Editing Virial theorem

{{Short description|Physics theorem}}

In [[mechanics]], the '''virial theorem''' provides a general equation that relates the average over time of the total [[kinetic energy]] of a stable system of discrete particles, bound by a [[conservative force]] (where the [[Work (physics)|work]] done is independent of path), with that of the total [[potential energy]] of the system. Mathematically, the [[theorem]] states that
<math display="block">
 \langle T \rangle = -\frac12\,\sum_{k=1}^N \langle\mathbf{F}_k \cdot \mathbf{r}_k\rangle,
</math>
where {{mvar|T}} is the total kinetic energy of the {{mvar|N}} particles, {{math|'''F'''<sub>''k''</sub>}} represents the [[force]] on the {{mvar|k}}th particle, which is located at position {{math|'''r'''<sub>''k''</sub>}}, and [[angle brackets]] represent the average over time of the enclosed quantity. The word '''virial''' for the right-hand side of the equation derives from {{lang|la|vis}}, the [[Latin]] word for "force" or "energy", and was given its technical definition by [[Rudolf Clausius]] in 1870.<ref>{{cite journal | last = Clausius | first = RJE | year = 1870 | title = On a Mechanical Theorem Applicable to Heat | journal = Philosophical Magazine |series=Series 4 | volume = 40 | issue = 265 | pages = 122–127 |doi=10.1080/14786447008640370}}</ref>

The significance of the virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution, such as those considered in [[statistical mechanics]]; this average total kinetic energy is related to the [[temperature]] of the system by the [[equipartition theorem]]. However, the virial theorem does not depend on the notion of temperature and holds even for systems that are not in [[thermal equilibrium]]. The virial theorem has been generalized in various ways, most notably to a [[tensor]] form.

If the force between any two particles of the system results from a [[potential energy]] {{math|1=''V''(''r'') = ''αr<sup>n</sup>''}} that is proportional to some power {{mvar|n}} of the [[mean inter-particle distance|interparticle distance]] {{mvar|r}}, the virial theorem takes the simple form
<math display="block">
 2 \langle T \rangle = n \langle V_\text{TOT} \rangle.
</math>

Thus, twice the average total kinetic energy {{math|{{angbr|''T''}}}} equals {{mvar|n}} times the average total potential energy {{math|{{angbr|''V''<sub>TOT</sub>}}}}. Whereas {{math|''V''(''r'')}} represents the potential energy between two particles of distance {{mvar|r}}, {{math|''V''<sub>TOT</sub>}} represents the total potential energy of the system, i.e., the sum of the potential energy {{math|''V''(''r'')}} over all pairs of particles in the system. A common example of such a system is a star held together by its own gravity, where {{nobr|{{math|''n'' {{=}} −1}}.}}

== History ==
In 1870, [[Rudolf Clausius]] delivered the lecture "On a Mechanical Theorem Applicable to Heat" to the Association for Natural and Medical Sciences of the Lower Rhine, following a 20-year study of thermodynamics. The lecture stated that the mean [[vis viva]] of the system is equal to its virial, or that the average kinetic energy is one half of the average potential energy. The virial theorem can be obtained directly from [[Lagrange's identity]]{{Moved resource|date=December 2023}} as applied in classical gravitational dynamics, the original form of which was included in Lagrange's "Essay on the Problem of Three Bodies" published in 1772. [[Carl Gustav Jacob Jacobi|Carl Jacobi's]] generalization of the identity to ''N''&nbsp;bodies and to the present form of Laplace's identity closely resembles the classical virial theorem. However, the interpretations leading to the development of the equations were very different, since at the time of development, statistical dynamics had not yet unified the separate studies of thermodynamics and classical dynamics.<ref>{{Cite book |last=Collins |first=G. W. |year=1978 |title=The Virial Theorem in Stellar Astrophysics |publisher=Pachart Press |url=http://ads.harvard.edu/books/1978vtsa.book/ |bibcode=1978vtsa.book.....C |isbn=978-0-912918-13-6 |chapter=Introduction}}</ref> The theorem was later utilized, popularized, generalized and further developed by [[James Clerk Maxwell]], [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]], [[Henri Poincaré]], [[Subrahmanyan Chandrasekhar]], [[Enrico Fermi]], [[Paul Ledoux]], [[Richard Bader]] and [[Eugene Parker]]. [[Fritz Zwicky]] was the first to use the virial theorem to deduce the existence of unseen matter, which is now called [[dark matter]]. [[Richard Bader]] showed that the charge distribution of a total system can be partitioned into its kinetic and potential energies that obey the virial theorem.<ref name=rfwbpmb1972>{{cite journal |author1-last=Bader |author1-first=R. F. W. |author1-link=Richard Bader |author2-last=Beddall |author2-first=P. M. |title=Virial Field Relationship for Molecular Charge Distributions and the Spatial Partitioning of Molecular Properties | journal=The Journal of Chemical Physics |volume=56 |issue=7 |url=https://aip.scitation.org/doi/pdf/10.1063/1.1677699 |year=1972 |pages=3320–3329 |doi=10.1063/1.1677699 |bibcode=1972JChPh..56.3320B}}</ref> As another example of its many applications, the virial theorem has been used to derive the [[Chandrasekhar limit]] for the stability of [[white dwarf]] [[star]]s.

