Editing VE

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=Global Energy Considerations=

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<font size="-1">[[H_BookTiledMenu#Context|<b>Global Energy<br />Considerations</b>]]</font>
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==Preface==
The astrophysics community relies heavily on the virial equations &#8212; most often in the context of the [[#Scalar_Virial_Theorem|scalar virial theorem]] &#8212; to ascertain the basic properties of equilibrium systems.  As is described below, fundamentally the virial equations are obtained by taking moments of the Euler equation.  By examining the balance among various relevant energy reservoirs, the mathematical expression that defines virial equilibrium provides a means by which, for example, the radius of a configuration can be estimated, given a total system mass and mean system temperature.  It can also be used to estimate a system's maximum allowed rotation frequency and whether or not the properties of the equilibrium configuration will be significantly modified if the system is embedded in a hot tenuous external medium.

As is also discussed, below, it can be even more informative to examine how a system's global, Gibbs-like free energy, <math>\mathfrak{G}</math>, varies under contraction or expansion.  Extrema in the free energy identify equilibrium configurations, for example.  For spherically symmetric systems, in particular, the [[#Scalar_Virial_Theorem|scalar virial theorem]] is "derived" by identifying under what conditions <math>d\mathfrak{G}/dR = 0</math>.  Furthermore, the sign of the second derivative, <math>d^2\mathfrak{G}/dR^2</math>, tells whether or not the equilibrium state is stable or unstable.  Here we define relevant energy reservoirs that contribute to a system's global free energy.  In separate chapters we use the free energy function to help identify the properties of equilibrium systems and to examine their relative stability.

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  <td align="center">
'''Why Bother?'''<br />

Excerpts drawn from the introductory chapter (p. 3) of<br />
{{ Collins78figure }}
  </td>
</tr>
<tr><td align="left">
<font color="#770000">'''Question'''</font>:  &nbsp; Why bother introducing the virial theorem and its allied free-energy expression, given that the astrophysical systems we are interested in analyzing can be fully described by solutions of the set of [[PGE#Principal_Governing_Equations|Principal Governing Equations]]?

<font color="#770000">'''Answer'''</font>: &nbsp; The [[PGE#Principal_Governing_Equations|Principal Governing Equations]] are, in general, <font color="#008899">non-linear, second-order, vector differential equations which exhibit closed form solutions only in special cases.  Although additional cases may be solved numerically, insight into the behavior of systems in general is very difficult to obtain in this manner.  The virial theorem</font> and its associated free-energy expression <font color="#008899">generally deals in scalar quantities and usually is applied on a global scale.  This reduction in complexity  &#8212; from a vector description to a scalar one &#8212; frequently enables us to solve the resulting equations</font> in closed form and to ascertain more straightforwardly what physical processes are most responsible for defining properties of the solution.

<font color="#770000">'''Caution'''</font>: &nbsp; We should always keep in mind that <font color="#008899">this reduction in complexity results in a concomitant loss of information and we cannot expect to obtain as complete a description of a physical system as would be possible from a full solution of the</font> [[PGE#Principal_Governing_Equations|Principal Governing Equations]].
</td></tr>
</table>

==Virial Equations (Inertial Frame)==
Most of the material presented here has been drawn from Chandrasekhar's ''Ellipsoidal Figures of Equilibrium'' &#8212; hereafter [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; first published in 1969.  Relying heavily on [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>'s] in-depth treatment of the topic, our aim is to highlight key aspects of the tensor-virial equations and to present them in a form that serves as a foundation for our separate discussions of the equilibrium and stability of self-gravitating fluid systems.  Strong parallels are drawn between the [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] presentation and our own so that it will be relatively straightforward for the reader to consult the [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] publication to obtain details of the various derivations. Text that appears in a green font has been drawn ''verbatim'' from this reference. 

