Editing VE (section)

==Virial Equations (Rotating Frame)==

As we have [[PGE/RotatingFrame#Euler_Equation_.28rotating_frame.29|explained elsewhere]], when examining the equilibrium, stability, and dynamical behavior of configurations that are rotating with angular velocity, <math>\vec\Omega_f</math>, it is useful to reference the

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<font color="#770000">'''Lagrangian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) \, .</math>
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Chandrasekhar also adopts this tactic.  In [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], the equivalent expression first appears in &sect;12 as equation (62) and has the form,
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  <td align="right">
<math>\rho \frac{du_i}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{\partial p}{\partial x_i} + \rho \frac{\partial \mathfrak{B}}{\partial x_i} 
+ 2\rho \epsilon_{i \ell m}u_\ell \Omega_m + \frac{1}{2} \rho \frac{\partial}{\partial x_i}|\vec\Omega \times \vec{x}|^2 \, ,</math>
  </td>
</tr>
</table>
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where, as noted in [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b> &sect;12, p. 25], the terms <math>|\vec\Omega \times \vec{x}|^2/2</math> and <math>2\vec{u} \times \vec\Omega</math> <font color="#007700">represent the centrifugal potential and the Coriolis acceleration, respectively</font> &#8212; also see [[PGE/RotatingFrame#Centrifugal_and_Coriolis_Accelerations|our related discussion of the centrifugal and Coriolis accelerations]].  As Chandrasekhar details, the Coriolis and centrifugal contributions introduce additional terms to the second-order virial, as follows:

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  <td align="right">
<math>\frac{d}{dt} \int\limits_V \rho v_i x_j d^3x </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2 \mathfrak{T}_{ij} + \delta_{ij}\Pi + \mathfrak{W}_{ij} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>+ 2\epsilon_{i \ell m} \Omega_m \int\limits_V \rho v_\ell x_j d^3x  
+ \Omega^2I_{ij} - \Omega_i \Omega_k I_{kj}\, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect; 11a, p. 25, Eq. (63) and Epilogue, p. 244, Eq. (1)
  </td>
</tr>
</table>
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<!--  DELETE THIS COMMENT.... The sign difference occurs just because the term appears on the "other" side of the equation; compare, for example, w/ Shu92 ...
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline in July 2014:  It is unclear to me why Weber's integral over the surface pressure is ''subtracted'' rather than ''added'' to the other terms on the right-hand side of his tensor virial expression.  Perhaps this is due to a definition of the unit normal vector that is different from the definition used by Shu.]]
END OF DELETED COMMENT -->

In his discussion of the ''Oscillation and Collapse of Interstellar Clouds,'' {{ Weber76full }} begins with this form of the second-order virial, but adds to it a contribution due to pressure-confinement by an external medium, as [[VE#Generalization|introduced above in the context of Shu's generalization]].  
Specifically, Weber opens up his discussion with the following form of the tensor virial equations:

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  <td align="right">
<math>\frac{dL_{ij}}{dt} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2 \mathfrak{T}_{ij} + \delta_{ij}\Pi + \mathfrak{W}_{ij} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>+ 2\epsilon_{i \ell m} \Omega_m L_{\ell j}  + I_{jm}(|\vec\Omega|^2 \delta_{im} - \Omega_i\Omega_m)
- \oint P_e x_j n_i dS \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Weber76 }}, Eq. (1)
  </td>
</tr>
</table>
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where,
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  <td align="right">
<math>L_{ij}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\int\limits_V \rho v_i  x_j d^3x \, .</math>
  </td>
</tr>
</table>
</div>