== Illustrative special case ==

Consider {{math|1=''N'' = 2}} particles with equal mass {{mvar|m}}, acted upon by mutually attractive forces. Suppose the particles are at diametrically opposite points of a circular orbit with radius {{mvar|r}}. The velocities are {{math|'''v'''<sub>1</sub>(''t'')}} and {{math|1='''v'''<sub>2</sub>(''t'') = −'''v'''<sub>1</sub>(''t'')}}, which are normal to forces {{math|'''F'''<sub>1</sub>(''t'')}} and {{math|1='''F'''<sub>2</sub>(''t'') = −'''F'''<sub>1</sub>(''t'')}}. The respective magnitudes are fixed at {{mvar|v}} and {{mvar|F}}. The average kinetic energy of the system in an interval of time from {{math|''t''<sub>1</sub>}} to {{math|''t''<sub>2</sub>}} is 
<math display="block">
 \langle T \rangle =
 \frac{1}{t_2 - t_1} \int_{t_1}^{t_2} \sum_{k=1}^N \frac12 m_k |\mathbf{v}_k(t)|^2 \,dt =
 \frac{1}{t_2 - t_1} \int_{t_1}^{t_2} \left( \frac12 m|\mathbf{v}_1(t)|^2 + \frac12 m|\mathbf{v}_2(t)|^2 \right) \,dt = mv^2.
</math>
Taking center of mass as the origin, the particles have positions {{math|'''r'''<sub>1</sub>(''t'')}} and {{math|1='''r'''<sub>2</sub>(''t'') = −'''r'''<sub>1</sub>(''t'')}} with fixed magnitude {{mvar|r}}. The attractive forces act in opposite directions as positions, so <!--<math>\mathbf F_1(t) \cdot \mathbf r_1(t) = \mathbf F_2(t) \mathbf r_2(t) = -Fr </math>-->{{math|1='''F'''<sub>1</sub>(''t'') ⋅ '''r'''<sub>1</sub>(''t'') = '''F'''<sub>2</sub>(''t'') ⋅ '''r'''<sub>2</sub>(''t'') = −''Fr''}}. Applying the [[centripetal force]] formula {{math|1=''F'' = ''mv''<sup>2</sup>/''r''}} results in
<math display="block">
 -\frac12 \sum_{k=1}^N \langle \mathbf{F}_k \cdot \mathbf{r}_k \rangle =
 -\frac12(-Fr - Fr) = Fr = \frac{mv^2}{r} \cdot r = mv^2 = \langle T \rangle,
</math>
as required. Note: If the origin is displaced, then we'd obtain the same result. This is because the dot product of the displacement with equal and opposite forces {{math|'''F'''<sub>1</sub>(''t'')}}, {{math|'''F'''<sub>2</sub>(''t'')}} results in net cancellation.

== 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.

== In quantum mechanics ==

Although originally derived for classical mechanics, the virial theorem also holds for quantum mechanics, as first shown by [[Vladimir Fock]]<ref>{{cite journal | last = Fock | first = V. | s2cid = 122502103 | year = 1930 | title = Bemerkung zum Virialsatz | journal = Zeitschrift für Physik A | volume = 63 | issue = 11 | pages = 855–858 | doi = 10.1007/BF01339281|bibcode = 1930ZPhy...63..855F }}</ref> using the [[Ehrenfest theorem]].

Evaluate the [[commutator]] of the [[Hamiltonian (quantum mechanics)|Hamiltonian]]
<math display="block">
 H = V\bigl(\{X_i\}\bigr) + \sum_n \frac{P_n^2}{2m_n}
</math>
with the position operator {{mvar|X<sub>n</sub>}} and the momentum operator
<math display="block">
 P_n = -i\hbar \frac{d}{dX_n}
</math>
of particle {{mvar|n}},
<math display="block">
 [H, X_n P_n] = X_n [H, P_n] + [H, X_n] P_n = i\hbar X_n \frac{dV}{dX_n} - i\hbar\frac{P_n^2}{m_n}.
</math>

Summing over all particles, one finds that for
<math display="block">
 Q = \sum_n X_n P_n
</math>
the commutator is
<math display="block">
 \frac{i}{\hbar} [H, Q] = 2 T - \sum_n X_n \frac{dV}{dX_n},
</math>
where <math display="inline">T = \sum_n P_n^2/2m_n</math> is the kinetic energy. The left-hand side of this equation is just {{math|''dQ''/''dt''}}, according to the [[Heisenberg equation]] of motion. The expectation value {{math|{{angbr|''dQ''/''dt''}}}} of this time derivative vanishes in a stationary state, leading to the '''''quantum virial theorem''''':
<math display="block">
 2\langle T\rangle = \sum_n \left\langle X_n \frac{dV}{dX_n}\right\rangle.
</math>

=== Pokhozhaev's identity ===
{{Unreferenced section|date=April 2020}}
In the field of quantum mechanics, there exists another form of the virial theorem, applicable to localized solutions to the stationary [[nonlinear Schrödinger equation]] or [[Klein–Gordon equation]], is [[Pokhozhaev's identity]],<ref>{{Cite journal |last1=Berestycki |first1=H. |last2=Lions |first2=P.-L. |year=1983 |title=Nonlinear scalar field equations, I existence of a ground state |url=https://link.springer.com/article/10.1007/BF00250555 |journal=Arch. Rational Mech. Anal. |volume=82 |issue=4 |pages=313&ndash;345 |doi=10.1007/BF00250555 |bibcode=1983ArRMA..82..313B |s2cid=123081616 }}</ref> also known as [[Derrick's theorem]].
Let <math>g(s)</math> be continuous and real-valued, with <math>g(0) = 0</math>.