===Setting the Stage===

[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>, &sect;8, p. 15] <font color="#007700">A standard technique for treating the integro-differential equations of mathematical physics is to take the moments of the equations concerned and consider suitably truncated sets of the resulting equations.  The ''virial method'' &hellip; is essentially the method of the moments applied to the solution of hydrodynamical problems in which the gravitational field of the prevailing distribution of matter is taken into account.  The ''virial equations'' of the various orders are, in fact, no more than the moments of the relevant hydrodynamical equations.</font>  In this context, Chandrasekhar's focus is on two of the four [[PGE#Principal_Governing_Equations|principal governing equations]] that serve as the foundation of our entire H_Book, namely, the

<div align="center">
<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />
('''Momentum Conservation''')

{{ Template:Math/EQ_Euler01 }}
</div>
and the
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{Template:Math/EQ_Poisson01}}
[[File:OriginButton.jpg|125px|link=PGE/PoissonOrigin]]
</div>

In [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], the Euler equation first appears in &sect;11 (p. 20) as equation (38) and is written as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <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} \, ,</math>
  </td>
</tr>
</table>
</div>
and the Poisson equation appears in &sect;10 (p. 20) &#8212; specifically, the left-most component of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>'s] equation (37) &#8212; as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\nabla^2 \mathfrak{B}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 4\pi G \rho \, .</math>
  </td>
</tr>
</table>
</div>
It is clear, therefore, that Chandrasekhar uses the variable <math>\vec{u}</math> instead of <math>\vec{v}</math> to represent the inertial velocity field.  More importantly, he adopts a different variable name ''and a different sign convention'' to represent the gravitational potential, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">
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  <td align="right">
<math>- \Phi = \mathfrak{B} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>G \int\limits_V \frac{\rho(\vec{x}^{~'})}{|\vec{x} - \vec{x}^{~'}|} d^3x^' \, .</math>
  </td>
</tr>
</table>
</div>
Hence, care must be taken to ensure that the signs on various mathematical terms are internally consistent when mapping derivations and resulting expressions from [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] into this H_Book.

===First-Order Virial Equations===
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>, &sect;11(a), p. 21] <font color="#007700">The [virial] equations of the first order are obtained by simply integrating [the Euler equation] over the instantaneous volume, <math>V</math>, occupied by the fluid</font>.  Specifically, using our H_Book variable notation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\int\limits_V \rho \frac{dv_i}{dt} d^3x</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \int\limits_V \frac{\partial P}{\partial x_i} d^3x - \int\limits_V \rho \frac{\partial \Phi}{\partial x_i} d^3x \, ,</math>
  </td>
</tr>
</table>
</div>
leads to (see [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{d^2 I_i}{dt^2} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, ,</math>
  </td>
</tr>
</table>
</div>
where the <span id="MomentOfInertia">moments of inertia</span> about the three separate principal axes <math>(i = 1,2,3)</math> are defined by the expressions,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>I_i</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\int\limits_V \rho x_i d^3x \, .</math>
  </td>
</tr>
</table>
</div>
Thus, the first-order virial equation(s) <font color="#007700">expresses the uniform motion of the center of mass of the system</font>.

===Second-Order Tensor Virial Equations===
In discussing the origin of the second-order (tensor) virial equation, [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] will continue to serve as our primary reference.  However, in &sect;4.3 of their widely referenced textbook titled, "Galactic Dyamics," Binney &amp; Tremaine (1987) &#8212; hereafter [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] &#8212; also present a detailed derivation of the second-order virial equation, which they refer to as the ''tensor virial theorem.''  Because their presentation is set in the context of discussions of the structure of ''stellar dynamic'' systems, the [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] derivation fundamentally originates from the collisionless Boltzmann equation. In what follows we will identify where various key equations appear in [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], as well as in [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], because it can sometimes be useful to compare derivations made from the stellar-dynamic versus the fluid-dynamic perspective.

