Editing Virial theorem (section)

== Statement and derivation ==
Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step.

For a collection of {{mvar|N}} point particles, the [[scalar (physics)|scalar]] [[moment of inertia]] {{mvar|I}} about the [[origin (mathematics)|origin]] is
<math display="block">
 I = \sum_{k=1}^N m_k |\mathbf{r}_k|^2 = \sum_{k=1}^N m_k r_k^2,
</math>
where {{math|''m''<sub>''k''</sub>}} and {{math|'''r'''<sub>''k''</sub>}} represent the mass and position of the {{mvar|k}}th particle. {{math|1=''r''<sub>''k''</sub> = {{abs|'''r'''<sub>''k''</sub>}}}} is the position vector magnitude. Consider the scalar
<math display="block">
 G = \sum_{k=1}^N \mathbf{p}_k \cdot \mathbf{r}_k,
</math>
where {{math|'''p'''<sub>''k''</sub>}} is the [[momentum]] [[vector (geometry)|vector]] of the {{mvar|k}}th particle.<ref name=":0">{{Cite book |last=Goldstein |first=Herbert |author-link=Herbert Goldstein |title=Classical mechanics |date=1980 |publisher=Addison-Wesley |isbn=0-201-02918-9 |edition=2nd |oclc=5675073}}</ref> Assuming that the masses are constant, {{mvar|G}} is one-half the time derivative of this moment of inertia:
<math display="block">\begin{align}
 \frac12 \frac{dI}{dt} &= \frac12 \frac{d}{dt} \sum_{k=1}^N m_k \mathbf{r}_k \cdot \mathbf{r}_k \\
                       &= \sum_{k=1}^N m_k \, \frac{d\mathbf{r}_k}{dt} \cdot \mathbf{r}_k \\
                       &= \sum_{k=1}^N \mathbf{p}_k \cdot \mathbf{r}_k = G.
\end{align}</math>
In turn, the time derivative of {{mvar|G}} is
<math display="block">\begin{align}
 \frac{dG}{dt} &= \sum_{k=1}^N \mathbf{p}_k \cdot \frac{d\mathbf{r}_k}{dt} +
                  \sum_{k=1}^N \frac{d\mathbf{p}_k}{dt} \cdot \mathbf{r}_k \\
               &= \sum_{k=1}^N m_k \frac{d\mathbf{r}_k}{dt} \cdot \frac{d\mathbf{r}_k}{dt} +
                  \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k \\
               &= 2 T + \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k,
\end{align}</math>
where {{math|''m''<sub>''k''</sub>}} is the mass of the {{mvar|k}}th particle, {{math|1='''F'''<sub>''k''</sub> = {{sfrac|''d'''''p'''<sub>''k''</sub>|''dt''}}}} is the net force on that particle, and {{mvar|T}} is the total [[kinetic energy]] of the system according to the {{math|1='''v'''<sub>''k''</sub> = {{sfrac|''d'''''r'''<sub>''k''</sub>|''dt''}}}} velocity of each particle,
<math display="block">
 T = \frac12 \sum_{k=1}^N m_k v_k^2 =
 \frac12 \sum_{k=1}^N m_k \frac{d\mathbf{r}_k}{dt} \cdot \frac{d\mathbf{r}_k}{dt}.
</math>

=== Connection with the potential energy between particles ===

The total force {{math|'''F'''<sub>''k''</sub>}} on particle {{mvar|k}} is the sum of all the forces from the other particles {{mvar|j}} in the system:
<math display="block">
 \mathbf{F}_k = \sum_{j=1}^N \mathbf{F}_{jk},
</math>
where {{math|'''F'''<sub>''jk''</sub>}} is the force applied by particle {{mvar|j}} on particle {{mvar|k}}. Hence, the virial can be written as
<math display="block">
 -\frac12\,\sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k =
 -\frac12\,\sum_{k=1}^N \sum_{j=1}^N \mathbf{F}_{jk} \cdot \mathbf{r}_k.
</math>