Denote <math display="inline">G(s) = \int_0^s g(t)\,dt</math>.
Let
<math display="block">
 u \in L^\infty_{\text{loc}}(\R^n), \quad
 \nabla u \in L^2(\R^n), \quad
 G(u(\cdot)) \in L^1(\R^n), \quad
 n \in \N
</math>
be a solution to the equation
<math display="block">
 -\nabla^2 u = g(u),
</math>
in the sense of [[Distribution (mathematics)|distributions]].
Then <math>u</math> satisfies the relation
<math display="block">
 \left(\frac{n - 2}{2}\right) \int_{\R^n} |\nabla u(x)|^2 \,dx = n \int_{\R^n} G\big(u(x)\big) \,dx.
</math>

== In special relativity ==
{{Unreferenced section|date=April 2020}}For a single particle in special relativity, it is not the case that {{math|1=''T'' = {{sfrac|1|2}}'''p''' · '''v'''}}. Instead, it is true that {{math|1=''T'' = (''γ'' − 1) ''mc''<sup>2</sup>}}, where {{mvar|γ}} is the [[Lorentz factor]]
<math display="block">
 \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}},
</math>
and {{math|1='''β''' = {{sfrac|'''v'''|''c''}}}}. We have
<math display="block">\begin{align}
 \frac 12 \mathbf{p} \cdot \mathbf{v}
  &= \frac 12 \boldsymbol{\beta} \gamma mc \cdot \boldsymbol{\beta} c \\
  &= \frac 12 \gamma \beta^2 mc^2 \\[5pt]
  &= \left(\frac{\gamma \beta^2}{2(\gamma - 1)}\right) T.
\end{align}</math>
The last expression can be simplified to
<math display="block">
 \left(\frac{1 + \sqrt{1 - \beta^2}}{2}\right) T = \left(\frac{\gamma + 1}{2 \gamma}\right) T.
</math>
Thus, under the conditions described in earlier sections (including [[Newton's third law of motion]], {{math|1='''F'''<sub>''jk''</sub> = −'''F'''<sub>''kj''</sub>}}, despite relativity), the time average for {{mvar|N}} particles with a power law potential is
<math display="block">
 \frac{n}{2} \left\langle V_\text{TOT} \right\rangle_\tau =
 \left\langle \sum_{k=1}^N \left(\tfrac{1 + \sqrt{1 - \beta_k^2}}{2}\right) T_k \right\rangle_\tau =
 \left\langle \sum_{k=1}^N \left(\frac{\gamma_k + 1}{2 \gamma_k}\right) T_k \right\rangle_\tau.
</math>
In particular, the ratio of kinetic energy to potential energy is no longer fixed, but necessarily falls into an interval:
<math display="block">
 \frac{2 \langle T_\text{TOT} \rangle}{n \langle V_\text{TOT} \rangle} \in [1, 2],</math>
where the more relativistic systems exhibit the larger ratios.

== Examples ==
The virial theorem has a particularly simple form for periodic motion. It can be used to perform perturbative calculation for nonlinear oscillators.<ref name=":2">{{Cite journal |last=Sivardiere |first=Jean |date=December 1986 |title=Using the virial theorem |url=https://pubs.aip.org/aapt/ajp/article/54/12/1100-1103/1041519 |journal=American Journal of Physics |language=en |volume=54 |issue=12 |pages=1100–1103 |doi=10.1119/1.14723 |bibcode=1986AmJPh..54.1100S |issn=0002-9505}}</ref>

It can also be used to study motion in a [[central potential]].<ref name=":0" /> If the central potential is of the form <math>U \propto r^n</math>, the virial theorem simplifies to <math>\langle T \rangle= \frac{n}{2} \langle U \rangle</math>.{{Citation needed|date=October 2023}} In particular, for gravitational or electrostatic ([[Coulomb law|Coulomb]]) attraction, <math>\langle T \rangle= -\frac{1}2 \langle U \rangle</math>.

=== Driven damped harmonic oscillator ===

Analysis based on Sivardiere, 1986.<ref name=":2" /> For a one-dimensional oscillator with mass <math>m</math>, position <math>x</math>, driving force <math>F\cos(\omega t)</math>, spring constant <math>k</math>, and damping coefficient <math>\gamma</math>, the equation of motion is 
<math display="block">
 m \underbrace{\frac{d^2x}{dt^2}}_{\text{acceleration}} = \underbrace{-kx \vphantom{\frac dd}}_\text{spring}\ \underbrace{-\ \gamma \frac{dx}{dt}}_\text{friction}\ \underbrace{+\ F\cos(\omega t) \vphantom{\frac dd}}_\text{external driving}.
</math>

When the oscillator has reached a steady state, it performs a stable oscillation <math>x = X\cos(\omega t + \varphi)</math>, where <math>X</math> is the amplitude, and <math>\varphi</math> is the phase angle.