====Derivation====
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>, &sect;11(b), p. 22] The second-order (tensor) virial equations <font color="#007700">are obtained by multiplying [the Euler equation] by <math>x_j</math> and integrating over the volume, <math>V</math></font>.  Specifically, again using our H_Book variable notation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\int\limits_V \rho \frac{dv_i}{dt} x_j d^3x</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \int\limits_V x_j \frac{\partial P}{\partial x_i} d^3x - \int\limits_V \rho x_j \frac{\partial \Phi}{\partial x_i} d^3x \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 211, Eq. (4-72)
  </td>
</tr>
</table>
</div>
or, separating the term on the left-hand side into two physically distinguishable components &#8212; see equation 44 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; this can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{d}{dt} \int\limits_V \rho v_i x_j d^3x - 2 \mathfrak{T}_{ij}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\delta_{ij}\Pi + \mathfrak{W}_{ij} \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 22, Eq. (47)
  </td>
</tr>
</table>
</div>
where, by definition,
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<tr>
  <td colspan="6">
&nbsp;
  </td>
  <th align="center">
References
  </th>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{T}_{ij}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \int\limits_V \rho v_i v_j  d^3x </math>
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="left">
is the (ordered) kinetic energy tensor
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 17, Eq. (9)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 212, Eq. (4-74b)
  </td>
</tr>

<tr>
  <td align="right">
<math>\Pi</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\int\limits_V P d^3x </math>
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="left">
is &#x2154; of the total thermal (''i.e.,'' random kinetic) energy
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 16, Eq. (7)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 212, Eq. (4-74b)
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{W}_{ij}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \int\limits_V \rho \Phi_{ij} d^3x </math>
  </td>
  <td align="center">
&nbsp;&nbsp;&nbsp;&nbsp;
  </td>
  <td align="left">
&nbsp;
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 17, Eq. (15)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 68, Eq. (2-126)
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \int\limits_V \rho x_i \frac{\partial \Phi}{\partial x_j} d^3x </math>
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="left">
is the gravitational potential energy tensor
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 18, Eq. (18)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 67, Eq. (2-123)
  </td>
</tr>
</table>
</div>
Note that, in the definition of the gravitational potential energy tensor, Chandrasekhar has introduced a tensor generalization of the gravitational potential [see his Eq. (14), p. 17], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>- \Phi_{ij} = \mathfrak{B}_{ij}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>G\int\limits_V \rho(\vec{x}^') \frac{ (x_i - x_i^')(x_j - x_j^') }{|\vec{x} - \vec{x}^{~'}|^3} d^3x^' \, ;</math>
  </td>
</tr>
</table>
</div>
this same potential energy tensor appears explicitly as part of the expression for <math>\mathfrak{W}_{ij}</math> that is presented as Equation (2-126), on p. 67 of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>].


The antisymmetric part of this tensor expression gives (see [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{d}{dt} \int\limits_V \rho (v_ix_j - v_j x_i) d^3x</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, ,</math>
  </td>
</tr>
</table>
</div>
which <font color="#007700">expresses simply the conservation of the angular momentum of the system</font>.  The symmetric part of the tensor expression gives what is generally referred to as (see [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details) the,
<div align="center">
<span id="PGE:TVE"><font color="#770000">'''Tensor Virial Equation'''</font></span><br />
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{1}{2} \frac{d^2 I_{ij}}{dt^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 23, Eq. (51)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-78)
  </td>
</tr>
</table>
</div>
<span id="MOItensor">where,</span>
<div align="center">
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<tr>
  <td colspan="6">
&nbsp;
  </td>
  <th align="center">
References
  </th>
</tr>

<tr>
  <td align="right">
<math>I_{ij}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\int\limits_V \rho x_i  x_j d^3x </math>
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="left">
is the moment of inertia tensor
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 16, Eq. (4)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 212, Eq. (4-76)
  </td>
</tr>
</table>
</div>

====Steady State (Virial Equilibrium)====
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b> &sect;11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \delta_{ij}\Pi \, .</math>
  </td>
</tr>
</table>
</div>
<font color="#007700">[This] provides six integral relations which must obtain whenever the conditions are stationary</font>.

