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=Origin of the Poisson Equation=
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<font size="-1">[[H_BookTiledMenu#Context|<b>Poisson</b>]]</font>
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We will follow closely the presentation found in §2.1 of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] in deriving the,
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{ Math/EQ_Poisson01 }}
</div>
 <br />

<font color="#007700">According to Isaac Newton's inverse-square law of gravitation,</font> the acceleration, <math>\vec{a}(\vec{x})</math>, felt at any point in space, {{ Template:Math/VAR_PositionVector01 }}, due to the gravitational attraction of a distribution of mass, <math>\rho(\vec{x})</math>, is obtained by integrating over the accelerations exerted by each small mass element, <math>\rho(\vec{x}^{~'}) d^3x'</math>, as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\vec{a}(\vec{x})</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x' \, ,
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-2)
</td>
</tr>
</table>
</div>
where, {{ Template:Math/C_GravitationalConstant }} is the universal gravitational constant.

==Step 1==
In the astrophysics literature, it is customary to adopt the following definition of the,
<div align="center" id="GravitationalPotential">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3">
<font color="#770000">'''Scalar Gravitational Potential'''</font>
</td>
</tr>
<tr>
<td align="right">
<math>\Phi(\vec{x})</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>-G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-3)<br />
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], §10, p. 17, Eq. (11)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], §4.2, p. 77, Eq. (12)
</td>
</tr>
</table>
</div>

(Note:   As we have detailed in a [[VE#Setting_the_Stage|separate discussion]], throughout [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] Chandrasekhar adopts a ''different sign convention'' as well as a different variable name to represent the gravitational potential.) Recognizing that the gradient of the function, <math>~|\vec{x}^{~'} - \vec{x}|^{-1}</math>, with respect to {{ Template:Math/VAR_PositionVector01 }} is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\nabla_x \biggl[ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3} \, ,
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-4)
</td>
</tr>
</table>
</div>
and given that, in the above expression for the gravitational acceleration, the integration is taken over the volume that is identified by the ''primed'' <math>(\vec{x}~{'})</math>, rather than the unprimed <math>(\vec{x})</math>, coordinate system, <font color="#007700">we find that we may write</font> the gravitational acceleration as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\vec{a}(\vec{x})</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\int G\rho(\vec{x}^{~'}) \nabla_x \biggl[ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr]d^3 x'
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\nabla_x \biggl\{ G \int \biggl[ \frac{\rho(\vec{x}^{~'}) }{|\vec{x}^{~'} - \vec{x}|} \biggr]d^3 x'\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-\nabla_x \Phi \, .</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-5)
</td>
</tr>
</table>
</div>

==Step 2==
Next, we realize that the divergence of the gravitational acceleration takes the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\nabla_x \cdot \vec{a}(\vec{x})</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\nabla_x \cdot \int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x'
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\int G\rho(\vec{x}^{~'}) \biggl\{ \nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] \biggr\} d^3 x' \, .
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-6)
</td>
</tr>
</table>
</div>
Examining the expression inside the curly braces, we find that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{|\vec{x}^{~'} - \vec{x}|^3}
+ 3 \biggl[ \frac{ (\vec{x}^{~'} - \vec{x}) \cdot (\vec{x}^{~'} - \vec{x}) }{|\vec{x}^{~'} - \vec{x}|^5}\biggr]
</math>
</td>
</tr>
</table>
</div>

<table border="0" align="center" width="80%" cellpadding="8"><tr><td align="left">
Note:   Ostensibly, this last expression is the same as equation 2-7 of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], but apparently there is a typesetting error in the BT87 publication. As printed, the denominator of the first term on the right-hand side is <math>|\vec{x}^{~'} - \vec{x}|^1</math>, whereas it should be <math>|\vec{x}^{~'} - \vec{x}|^3</math> as written here. In an [https://www-thphys.physics.ox.ac.uk/people/JamesBinney/web/index_files/book%201%20errors.pdf ''Errata'' to <b><font color="red">BT87</font></b>], the authors have identified this error along with its correction as the first among a list of ''innocuous errors''.
</td></tr></table>

