exterior (see Laplace equation ).

Vector and scalar field

The potential theory is based on the fact that for every conservative vector field there is a scalar potential field , that is, that in every point the vector field through the gradient of the potential field according to ${\displaystyle {\vec {a}}({\vec {r}})}$ ${\displaystyle \Phi ({\vec {r}})}$

${\displaystyle {\vec {a}}({\vec {r}})=\operatorname {grad} \,\Phi ({\vec {r}})={\vec {\nabla }}\,\Phi ({\vec {r}})}$

is given with the Nabla operator (one speaks therefore of the gradient field ). At the same time, by forming the divergence of, the sources and sinks of the field can be determined (for example the electric charges in an electric field): ${\displaystyle \nabla }$ ${\displaystyle {\vec {a}}}$

${\displaystyle \rho ({\vec {r}})=\operatorname {div} \,{\vec {a}}({\vec {r}})=\operatorname {div} \,(\operatorname {grad} \,\Phi ({\vec {r}}))={\vec {\nabla }}\cdot \,({\vec {\nabla }}\,\Phi ({\vec {r}}))=\Delta \Phi ({\vec {r}})}$

with the Laplace operator . The potential theory now deals with how, for a given quantity, e.g. B. the source field , calculate the corresponding other sizes. Depending on the respective question, one speaks of various "problems". ${\displaystyle \Delta }$ ${\displaystyle \rho ({\vec {r}})}$

Poisson problem

The Poisson equation applies to the potential

${\displaystyle \Delta \Phi ({\vec {r}})=\rho ({\vec {r}}).}$

If the source field is given, the potential can be determined by integration: There is a single point source of strength at the point the potential ${\displaystyle \rho ({\vec {r}})}$ ${\displaystyle q}$ ${\displaystyle {\vec {r}}'}$

${\displaystyle \Phi ({\vec {r}})={\frac {1}{4\pi }}{\frac {q}{|{\vec {r}}-{\vec {r}}'|}}\quad {\text{bzw.}}\quad \mathrm {d} \Phi ({\vec {r}})={\frac {1}{4\pi }}{\frac {\mathrm {d} q}{|{\vec {r}}-{\vec {r}}'|}}={\frac {1}{4\pi }}{\frac {\rho ({\vec {r}}')}{|{\vec {r}}-{\vec {r}}'|}}\ \mathrm {d} V'}$

generated, results from totaling up or integration

${\displaystyle \Phi ({\vec {r}})={\frac {1}{4\pi }}\int {\frac {\rho ({\vec {r}}')}{|{\vec {r}}-{\vec {r}}'|}}\ \mathrm {d} V'.}$

Dirichlet problem

Often the source fields cannot be measured directly in physics, but their potential field can be measured in a certain spatial area. One such case is the exploration of the earth's interior using geodetic or geophysical methods:

You cannot drill deep into the earth's interior to determine the density there - but you can measure its effect on the earth's surface in the form of gravitational acceleration and deviation from the vertical.

In such a case, part of the room is determined, but the source field itself is unknown. It is only unambiguous under certain constraints and generally allows several solutions (see also the inversion problem of potential theory ). An elegant mathematical solution to the Dirichlet problem is possible with the help of Green's functions . ${\displaystyle \Phi ({\vec {r}})}$

Simple Layer Potential

One difficulty with practical calculations in potential theory is often the large amount of data to be processed , for example for harmonic spherical function developments to determine the gravitational field and geoid . For example, in order to calculate 50,000 mass functions of the earth's body from orbital disturbances of satellites , Neumann's method requires approx. 100,000 data sets and the inversion of huge matrices ( systems of equations ).

For this problem of satellite geodesy , the Bonn geodesist Karl Rudolf Koch developed a so-called robust , very effective calculation method under the name " Potential of the simple layer " , in which the interference potential is not through harmonic functions , but as an area on the earth's surface is pictured. These fictitious thin layers replace the source or mass distribution in the deeper interior of the earth and in the earth's crust , which is unknown in detail . The calculation method, which is in principle discontinuous at the model edges, has proven itself immensely in practice and was able to reduce the calculation times of the large computers to a fraction.

literature

• RJE Clausius: The potential function and the potential . Leipzig: Barth, 1859.
• T. Wand: The principles of mathematical physics and potential theory together with their most excellent applications are shown in the plan . Leipzig: BG Teubner, 1871.
• C. Neumann: Investigations into the logarithmic and Newtonian potential . Leipzig: BG Teubner, 1877.
• Max Doehler: Contribution to the theory of potential: Green's function for the ellipsoid of revolution, the infinite circular cylinder and shells that are formed by two confocal ellipsoids of revolution, respectively. two coaxial infinite circular cylinders are limited . Matthes, Brandenburg 1889 ( digitized version ).
• A. Wangerin: Theory of the potential and the spherical functions (2nd volume) . Leipzig: GJ Göschen, 1909–1921.
• R. Courant, D. Hilbert: Methods of mathematical physics Second volume . Berlin: Springer, 1924.
• S. Axler, P. Bourdon, W. Ramey: Harmonic Function Theory . 2nd edition, Springer-Verlag, 2001, ISBN 0387952187 .
• OD Kellogg : Foundations of Potential Theory . Dover Publications, 1967, ISBN 0486601447 .
• Karl Ledersteger : Handbook of Surveying Volume V Astronomical and Physical Geodesy . JB Metzler, Stuttgart 1969 ( theorems and gravitational field, equilibrium figures and the principle of defoliation , isostasy, etc.).
• Rudolf Sigl : Introduction to Potential Theory . Wichmann-Verlag, 1973, ISBN 3-87907-031-8 .

Individual evidence

1. ^ Walter Gellert, Herbert Küstner, Manfred Hellwich, Herbert Kästner (Eds.): Small encyclopedia of mathematics. Leipzig 1970, p. 741.
2. ↑ Grimsehl: Textbook of Physics, Vol. I ; Leipzig 1954, p. 160.