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<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Scalar<br />Virial<br />Theorem</b>]]</font>
|}
===Scalar Virial Theorem===
====Standard Presentation [the Virial of Clausius (1870)]====
The trace of the tensor virial equation (TVE), which is obtained by identifying the trace of each term in the TVE, produces the scalar virial equation, which is widely referenced and used by the astrophysics community.  More specifically, setting, 
<div align="center">
<table border="0" cellpadding="2" align="center">
<tr>
  <td colspan="4">
&nbsp;
  </td>
  <th align="center">
Description
  </th>
  <td colspan="1">
&nbsp;
  </td>
  <th align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]
Reference
  </th>
</tr>

<tr>
  <td align="right">
<math>I = \sum\limits_{i=1,3} I_{ii}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int\limits_V \rho (\vec{x}) |\vec{x}|^2 d^3x  </math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
scalar moment of inertia
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[Eqs. (3) &amp; (5), p. 16]
  </td>
</tr>

<tr>
  <td align="right">
<math>T_\mathrm{kin} = \sum\limits_{i=1,3} \mathfrak{T}_{ii}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \int\limits_V \rho |\vec{v}|^2  d^3x </math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
total (ordered) kinetic energy
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[Eq. (8), p. 16]
  </td>
</tr>

<tr>
  <td align="right">
<math>W_\mathrm{grav} = \sum\limits_{i=1,3} \mathfrak{W}_{ii}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>- \int\limits_V \rho x_i \frac{\partial \Phi}{\partial x_i} d^3x </math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
gravitational potential energy
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[Eq. (18), p. 18]
  </td>
</tr>

<tr>
  <td align="right">
<math>S_\mathrm{therm} = \frac{1}{2} \sum\limits_{i=1,3} \delta_{ii}\Pi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{2} \int\limits_V P d^3x </math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
total thermal (random kinetic) energy
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[Eq. (7), p. 16]
  </td>
</tr>
</table>
</div>
the scalar virial equation is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{1}{2} \frac{d^2 I}{dt^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) +  W_\mathrm{grav} \, ;</math>
  </td>
</tr>
</table>
</div>
and, for a stationary state, we have the equilibrium condition that is broadly referred to as the,
<div align="center">
<span id="TVE"><font color="#770000">'''Scalar Virial Theorm'''</font></span><br />
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) +  W_\mathrm{grav} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-79)
  </td>
</tr>
</table>
</div>
(In a footnote to their Equation 4-79, [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] point out that the ''scalar virial theorem'' was first proved by R. Clausius in 1870; see various links to this work under our [[VE#Related_Discussions|"Related Discussions" subsection, below]].)

====Generalization====
Chapter 24 in Volume II (''Gas Dynamics'') of [<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] presents a generalization of the scalar virial theorem that includes the effects of (a) a magnetic field that threads through a self-gravitating fluid system, and (b) an imposed surface pressure, <math>P_e</math>, when the configuration is embedded in a hot, tenuous external medium.  Text that appears in an orange font in the following paragraph has been drawn ''verbatim'' from this reference.