Applying the virial theorem, we have <math>m \langle \dot x \dot x \rangle = k\langle xx \rangle + \gamma \langle x\dot x \rangle - F \langle \cos(\omega t) x \rangle</math>, which simplifies to <math>F\cos(\varphi) = m(\omega_0^2 - \omega^2)X</math>, where <math>\omega_0 = \sqrt{k/m}</math> is the natural frequency of the oscillator.

To solve the two unknowns, we need another equation. In steady state, the power lost per cycle is equal to the power gained per cycle:
<math display="block">
 \underbrace{\langle \dot x \, \gamma \dot x\rangle}_\text{power dissipated} =
 \underbrace{\langle \dot x \, F \cos \omega t \rangle}_\text{power input},</math>
which simplifies to <math>\sin \varphi = -\frac{\gamma X \omega}{F}</math>.

Now we have two equations that yield the solution
<math display="block">\begin{cases}
 X = \sqrt{\dfrac{F^2}{\gamma^2 \omega^2 + m^2 (\omega_0^2 - \omega^2)^2}}, \\
 \tan\varphi = -\dfrac{\gamma \omega}{m(\omega_0^2 - \omega^2)}.
\end{cases}</math>

=== Ideal-gas law ===
Consider a container filled with an ideal gas consisting of point masses. The only forces applied to the point masses are due to the container walls. In this case, the expression in the virial theorem equals
<math display="block">
 \Big \langle \sum_i \mathbf{F}_i \cdot \mathbf{r}_i \Big \rangle =
- P \oint \hat{\mathbf{n}} \cdot \mathbf{r} \,dA,
</math>
since, by definition, the pressure ''P'' is the average force per area exerted by the gas upon the walls, which is normal to the wall. There is a minus sign because 
<math>\hat{\mathbf{n}}</math> is the unit normal vector pointing outwards, and the force to be used is the one upon the particles by the wall.


Then the virial theorem states that
<math display="block">
 \langle T \rangle =
 \frac{P}{2} \oint \hat{\mathbf{n}} \cdot \mathbf{r} \,dA.
</math>
By the [[divergence theorem]], <math display="inline">\oint \hat{\mathbf{n}} \cdot \mathbf{r} \,dA = \int \nabla \cdot \mathbf{r} \,dV = 3 \int dV = 3V</math>. 

From [[equipartition]], the average total kinetic energy <math display="inline">\langle T \rangle = N \big\langle \frac 12 mv^2 \big\rangle = N \cdot \frac 32 kT</math>. Hence, <math>PV = NkT</math>, the [[ideal gas law]].<ref>{{Cite web |date=2018-03-22 |title=2.11: Virial Theorem |url=https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/02%3A_Review_of_Newtonian_Mechanics/2.11%3A_Virial_Theorem |access-date=2023-06-07 |website=Physics LibreTexts |language=en}}</ref>

=== Dark matter ===
In 1933, Fritz Zwicky applied the virial theorem to estimate the mass of [[Coma Cluster]], and discovered a discrepancy of mass of about 450, which he explained as due to "dark matter".<ref name=":1">{{Cite journal |last=Zwicky |first=Fritz |date=1933 |others=Translated by Heinz Andernach |title=The Redshift of Extragalactic Nebulae |url=https://resolver.caltech.edu/CaltechAUTHORS:20190108-092708978 |journal=Helvetica Physica Acta |language=en |volume=6 |pages=110–127 |issn=0018-0238}}</ref> He refined the analysis in 1937, finding a discrepancy of about 500.<ref>{{Cite journal |last=Zwicky |first=F. |date=October 1937 |title=On the Masses of Nebulae and of Clusters of Nebulae |journal=The Astrophysical Journal |volume=86 |pages=217 |doi=10.1086/143864 |bibcode=1937ApJ....86..217Z |issn=0004-637X|doi-access=free }}</ref><ref>{{Cite journal |last1=Bertone |first1=Gianfranco |last2=Hooper |first2=Dan |date=2018-10-15 |title=History of dark matter |url=https://link.aps.org/doi/10.1103/RevModPhys.90.045002 |journal=Reviews of Modern Physics |language=en |volume=90 |issue=4 |page=045002 |doi=10.1103/RevModPhys.90.045002 |arxiv=1605.04909 |bibcode=2018RvMP...90d5002B |s2cid=18596513 |issn=0034-6861}}</ref>

==== Theoretical analysis ====
He approximated the Coma cluster as a spherical "gas" of <math>N</math> stars of roughly equal mass <math>m</math>, which gives <math display="inline">\langle T \rangle= \frac 12 Nm \langle v^2 \rangle</math>. The total gravitational potential energy of the cluster is <math>U = -\sum_{i < j} \frac{Gm^2}{r_{i,j}}</math>, giving <math display="inline">\langle U\rangle = -Gm^2 \sum_{i < j} \langle {1}/{r_{i,j}}\rangle</math>. Assuming the motion of the stars are all the same over a long enough time ([[ergodicity]]), <math display="inline">\langle U\rangle = -\frac{1}2 N^2 Gm^2\langle {1}/{r}\rangle</math>.

