View source for Multi-pole potential

''potential of a multi-pole''

A [[Harmonic function|harmonic function]] on the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m0652101.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m0652102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m0652103.png" />, which is a partial derivative of some order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m0652104.png" /> of the main fundamental solution of the [[Laplace equation|Laplace equation]]; that is, a function of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m0652105.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m0652106.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m0652107.png" /></td> </tr></table>

For brevity, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m0652108.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m0652109.png" /> dipole potentials have the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521010.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521013.png" /> are the direction cosines of the radius vector of the point of observation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521014.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521015.png" />, for example, is interpreted as a dipole potential with moment 1 and axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521016.png" />, that is, the limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521017.png" /> of the sum of the Newton potentials of a mass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521018.png" /> placed at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521019.png" />, and a mass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521020.png" /> placed at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521021.png" />; otherwise this function can be represented as the magnetic potential of a small magnet placed at the origin along the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521022.png" />. Similarly, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521024.png" /> are dipole potentials with axes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521026.png" />, respectively. By taking linear combinations of these functions it is possible to obtain the potential of an arbitrarily oriented dipole with any moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521027.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521028.png" /> a quadrupole potential arises, obtained by a limit transition from a fixed system of four point masses whose total mass is always equal to zero, etc.

The [[Newton potential|Newton potential]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521029.png" /> of a bounded body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521030.png" /> of density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521031.png" />, situated so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521032.png" />, can be expanded in a series of multi-pole potentials:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521034.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521035.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521036.png" /></td> </tr></table>

is the total mass of the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521037.png" />, and the coefficients

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521038.png" /></td> </tr></table>

are called dipole moments for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521039.png" />, quadrupole moments for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521040.png" />, and, in general, multi-pole moments for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521041.png" />. The series (1) differs from the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521042.png" /> in spherical functions,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

by rearrangement of the terms; the terms of (2) can also be interpreted as potentials of multi-poles that are oriented in a special way (see [[#References|[1]]]). Therefore the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521044.png" /> are also often called, respectively, dipole, quadrupole and, more generally, multi-pole moments.

Expansions of the type (1) and (2) are used in the description and approximate representation of scalar and vector potentials, not only in connection with the fundamental solution of the Laplace equation, but also of the [[Helmholtz equation|Helmholtz equation]] (see [[#References|[2]]]).

In the hydrodynamics of planar flows of an ideal incompressible fluid there are also applications of complex multi-pole potentials of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521045.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521046.png" /> is a complex variable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521048.png" /> are, respectively, the moment and angle of orientation of the multi-pole. The dipole potential obtained for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521051.png" /> can be interpreted as the limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521052.png" /> of the sum of the complex potentials of a source of capacity 1 at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521053.png" /> and a sink of capacity 1 at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521054.png" />. The expansion (1) here corresponds to the expansion of the complex potential of the velocity of the flow of a streamlined planar body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521055.png" />, in a neighbourhood of the point at infinity:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521056.png" /></td> </tr></table>

Here the action of the streamlined body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065210/m06521057.png" /> is replaced by the resultant action of multi-pole potentials placed at the origin (see [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F.M. Morse,   "Methods of theoretical physics" , '''2''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.D. Jackson,   "Classical electrodynamics" , Wiley  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.M. Milne-Thomson,   "Theoretical hydrodynamics" , Macmillan  (1950)</TD></TR></table>