[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] begins by adding a term to the Euler equation that accounts for <font color="orange">the Maxwell stress tensor, <math>T_{ik}</math>, associated with the ambient magnetic field</font>, <math>~\vec{B}</math>, where,
<div align="center">
<math>
T_{ik} = \frac{B_i B_k}{4\pi} - \frac{|\vec{B}|^2}{8\pi} \delta_{ik} \, .
</math>
<br />&nbsp;<br />
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, p. 329, Eq. (24.3)
</div>
Drawing from Equation (24.1), the associated modified Euler equation is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\rho \frac{dv_i}{dt} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{\partial P}{\partial x_i} - \rho \frac{\partial \Phi}{\partial x_i} + \frac{\partial T_{ik}}{\partial x_k}  \, .</math>
  </td>
</tr>
</table>
</div>
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>, Vol. II, pp. 329-330] <font color="orange">If we were to multiply [this modified Euler equation] by <math>~x_m</math> and integrate over volume <math>V</math>, we would get the [appropriately modified] ''tensor virial theorem'', the off-diagonal elements of which carry information concerning angular-momentum conservation (see </font> [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] <font color="orange">for an exposition).  [Here] we shall be more interested in the trace of the tensor equation, which we may derive by simply multiplying [the modified Euler equation] by <math>~x_i</math> (with an implicit summation over repeated indices) and integrating over <math>V</math></font>.  The resulting relation governing the equilibrium of stationary states (see [<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] for derivation details), as we shall reference it, is the
<div align="center">
<span id="GenTVE"><font color="#770000">'''Generalized Scalar Virial Theorem'''</font></span><br />
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) +  W_\mathrm{grav} + \mathcal{M}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> P_e \oint \vec{x}\cdot \hat{n} dA - \oint \vec{x}\cdot \overrightarrow{T}\hat{n} dA   \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, p. 331, Eq. (24.12)
  </td>
</tr>
</table>
</div>
<font color="orange">where <math>\mathcal{M}</math> equals the magnetic energy contained in volume <math>V</math></font>,
<div align="center">
<math>
\mathcal{M} \equiv \int\limits_V \frac{|\vec{B}|^2}{8\pi} d^3x \, .
</math>
<br />&nbsp;<br />
[<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], p. 165, Eq. (7.1.18)<br />
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, p. 330, Eq. (24.9)
</div>

[It should be noted that {{ CF53full }} and {{ MS56full }} provide early discussions of virial equilibrium conditions that take into account the energy associated with a magnetic field.]

==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

<div align="center">
<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>
</div>


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,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <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>
</div>
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:

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <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>
</div>
<!--  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:

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <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>
</div>
where,
<div align="center">
<table border="0" cellpadding="2" align="center">

<tr>
  <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>

==Free Energy Expression==
Associated with any isolated, self-gravitating, gaseous configuration we can identify a total Gibbs-like free energy, <math>\mathfrak{G}</math>, given by the sum of the relevant contributions to the total energy of the configuration,
<div align="center">
<math>
\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + T_\mathrm{kin} + P_e V + \cdots
</math>
</div> 
Here, we have explicitly included the gravitational potential energy, <math>W_\mathrm{grav}</math>, the ordered kinetic energy, <math>T_\mathrm{kin}</math>, a term that accounts for surface effects if the configuration of volume <math>V</math> is embedded in an external medium of pressure <math>P_e</math>, and <math>\mathfrak{S}_\mathrm{therm}</math>, the reservoir of thermodynamic energy that is available to perform work as the system expands or contracts.  Our above discussion of the [[#Scalar_Virial_Theorem|scalar virial theorem]] provides mathematical definitions of each of these energy terms, except for <math>\mathfrak{S}_\mathrm{therm}</math>, which we discuss now.

===Reservoir of Thermodynamic Energy===
<math>\mathfrak{S}_\mathrm{therm}</math> derives from the differential, "PdV" work that is often discussed in the context of thermodynamic systems.  It should be made clear that, here, "dV" refers to the differential volume ''per unit mass,''  so it should be written as "<math>~d(\rho^{-1})</math>", to be consistent with the notation used throughout this H_Book.  Therefore, the differential thermodynamic work is,
<div align="center">
<math>d\mathfrak{w} = Pd(1/\rho) = -  \biggl( \frac{P}{\rho^2} \biggr) d\rho \, .</math>
</div>
After an ''evolutionary'' equation of state has been adopted, this differential relationship can be integrated to give an expression for the energy per unit mass, <math>\mathfrak{w}</math>, that is potentially available for work.  Then we define the thermodynamic energy reservoir as,
<div align="center">
<math>\mathfrak{S}_\mathrm{therm} \equiv - \int \mathfrak{w} ~dm \, .</math>
</div>