Zwicky estimated <math>\langle U\rangle</math> as the gravitational potential of a uniform ball of constant density, giving <math display="inline">\langle U\rangle = -\frac 35 \frac{GN^2m^2}{R}</math>.

So by the virial theorem, the total mass of the cluster is<math display="block">Nm = \frac{5\langle v^2\rangle}{3G\langle \frac{1}{r}\rangle}</math>

==== Data ====
Zwicky<math>_{1933}</math><ref name=":1" /> estimated that there are <math>N = 800</math> galaxies in the cluster, each having observed stellar mass <math>m = 10^9 M_{\odot}</math> (suggested by Hubble), and the cluster has radius <math>R = 10^6 \text{ly}</math>. He also measured the radial velocities of the galaxies by doppler shifts in galactic spectra to be <math>\langle v_r^2\rangle = (1000 \text{km/s})^2</math>. Assuming [[Equipartition theorem|equipartition]] of kinetic energy, <math>\langle v^2\rangle = 3 \langle v_r^2\rangle</math>.

By the virial theorem, the total mass of the cluster should be <math>\frac{5R \langle v_r^2\rangle}{G} \approx 3.6\times 10^{14} M_\odot</math>. However, the observed mass is <math>Nm = 8 \times 10^{11} M_\odot</math>, meaning the total mass is 450 times that of observed mass.

== Generalizations ==

Lord Rayleigh published a generalization of the virial theorem in 1900,<ref>{{cite journal|doi=10.1080/14786440009463903 |title=XV. On a theorem analogous to the virial theorem |date=August 1900 |author=Lord Rayleigh |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |series=5 |volume=50 |issue=303 |pages=210–213 |url=https://zenodo.org/record/1871519 }}</ref> which was partially reprinted in 1903.<ref>{{Cite book |url=https://books.google.com/books?id=S-sPAAAAYAAJ&pg=PA491 |title=Scientific Papers: 1892–1901 |date=1903 |author=Lord Rayleigh |publisher=Cambridge: Cambridge University Press |pages=491–493 }}</ref> [[Henri Poincaré]] proved and applied a form of the virial theorem in 1911 to the problem of formation of the Solar System from a proto-stellar cloud (then known as [[cosmogony]]).<ref>{{cite book
| last = Poincaré | first = Henri | author-link = Henri Poincaré | title = Leçons sur les hypothèses cosmogoniques |lang=fr |trans-title=Lectures on Theories of Cosmogony | publisher = Hermann | year = 1911 | pages = 90–91 et seq | location = Paris }}</ref> A variational form of the virial theorem was developed in 1945 by Ledoux.<ref>{{cite journal
 | last = Ledoux
 | first = P.
 | year = 1945
 | title = On the Radial Pulsation of Gaseous Stars
 | journal = The Astrophysical Journal
 | volume = 102
 | pages = 143–153
 | doi = 10.1086/144747
 | bibcode = 1945ApJ...102..143L
| doi-access = free
 }}</ref> A [[tensor]] form of the virial theorem was developed by Parker,<ref>{{cite journal
 | last = Parker
 | first = E. N.
 | year = 1954
 | title = Tensor Virial Equations
 | journal = Physical Review
 | volume = 96
 | issue = 6
 | pages = 1686–1689
 | doi = 10.1103/PhysRev.96.1686
 | bibcode = 1954PhRv...96.1686P
}}</ref> Chandrasekhar<ref>{{cite journal
 | last1 = Chandrasekhar
 | first1 = S.
 | author1-link = Subrahmanyan Chandrasekhar
 | last2 = Lebovitz
 | first2 = N. R.
 | year = 1962
 | title = The Potentials and the Superpotentials of Homogeneous Ellipsoids
 | journal = Astrophys. J.
 | volume = 136
 | pages = 1037–1047
 | doi = 10.1086/147456
 | bibcode = 1962ApJ...136.1037C
| doi-access = free
 }}</ref> and Fermi.<ref>{{cite journal
 | last1 = Chandrasekhar
 | first1 = S.
 | author1-link = Subrahmanyan Chandrasekhar
 | last2 = Fermi
 | first2 = E.
 | year = 1953
 | title = Problems of Gravitational Stability in the Presence of a Magnetic Field
 | journal = Astrophys. J.
 | volume = 118
 | pages = 116
 | doi = 10.1086/145732
 | bibcode = 1953ApJ...118..116C
}}</ref> The following generalization of the virial theorem has been established by Pollard in 1964 for the case of the inverse square law:<ref>{{cite journal
 | last = Pollard
 | first= H.
 | year = 1964
 | title = A sharp form of the virial theorem
 | journal = Bull. Amer. Math. Soc.
 | volume = LXX
 | pages = 703–705
 | doi = 10.1090/S0002-9904-1964-11175-7
 | issue = 5
| doi-access = free
 }}</ref><ref>{{cite book
 | last = Pollard
 | first = Harry
 | title = Mathematical Introduction to Celestial Mechanics
 | publisher = Prentice–Hall, Inc.
 | location = Englewood Cliffs, NJ
 | year = 1966
 | isbn=978-0-13-561068-8
}}</ref>{{Failed verification|date=December 2023}}
<math display="block">
 2\lim_{\tau\to+\infty} \langle T\rangle_\tau =
 \lim_{\tau\to+\infty} \langle U\rangle_\tau \quad \text{if and only if} \quad \lim_{\tau\to+\infty}{\tau}^{-2}I(\tau) = 0.
</math>
A ''boundary'' term otherwise must be added.<ref>{{cite journal
 | last1 = Kolár
 | first1 = M.
 | last2 = O'Shea
 | first2 = S. F.
 | date = July 1996
 | title = A high-temperature approximation for the path-integral quantum Monte Carlo method
 | journal = Journal of Physics A: Mathematical and General
 | volume = 29
 | issue = 13
 | pages = 3471–3494
 | bibcode = 1996JPhA...29.3471K
 | doi = 10.1088/0305-4470/29/13/018
}}</ref>