<font color="#007700">When <math>(\vec{x}^{~'} - \vec{x}) \ne 0</math>, we may cancel the factor <math>|\vec{x}^{~'} - \vec{x}|^2</math> from top and bottom of the last term in this equation to conclude that</font>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] = 0</math>
</td>
<td align="center">
      when,      
</td>
<td align="left">
<math>
(\vec{x}^{~'} \ne \vec{x}) \, .
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-8)
</td>
</tr>
</table>
</div>
<font color="#007700">Therefore, any contribution to the integral must come from the point <math>\vec{x}^{~'} = \vec{x}</math>, and we may restrict the volume of integration to a small sphere … centered on this point. Since</font>, for a sufficiently small sphere, <font color="#007700">the density will be almost constant through this volume, we can take <math>\rho(\vec{x}~{'}) = \rho(\vec{x})</math> out of the integral.</font> Via the divergence theorem (for details, see appendix 1.B — specifically, equation 1B-42 — of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>]), the remaining volume integral may be converted into a surface integral over the small volume centered on the point <math>\vec{x}^{~'} = \vec{x}</math> and, in turn, this surface integral may be written in terms of an integral over the solid angle, <math>d^2\Omega</math>, to give:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\nabla_x \cdot \vec{a}(\vec{x})</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-G\rho(\vec{x}) \int d^2\Omega
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-4\pi G\rho(\vec{x}) \, .
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 32, Eq. (2-9b)
</td>
</tr>
</table>
</div>

==Step 3==
Finally, combining the results of ''Step 1'' and ''Step 2'' gives the desired,
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{ Template:Math/EQ_Poisson01 }}
</div>
which serves as one of the [[PGE#Principal_Governing_Equations|principal governing equations]] in our examination of the '''Structure, Stability, & Dynamics of Self-Gravitating Fluids'''.

=See Also=
<ul>
<li>Ulrich D. Jentschura & Jonathan Sapirstein (April, 2018), arXiv:1801.10224v2 [math-ph], ''Green Function of the Poisson Equation: <math>D = 2, 3, 4</math>''</li>
<li>
Mark Viola (April 2021) ''[https://math.stackexchange.com/questions/4103885/generalizations-of-poissons-equation Generalizations of Poisson's Equation -- Math Stack Exchange]''
<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
"<font color="darkgreen">… we find the Green function for Poisson's equation, <math>\nabla^2G_0(\vec{x}|\vec{y}) = -\delta(\vec{x}-\vec{y})</math> is given by </font>"
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>G_0(\vec{x}|\vec{y})</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>
\frac{\Gamma(n/2-1)}{4\pi^{n/2}|\vec{x} - \vec{y}|^{n-2}} \, ,
</math>
</td>
</tr>
</table>
<font color="darkgreen">where <math>\delta(\vec{x})</math> is the Dirac Delta.</font> Hence, when the right-hand-side source function is spatially extended, <font color="darkgreen">… the solution of the Poisson's equation <math>\nabla^2u(\vec{x}) = p(\vec{x})</math> can be written as</font>
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>u(\vec{x})</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>
\int p(\vec{x}+\vec{y}) \biggl[ \frac{\Gamma(n/2-1)}{4\pi^{n/2}|\vec{y}|^{n-2}}\biggr] d^n\vec{y} \, .
</math>
</td>
</tr>
</table>

Example 1 [<math>n=3, \Gamma(n/2-1)=\pi^{1 / 2}</math>]:
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>u(\vec{x})</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>
\int \biggl[ \frac{p(\vec{x}+\vec{y}) }{4\pi|\vec{y}|}\biggr] d^3\vec{y} \, ;
</math>
</td>
</tr>
</table>

Example 2 [<math>n=4, \Gamma(n/2-1)=1</math>]:
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>u(\vec{x})</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>
\int \biggl[ \frac{p(\vec{x}+\vec{y}) }{4\pi^2|\vec{y}|^2}\biggr] d^4\vec{y} \, ;
</math>
</td>
</tr>
</table>

Example 3 [<math>n=5, \Gamma(n/2-1)=\sqrt{\pi}/2</math>]:
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>u(\vec{x})</math></td>
<td align="center"><math>=</math></td>
<td align="left"><math>
\int \biggl[ \frac{p(\vec{x}+\vec{y}) }{8\pi^2|\vec{y}|^3}\biggr] d^5\vec{y} \, .
</math>
</td>
</tr>
</table>

</td></tr></table>
</li>
</ul>

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