====Isothermal Systems====
If each element of gas maintains its temperature when the system undergoes compression or expansion &#8212; that is, if the compression/expansion is isothermal &#8212; then, the relevant evolutionary equation of state is,
<div align="center">
<math>P = c_s^2 \rho \, ,</math>
</div>
where the constant, <math>c_s</math>, is the isothermal sound speed.  In this case, the expression for the differential thermodynamic work becomes,
<div align="center">
<math>d\mathfrak{w} =  -  \biggl( \frac{c_s^2}{\rho} \biggr) d\rho =  -  c_s^2 d\ln\rho \, .</math>
</div>
Hence, to within an additive constant, we have,
<div align="center">
<math>\mathfrak{w} = -  c_s^2 \ln \biggl(\frac{\rho}{\rho_0}\biggr) \, ,</math>
</div>
where, <math>\rho_0</math> is a (as yet unspecified) reference density, and integration throughout the configuration gives (for the isothermal case),
<div align="center">
<math>\mathfrak{S}_\mathrm{therm} = +  \int c_s^2 \ln \biggl(\frac{\rho}{\rho_0}\biggr) dm \, .</math>
</div>

====Adiabatic Systems====
If, upon compression or expansion, the gaseous configuration evolves adiabatically, the pressure will vary with density as,
<div align="center">
<math>P = K \rho^{\gamma_g} \, ,</math> 
</div>
where, <math>K</math> specifies the specific entropy of the gas and {{ Template:Math/MP_AdiabaticIndex }} is the ratio of specific heats that is relevant to the phase of compression/expansion.  In this case, the expression for the differential thermodynamic work becomes,
<div align="center">
<math>d\mathfrak{w} =  -  K \rho^{{\gamma_g}-2} d\rho =  -  \frac{K}{({\gamma_g}-1)} d\rho^{{\gamma_g}-1} \, .</math>
</div>
Hence, to within an additive constant, we have,
<div align="center">
<math>\mathfrak{w} = -  \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr) \, ,</math>
</div>
and integration throughout the configuration gives (for the adiabatic case),
<div align="center">
<math>\mathfrak{S}_\mathrm{therm} = +  \int \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr)  dm 
= \frac{2}{3({\gamma_g}-1)}  \int \frac{3}{2} \biggl( \frac{P}{\rho} \biggr)  dm
= \frac{2}{3({\gamma_g}-1)} S_\mathrm{therm} \, ,</math>
</div>
where, as introduced in our above discussion of the [[#Scalar_Virial_Theorem|scalar virial theorem]], <math>S_\mathrm{therm}</math> is the system's total thermal (''i.e.,'' random kinetic) energy.

====Relationship to the System's Internal Energy====
It is instructive to tie this introductory material to the classic discussion of thermodynamic systems, which relates a change in the system's internal energy per unit mass, <math>\Delta u_\mathrm{int}</math>, to the differential work, <math>\Delta \mathfrak{w}</math>, via the expression,
<div align="center">
<math>\Delta u_\mathrm{int} = \Delta Q - \Delta \mathfrak{w} \, ,</math>
</div>
where, <math>\Delta Q</math> is the change in heat content of the system.  

'''Isothermal Evolutions''':  Because the internal energy is only a function of the temperature, we can set <math>\Delta u_\mathrm{int} = 0</math> for expansions or contractions that occur isothermally.  Hence, for isothermal evolutions the change in heat content can immediately be deduced from the expression derived for the differential work; specifically, <math>\Delta Q = \Delta \mathfrak{w}</math>.

'''Adiabatic Evolutions''':  By definition, <math>\Delta Q = 0</math> for adiabatic evolutions, in which case we find <math>\Delta u_\mathrm{int} = - \Delta \mathfrak{w}</math>.  The definition of the thermodynamic energy reservoir can therefore be rewritten as,
<div align="center">
<math>\mathfrak{S}_\mathrm{therm} = - \int \mathfrak{w} ~dm = + \int u_\mathrm{int} ~dm = U_\mathrm{int} \, .</math>
</div>
Quite generally, then &#8212; in sync with the above derivation &#8212; we can replace <math>\mathfrak{S}_\mathrm{therm}</math> by,
<div align="center">
<math>~U_\mathrm{int} = \frac{2}{3(\gamma_g-1)} S_\mathrm{therm} \, ,</math> 
</div>
in the expression for the free energy when analyzing adiabatic evolutions.