Since no particle acts on itself (i.e., {{math|1='''F'''<sub>''jj''</sub> = 0}} for {{math|1 ≤ ''j'' ≤ ''N''}}), we split the sum in terms below and above this diagonal and add them together in pairs:
<math display="block">\begin{align}
 \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k
  &= \sum_{k=1}^N \sum_{j=1}^N \mathbf{F}_{jk} \cdot \mathbf{r}_k =
     \sum_{k=2}^N \sum_{j=1}^{k-1} \mathbf{F}_{jk} \cdot \mathbf{r}_k + \sum_{k=1}^{N-1} \sum_{j=k+1}^{N} \mathbf{F}_{jk} \cdot \mathbf{r}_k \\
  &= \sum_{k=2}^N \sum_{j=1}^{k-1} \mathbf{F}_{jk} \cdot \mathbf{r}_k + \sum_{j=2}^N \sum_{k=1}^{j-1} \mathbf{F}_{jk} \cdot \mathbf{r}_k = 
     \sum_{k=2}^N \sum_{j=1}^{k-1} (\mathbf{F}_{jk} \cdot \mathbf{r}_k + \mathbf{F}_{kj} \cdot \mathbf{r}_j) \\
  &= \sum_{k=2}^N \sum_{j=1}^{k-1} (\mathbf{F}_{jk} \cdot \mathbf{r}_k - \mathbf{F}_{jk} \cdot \mathbf{r}_j) =
     \sum_{k=2}^N \sum_{j=1}^{k-1} \mathbf{F}_{jk} \cdot (\mathbf{r}_k - \mathbf{r}_j),
\end{align}</math>
where we have used [[Newton's laws of motion|Newton's third law of motion]], i.e., {{math|1='''F'''<sub>''jk''</sub> = −'''F'''<sub>''kj''</sub>}} (equal and opposite reaction).

It often happens that the forces can be derived from a potential energy {{mvar|''V''<sub>''jk''</sub>}} that is a function only of the distance {{math|''r''<sub>''jk''</sub>}} between the point particles {{mvar|j}} and {{mvar|k}}. Since the force is the negative gradient of the potential energy, we have in this case
<math display="block">
 \mathbf{F}_{jk} = -\nabla_{\mathbf{r}_k} V_{jk} =
 -\frac{dV_{jk}}{dr_{jk}} \left(\frac{\mathbf{r}_k - \mathbf{r}_j}{r_{jk}}\right),
</math>
which is equal and opposite to {{math|1='''F'''<sub>''kj''</sub> = −∇<sub>'''r'''<sub>''j''</sub></sub>''V''<sub>''kj''</sub> = −∇<sub>'''r'''<sub>''j''</sub></sub>''V''<sub>''jk''</sub>}}, the force applied by particle {{mvar|k}} on particle {{mvar|j}}, as may be confirmed by explicit calculation. Hence,
<math display="block">\begin{align}
 \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k
  &= \sum_{k=2}^N \sum_{j=1}^{k-1} \mathbf{F}_{jk} \cdot (\mathbf{r}_k - \mathbf{r}_j) \\
  &= -\sum_{k=2}^N \sum_{j=1}^{k-1} \frac{dV_{jk}}{dr_{jk}} \frac{|\mathbf{r}_k - \mathbf{r}_j|^2}{r_{jk}} \\
  & =-\sum_{k=2}^N \sum_{j=1}^{k-1} \frac{dV_{jk}}{dr_{jk}} r_{jk}.
\end{align}</math>

Thus
<math display="block">
 \frac{dG}{dt} = 2 T + \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k =
 2 T - \sum_{k=2}^N \sum_{j=1}^{k-1} \frac{dV_{jk}}{dr_{jk}} r_{jk}.
</math>

=== Special case of power-law forces ===
In a common special case, the potential energy {{mvar|V}} between two particles is proportional to a power {{mvar|n}} of their distance {{mvar|r<sub>ij</sub>}}:
<math display="block">
 V_{jk} = \alpha r_{jk}^n,
</math>
where the coefficient {{mvar|α}} and the exponent {{mvar|n}} are constants. In such cases, the virial is
<math display="block">\begin{align}
 -\frac12\,\sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k
  &= \frac12\,\sum_{k=1}^N \sum_{j<k} \frac{dV_{jk}}{dr_{jk}} r_{jk} \\
  &= \frac12\,\sum_{k=1}^N \sum_{j<k} n \alpha r_{jk}^{n-1} r_{jk} \\
  &= \frac12\,\sum_{k=1}^N \sum_{j<k} n V_{jk} = \frac{n}{2}\, V_\text{TOT},
\end{align}</math>
where
<math display="block">
 V_\text{TOT} = \sum_{k=1}^N \sum_{j<k} V_{jk}
</math>
is the total potential energy of the system.