== Inclusion of electromagnetic fields ==

The virial theorem can be extended to include electric and magnetic fields. The result is<ref>{{cite book |first=George |last=Schmidt |title=Physics of High Temperature Plasmas |edition=Second |publisher=Academic Press |year=1979 |pages=72}}</ref>

<math display="block">
\frac12\frac{d^2I}{dt^2}
+ \int_Vx_k\frac{\partial G_k}{\partial t} \, d^3r
= 2(T+U) + W^\mathrm{E} + W^\mathrm{M} - \int x_k(p_{ik}+T_{ik}) \, dS_i,
</math>

where {{mvar|I}} is the [[moment of inertia]], {{mvar|G}} is the [[Poynting vector|momentum density of the electromagnetic field]], {{mvar|T}} is the [[kinetic energy]] of the "fluid", {{mvar|U}} is the random "thermal" energy of the particles, {{math|''W''{{isup|E}}}} and {{math|''W''{{isup|M}}}} are the electric and magnetic energy content of the volume considered. Finally, {{math|''p<sub>ik</sub>''}} is the fluid-pressure tensor expressed in the local moving coordinate system

<math display="block">
p_{ik}
= \Sigma n^\sigma m^\sigma \langle v_iv_k\rangle^\sigma
- V_iV_k\Sigma m^\sigma n^\sigma,
</math>

and {{math|''T<sub>ik</sub>''}} is the [[Maxwell stress tensor|electromagnetic stress tensor]],

<math display="block">
T_{ik}
= \left( \frac{\varepsilon_0E^2}{2} + \frac{B^2}{2\mu_0} \right) \delta_{ik}
- \left( \varepsilon_0E_iE_k + \frac{B_iB_k}{\mu_0} \right).
</math>

A [[plasmoid]] is a finite configuration of magnetic fields and plasma. With the virial theorem it is easy to see that any such configuration will expand if not contained by external forces. In a finite configuration without pressure-bearing walls or magnetic coils, the surface integral will vanish. Since all the other terms on the right hand side are positive, the acceleration of the moment of inertia will also be positive. It is also easy to estimate the expansion time {{mvar|τ}}. If a total mass {{mvar|M}} is confined within a radius {{mvar|R}}, then the moment of inertia is roughly {{math|''MR''<sup>2</sup>}}, and the left hand side of the virial theorem is {{math|{{sfrac|''MR''<sup>2</sup>|''τ''<sup>2</sup>}}}}. The terms on the right hand side add up to about {{math|''pR''<sup>3</sup>}}, where {{mvar|p}} is the larger of the plasma pressure or the magnetic pressure. Equating these two terms and solving for {{mvar|τ}}, we find

<math display="block">\tau\,\sim \frac{R}{c_\mathrm{s}},</math>

where {{math|''c''<sub>s</sub>}} is the speed of the [[ion acoustic wave]] (or the [[Alfvén wave]], if the magnetic pressure is higher than the plasma pressure). Thus the lifetime of a plasmoid is expected to be on the order of the acoustic (or Alfvén) transit time.

== Relativistic uniform system ==
In case when in the physical system the pressure field, the electromagnetic and gravitational fields are taken into account, as well as the field of particles’ acceleration, the virial theorem is written in the relativistic form as follows:<ref>{{cite journal |last=Fedosin|first=S. G.|s2cid=53692146|title=The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept|journal=Continuum Mechanics and Thermodynamics|volume=29|issue=2|pages=361–371| date=2016| doi=10.1007/s00161-016-0536-8|arxiv=1801.06453|bibcode=2017CMT....29..361F}}</ref>

<math display="block"> \left\langle W_k \right\rangle \approx - 0.6 \sum_{k=1}^N\langle\mathbf{F}_k\cdot\mathbf{r}_k\rangle ,</math>

where the value {{math|''W<sub>k</sub>'' ≈ ''γ<sub>c</sub>T''}} exceeds the kinetic energy of the particles {{mvar|T}} by a factor equal to the Lorentz factor {{math|''γ<sub>c</sub>''}} of the particles at the center of the system. Under normal conditions we can assume that {{math|''γ<sub>c</sub>'' ≈ 1}}, then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient {{sfrac|1|2}}, but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering the pressure field and the field of particles’ acceleration inside the system, while the derivative of the scalar {{mvar|G}} is not equal to zero and should be considered as the [[material derivative]].