===Illustration===
As is derived in [[SSCpt1/Virial#Virial_Equilibrium|an accompanying discussion]], for a uniform-density, uniformly rotating, spherically symmetric configuration of mass <math>M</math> and radius <math>R</math>, 
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
W_\mathrm{grav}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \frac{3}{5} \frac{GM^2}{R_0} \biggl( \frac{R}{R_0} \biggr)^{-1} \, , 
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
T_\mathrm{kin}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{5}{4} \frac{J^2}{MR_0^2} \biggl( \frac{R}{R_0} \biggr)^{-2} \, , 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
V
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{4}{3} \pi R_0^3 \biggl( \frac{R}{R_0} \biggr)^{3} \, , 
</math>
  </td>
</tr>

</table>
</div> 
where, <math>J</math> is the system's total angular momentum and <math>R_0</math> is a reference length scale.

'''Adiabatic Systems''':  If, upon compression or expansion, the gaseous configuration behaves adiabatically, the reservoir of thermodynamic energy is,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\mathfrak{S}_\mathrm{therm} = U_\mathrm{int} = \frac{M K \rho^{\gamma_g-1}}{(\gamma_g - 1)} 
= \frac{M K }{(\gamma_g - 1)} \biggl( \frac{3M}{4\pi R_0^3} \biggr)^{\gamma_g-1} \biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} \, .
</math>
  </td>
</tr>

</table>
</div> 
Hence, the adiabatic free energy can be written as,
<div align="center">
<math>
\mathfrak{G} = -A\biggl( \frac{R}{R_0} \biggr)^{-1} +B\biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} + C \biggl( \frac{R}{R_0} \biggr)^{-2} + D\biggl( \frac{R}{R_0} \biggr)^3 \, ,
</math>
</div>
where, 
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{3}{5} \frac{GM^2}{R_0} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{K}{(\gamma_g-1)} \biggl( \frac{3}{4\pi R_0^3} \biggr)^{\gamma_g - 1} \biggr] M^{\gamma_g} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>C</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{5J^2}{4MR_0^2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>D</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{4}{3} \pi R_0^3 P_e \, .
</math>
  </td>
</tr>
</table>
</div>

'''Isothermal Systems''':  If, upon compression or expansion, the configuration remains isothermal &#8212; also see Appendix A of {{ Stahler83full }} &#8212; the reservoir of thermal energy is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\mathfrak{S}_\mathrm{therm} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
M c_s^2\ln \biggl( \frac{\rho}{\rho_0} \biggr) = - 3 M c_s^2 \biggl( \frac{R}{R_0} \biggr) \, . 
</math>
  </td>
</tr>

</table>
</div> 
Hence, the isothermal free energy can be written as,
<div align="center">
<math>
\mathfrak{G} = -A \biggl( \frac{R}{R_0} \biggr)^{-1} - B_I \ln \biggl( \frac{R}{R_0} \biggr) + C \biggl( \frac{R}{R_0} \biggr)^{-2} + D\biggl( \frac{R}{R_0} \biggr)^3 \, ,
</math>
</div>
where, aside from the coefficient definitions provided above in association with the adiabatic case,
<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>B_I</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
3Mc_s^2 \, .
</math>
  </td>
</tr>