Thus
<math display="block">
 \frac{dG}{dt} = 2 T + \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k = 2 T - n V_\text{TOT}.
</math>

For gravitating systems the exponent {{mvar|n}} equals −1, giving '''Lagrange's identity'''
<math display="block">
 \frac{dG}{dt} = \frac12 \frac{d^2 I}{dt^2} = 2 T + V_\text{TOT},
</math>
which was derived by [[Joseph-Louis Lagrange]] and extended by [[Carl Gustav Jacob Jacobi|Carl Jacobi]].

=== Time averaging ===

The average of this derivative over a duration {{mvar|τ}} is defined as
<math display="block">
 \left\langle \frac{dG}{dt} \right\rangle_\tau = \frac{1}{\tau} \int_0^\tau \frac{dG}{dt} \,dt = \frac{1}{\tau} \int_{G(0)}^{G(\tau)} \,dG = \frac{G(\tau) - G(0)}{\tau},
</math>
from which we obtain the exact equation
<math display="block">
 \left\langle \frac{dG}{dt} \right\rangle_\tau =
 2 \langle T \rangle_\tau + \sum_{k=1}^N \langle \mathbf{F}_k \cdot \mathbf{r}_k \rangle_\tau.
</math>

The '''virial theorem''' states that if {{math|1={{angbr|''dG''/''dt''}}{{sub|''τ''}} = 0}}, then
<math display="block">
 2 \langle T \rangle_\tau = -\sum_{k=1}^N \langle \mathbf{F}_k \cdot \mathbf{r}_k \rangle_\tau.
</math>

There are many reasons why the average of the time derivative might vanish. One often-cited reason applies to stably bound systems, that is, to systems that hang together forever and whose parameters are finite. In this case, velocities and coordinates of the particles of the system have upper and lower limits, so that {{math|''G''<sup>bound</sup>}} is bounded between two extremes, {{math|''G''<sub>min</sub>}} and {{math|''G''<sub>max</sub>}}, and the average goes to zero in the limit of infinite {{mvar|τ}}:
<math display="block">
 \lim_{\tau \to \infty} \left| \left\langle \frac{dG^{\text{bound}}}{dt} \right\rangle_\tau \right| =
 \lim_{\tau \to \infty} \left| \frac{G(\tau) - G(0)}{\tau} \right| \le
 \lim_{\tau \to \infty} \frac{G_\max - G_\min}{\tau} = 0.
</math>

Even if the average of the time derivative of {{mvar|G}} is only approximately zero, the virial theorem holds to the same degree of approximation.

For power-law forces with an exponent {{mvar|n}}, the general equation holds:
<math display="block">
 \langle T \rangle_\tau = -\frac12 \sum_{k=1}^N \langle \mathbf{F}_k \cdot \mathbf{r}_k \rangle_\tau
 = \frac{n}{2} \langle V_\text{TOT} \rangle_\tau.
</math>

For [[gravitation]]al attraction, {{math|''n'' {{=}} −1}}, and the average kinetic energy equals half of the average negative potential energy:
<math display="block">
 \langle T \rangle_\tau = -\frac12 \langle V_\text{TOT} \rangle_\tau.
</math>

This general result is useful for complex gravitating systems such as [[planetary system]]s or [[galaxy|galaxies]].

A simple application of the virial theorem concerns [[galaxy clusters]]. If a region of space is unusually full of galaxies, it is safe to assume that they have been together for a long time, and the virial theorem can be applied. [[Doppler effect]] measurements give lower bounds for their relative velocities, and the virial theorem gives a lower bound for the total mass of the cluster, including any dark matter.

If the [[Ergodicity|ergodic hypothesis]] holds for the system under consideration, the averaging need not be taken over time; an [[ensemble average]] can also be taken, with equivalent results.