An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature:<ref>{{Cite journal |last=Fedosin |first=Sergey G. |s2cid=125180719 |date=2018-09-24 |title=The integral theorem of generalized virial in the relativistic uniform model |url=http://em.rdcu.be/wf/click?upn=lMZy1lernSJ7apc5DgYM8f7AyOIJlVFO4uFv7zUQtzk-3D_DUeisO4Ue44lkDmCnrWVhK-2BAxKrUexyqlYtsmkyhvEp5zr527MDdThwbadScvhwZehXbanab8i5hqRa42b-2FKYwacOeM4LKDJeJuGA15M9FWvYOfBgfon7Bqg2f55NFYGJfVGaGhl0ghU-2BkIJ9Hz4M6SMBYS-2Fr-2FWWaj9eTxv23CKo9d8nFmYAbMtBBskFuW9fupsvIvN5eyv-2Fk-2BUc7hiS15rRISs1jpNnRQpDtk2OE9Hr6mYYe5Y-2B8lunO9GwVRw07Y1mdAqqtEZ-2BQjk5xUwPnA-3D-3D |journal=Continuum Mechanics and Thermodynamics |volume=31|issue=3|pages=627–638|language=en |doi=10.1007/s00161-018-0715-x |issn=1432-0959 |bibcode=2019CMT....31..627F |arxiv=1912.08683 }}</ref>

<math display="block"> v_\mathrm{rms} = c \sqrt{1- \frac {4 \pi \eta \rho_0 r^2}{c^2 \gamma^2_c \sin^2 \left( \frac {r}{c} \sqrt {4 \pi \eta \rho_0} \right) } } ,</math>

where <math>~ c </math> is the speed of light, <math>~ \eta </math> is the acceleration field constant, <math>~ \rho_0 </math> is the mass density of particles, <math>~ r </math> is the current radius.

Unlike the virial theorem for particles, for the electromagnetic field the virial theorem is written as follows:<ref>{{cite journal|last=Fedosin |first=S.G. |url= https://dergipark.org.tr/en/pub/gujs/issue/45480/435567 |title=The Integral Theorem of the Field Energy |journal= Gazi University Journal of Science |volume=32 |issue= 2 |pages= 686–703 |year=2019 |doi=10.5281/zenodo.3252783|s2cid= 197487015 |doi-access=free }}</ref>
<math display="block">~ E_{kf} + 2 W_f =0 , </math>
where the energy <math display="inline">~ E_{kf} = \int A_\alpha j^\alpha \sqrt {-g} \,dx^1 \,dx^2 \,dx^3 </math> considered as the kinetic field energy associated with four-current <math> j^\alpha </math>, and
<math display="block">~ W_f = \frac {1}{4 \mu_0 } \int F_{\alpha \beta} F^{\alpha \beta} \sqrt {-g} \,dx^1 \,dx^2 \,dx^3 </math>
sets the potential field energy found through the components of the electromagnetic tensor.

== In astrophysics ==
The virial theorem is frequently applied in astrophysics, especially relating the [[gravitational energy|gravitational potential energy]] of a system to its [[kinetic energy|kinetic]] or [[thermal energy]]. Some common virial relations are {{Citation needed|date=December 2019}}
<math display="block">\frac35 \frac{GM}{R} = \frac32 \frac{k_\mathrm{B} T}{m_\mathrm{p}} = \frac12 v^2 </math>
for a mass {{mvar|M}}, radius {{mvar|R}}, velocity {{mvar|v}}, and temperature {{mvar|T}}. The constants are [[Gravitational constant|Newton's constant]] {{mvar|G}}, the [[Boltzmann constant]] {{math|''k''<sub>B</sub>}}, and proton mass {{math|''m''<sub>p</sub>}}. Note that these relations are only approximate, and often the leading numerical factors (e.g. {{sfrac|3|5}} or {{sfrac|1|2}}) are neglected entirely.

=== Galaxies and cosmology (virial mass and radius) ===
{{Main|Virial mass}}
In [[astronomy]], the mass and size of a galaxy (or general overdensity) is often defined in terms of the "[[virial mass]]" and "[[virial radius]]" respectively. Because galaxies and overdensities in continuous fluids can be highly extended (even to infinity in some models, such as an [[singular isothermal sphere|isothermal sphere]]), it can be hard to define specific, finite measures of their mass and size. The virial theorem, and related concepts, provide an often convenient means by which to quantify these properties.