</table>
</div>

'''Summary''': We can combine the two cases &#8212; adiabatic and isothermal &#8212; into a single expression for <math>\mathfrak{G}</math> through a strategic use of the Kroniker delta function, <math>\delta_{1\gamma_g}</math>, as follows:
<div align="center">
<math>
\mathfrak{G} = -A\biggl( \frac{R}{R_0} \biggr)^{-1} +~ (1-\delta_{1\gamma_g})B\biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} -~ \delta_{1\gamma_g} B_I \ln \biggl( \frac{R}{R_0} \biggr) +~ C \biggl( \frac{R}{R_0} \biggr)^{-2} +~ D\biggl( \frac{R}{R_0} \biggr)^3 \, ,
</math>
</div>
Once the pressure exerted by the external medium (<math>P_e</math>), and the configuration's mass (<math>M</math>), angular momentum (<math>J</math>), and specific entropy (via <math>K</math>) &#8212; or, in the isothermal case, sound speed (<math>c_s</math>) &#8212;  have been specified, the values of all of the coefficients are known and this algebraic expression for <math>\mathfrak{G}</math> describes how the free energy of the configuration will vary with the configuration's relative size (<math>R/R_0</math>) for a given choice of <math>\gamma_g</math>.

==Whitworth (1981) and Stahler (1983)==
The above formulation of a [[#Free_Energy_Expression|Gibbs-like free energy]] has been motivated by the {{ Stahler83 }} analysis of stability of isothermal gas clouds, and it closely parallels the discussion by {{ Whitworth81full }} of "global gravitational stability for one-dimensional polytropes."  Whitworth introduces a "global potential function," <math>\mathfrak{u}</math>, that is the sum of three "internal conserved energy modes,"
<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>
\mathfrak{u}
</math>
  </td>
  <td align="center">
<math>
= 
</math>
  </td>
  <td align="left">
<math>
\mathfrak{g} + \mathfrak{B}_\mathrm{in} + \mathfrak{B}_\mathrm{ex} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
&nbsp;
<math>
- \frac{3}{5} \frac{GM_0^2}{R_0} \biggl(\frac{R}{R_0} \biggr)^{-1} 
+ (1-\delta_{1\eta})\biggl[ \frac{KM_0^\eta}{(\eta - 1)} V_0^{(1-\eta)} \biggr] \biggl(\frac{R}{R_0}\biggr)^{3(1-\eta)}
- \delta_{1\eta} \biggl[ 3KM_0 \ln\biggl(\frac{R}{R_0} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp; 
  </td>
  <td align="left">
&nbsp;
<math>
+ P_\mathrm{ex} V_0 \biggl( \frac{R}{R_0} \biggr)^{3}
</math>
  </td>
</tr>

</table>
</div>
Clearly Whitworth's global potential function, <math>\mathfrak{u}</math>, is what we have referred to as the configuration's Gibbs-like free energy, with <math>\eta</math> being used rather than <math>\gamma_g</math> to represent the ratio of specific heats in the adiabatic case.  Our expression for <math>\mathfrak{G}</math> would precisely match his expression for <math>\mathfrak{u}</math> if we chose to examine the free energy of a nonrotating configuration, that is, if we set <math>C=J=0</math>.

=See Also=
<ul>
  <li>{{ Clausius1870full }},  ''On a Mechanical Theorem Applicable to Heat''</li>
  <ul>
    <li>[http://books.google.com/books?id=Zk0wAAAAIAAJ&pg=PA122&lpg=PA122&dq=Mechanical+Theorem+Applicable+to+Heat&source=bl&ots=dhToAIe8k9&sig=0TaKVTmMnZ5qWvxLkdzuExMoU-U&hl=en&sa=X&ei=YFd-U5qiGdizyASS74CIAw&ved=0CDYQ6AEwAQ#v=onepage&q=Mechanical%20Theorem%20Applicable%20to%20Heat&f=false Google Book Reference]
    </li>
    <li>
[http://www.elastic-plastic.de/Clausius1870.pdf Alternative, stand-alone document]
    </li>
  </ul>
  <li>[http://en.wikipedia.org/wiki/Virial_theorem Wikipedia discussion of the virial theorem]</li>
  <li>{{ Collins78full }}</li>
<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
</ul>


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