Editing StabilityVariationalPrinciple (section)

==Most General Case==
Consider a free-energy function of the form,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{G}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{-3/j} + \mathcal{G}_0 \, ,</math>
  </td>
</tr>
</table>
</div>
where, <math>~a, b, c,</math> and <math>~\mathcal{G}_0</math> are constants, and the dimensionless configuration radius,
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<math>~\chi \equiv \frac{R}{R_0} \, ,</math>
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is defined in terms of a characteristic length, <math>~R_0</math>, which is likely to be different for each type of problem.  

===Virial Equilibrium===
The first variation (first derivative) of this function with respect to the configuration's radius is,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~\frac{d\mathcal{G}}{d\chi}</math>
  </td>
  <td align="center">
<math>~=</math>
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  <td align="left">
<math>~a\chi^{-2} - \biggl(\frac{3b}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 c}{j}\biggr) \chi^{-3/j -1} \, .</math>
  </td>
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</table>
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According to the virial theorem, the radius of an equilibrium configuration is obtained by setting <math>~d\mathcal{G}/d\chi = 0</math> and identifying the roots of the resulting equation.  For example, identifying roots of the polynomial expression,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~0</math>
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  <td align="center">
<math>~=</math>
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<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} - \biggl(\frac{1}{j}\biggr) \chi_\mathrm{eq}^{(j-3)/j } \, .</math>
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</tr>
</table>
</div>

===Stability===
Let's rewrite the first variation of the free-energy function in terms of three coefficients <math>~(e,f,g)</math> which, in general, we will permit to have different values from the original three <math>~(a,b,c)</math>,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{G}^'</math>
  </td>
  <td align="center">
<math>~=</math>
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  <td align="left">
<math>~e\chi^{-2} - \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -1} \, .</math>
  </td>
</tr>
</table>
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The first variation (first derivative) of ''this'' function with respect to the configuration's radius &#8212; which, in effect, represents the second variation of the free-energy function &#8212; gives,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\mathcal{G}^'}{d\chi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + \biggl(\frac{3}{j} + 1\biggr) \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -2} \, .</math>
  </td>
</tr>
</table>
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If we evaluate this function by setting <math>~\chi = \chi_\mathrm{eq}</math>, the sign of the resulting expression should indicate stability (positive) or dynamical instability (negative); and the marginally unstable configuration is identified by the value of <math>~\chi_\mathrm{eq}</math> for which <math>~d\mathcal{G}^'/d\chi = 0</math>.