In galaxy dynamics, the mass of a galaxy is often inferred by measuring the [[rotation velocity]] of its gas and stars, assuming [[circular orbit|circular Keplerian orbits]]. Using the virial theorem, the [[velocity dispersion]] {{mvar|σ}} can be used in a similar way. Taking the kinetic energy (per particle) of the system as {{math|1=''T'' = {{sfrac|1|2}}''v''<sup>2</sup> ~ {{sfrac|3|2}}''σ''<sup>2</sup>}}, and the potential energy (per particle) as {{math|''U'' ~ {{sfrac|3|5}} {{sfrac|''GM''|''R''}}}} we can write

<math display="block"> \frac{GM}{R} \approx \sigma^2. </math>

Here <math>R</math> is the radius at which the velocity dispersion is being measured, and {{mvar|M}} is the mass within that radius. The virial mass and radius are generally defined for the radius at which the velocity dispersion is a maximum, i.e.

<math display="block"> \frac{GM_\text{vir}}{R_\text{vir}} \approx \sigma_\max^2. </math>

As numerous approximations have been made, in addition to the approximate nature of these definitions, order-unity proportionality constants are often omitted (as in the above equations). These relations are thus only accurate in an [[order of magnitude]] sense, or when used self-consistently.

An alternate definition of the virial mass and radius is often used in cosmology where it is used to refer to the radius of a sphere, centered on a [[galaxy]] or a [[galaxy cluster]], within which virial equilibrium holds. Since this radius is difficult to determine observationally, it is often approximated as the radius within which the average density is greater, by a specified factor, than the [[Critical density (cosmology)|critical density]]
<math display="block">\rho_\text{crit}=\frac{3H^2}{8\pi G}</math>
where {{mvar|H}} is the [[Hubble's law|Hubble parameter]] and {{mvar|G}} is the [[gravitational constant]]. A common choice for the factor is 200, which corresponds roughly to the typical over-density in spherical top-hat collapse (see [[Virial mass]]), in which case the virial radius is approximated as
<math display="block">r_\text{vir} \approx r_{200}= r, \qquad \rho = 200 \cdot \rho_\text{crit}.</math>
The virial mass is then defined relative to this radius as
<math display="block">M_\text{vir} \approx M_{200} = \frac43\pi r_{200}^3 \cdot 200 \rho_\text{crit} .</math>

=== Stars ===
The virial theorem is applicable to the cores of stars, by establishing a relation between gravitational potential energy and thermal kinetic energy (i.e. temperature). As stars on the [[main sequence]] convert hydrogen into helium in their cores, the mean molecular weight of the core increases and it must contract to maintain enough pressure to support its own weight. This contraction decreases its potential energy and, the virial theorem states, increases its thermal energy. The core temperature increases even as energy is lost, effectively a negative [[specific heat]].<ref name="BASUCHATTOPADHYAY2010">{{cite book|author1=BAIDYANATH BASU|author2=TANUKA CHATTOPADHYAY|author3=SUDHINDRA NATH BISWAS|title=AN INTRODUCTION TO ASTROPHYSICS|url=https://books.google.com/books?id=WG-HkqCXhKgC&pg=PA365|date=1 January 2010|publisher=PHI Learning Pvt. Ltd.|isbn=978-81-203-4071-8|pages=365–}}</ref> This continues beyond the main sequence, unless the core becomes degenerate since that causes the pressure to become independent of temperature and the virial relation with {{mvar|n}} equals −1 no longer holds.<ref name="Rose1998">{{cite book|author=William K. Rose|title=Advanced Stellar Astrophysics|url=https://books.google.com/books?id=yaX0etDmbXMC&pg=PA242|date=16 April 1998|publisher=Cambridge University Press|isbn=978-0-521-58833-1|pages=242–}}</ref>

== See also ==
* [[Virial coefficient]]
* [[Virial stress]]
* [[Virial mass]]
* [[Chandrasekhar potential energy tensor|Chandrasekhar tensor]]
* [[Chandrasekhar virial equations]]
* [[Derrick's theorem]]
* [[Equipartition theorem]]
* [[Ehrenfest theorem]]
* [[Pokhozhaev's identity]]
* [[Statistical mechanics]]

== References ==
{{Reflist}}

== Further reading ==
* {{Cite book |last=Goldstein |first=H. |year=1980 |title=Classical Mechanics |edition=2nd |publisher=Addison–Wesley |isbn=978-0-201-02918-5}}
* {{Cite book |last=Collins |first=G. W. |year=1978 |title=The Virial Theorem in Stellar Astrophysics |publisher=Pachart Press |url=http://ads.harvard.edu/books/1978vtsa.book/ |bibcode=1978vtsa.book.....C |isbn=978-0-912918-13-6 }}
* {{cite journal |doi=10.1088/0143-0807/37/4/045405 |title=An elementary derivation of the quantum virial theorem from Hellmann–Feynman theorem |year=2016 |last1=i̇Pekoğlu |first1=Y. |last2=Turgut |first2=S. |journal=European Journal of Physics |volume=37 |issue=4 |page=045405 |s2cid=125030620 |bibcode=2016EJPh...37d5405I }}

== External links ==
* [http://www.mathpages.com/home/kmath572/kmath572.htm The Virial Theorem] at MathPages
* [http://hyperphysics.phy-astr.gsu.edu/hbase/astro/gravc.html#c2 ''Gravitational Contraction and Star Formation''], Georgia State University

[[Category:Physics theorems]]
[[Category:Dynamics (mechanics)]]
[[Category:Solid mechanics]]
[[Category:Concepts in physics]]
[[Category:Equations of astronomy]]