Virial theorem

A theorem according to which the kinetic energy $ \overline{T}\; $ of a mechanical system, averaged over an infinite period of time, is equal to the virial of the forces averaged over the same interval, i.e.

$$ \tag{1 } \overline{T}\; = - { \frac{1}{2} } \sum _ {i = 1 } ^ { N } \overline{ {\mathbf F _ {i} \mathbf r _ {i} }}\; , $$

where $ N $ is the number of material points of the system, $ \mathbf F _ {i} $ is the force acting on the $ i $- th point of the system and $ \mathbf r _ {i} $ is the radius vector of this point. The bars over the symbols denote that the respective function is averaged over an infinite period of time.

The virial theorem was established in 1870 by R. Clausius, and is a consequence of the equations of motion of a mechanical system on the condition that the motion of the system takes place in a bounded domain in space and with bounded (in modulus) velocities of the points. If the forces acting on the points of the system derive from a potential (are conservative), equation (1) assumes the form

$$ \tag{2 } \overline{T}\; = { \frac{1}{2} } \sum _ {i = 1 } ^ { N } \overline{ {( \mathbf r _ {i} \nabla _ {i} ) U }}\; . $$

If one imposes an additional condition — viz., that the potential energy is homogeneous of degree $ \nu $ in the coordinates of the material points — (2) implies a relationship between the average values of kinetic and potential energies of the system:

$$ \tag{3 } \overline{T}\; = { \frac \nu {2} } \overline{U}\; , $$

which is important in practice. For instance, for a harmonic oscillator ( $ U \sim r ^ {2} $, $ \nu = 2 $), $ \overline{T}\; = \overline{U}\; $, while for a point moving in a Newtonian gravity field ( $ U \sim 1/r $, $ \nu = - 1 $), $ \overline{T}\; = - \overline{U}\; /2 $.

The virial theorem is utilized in mechanics, statistical mechanics, astronomy, and atomic physics (e.g. in demonstrating equations of state and in the determination of the constant intermolecular interactions). The theorem in the forms (2) and (3) is also used in quantum mechanics (with appropriate generalizations of the averaging operation and of other notions employed in (2) and (3)).

References

[1]	L.D. Landau, E.M. Lifshits, "Mechanics" , Pergamon (1965) (Translated from Russian)
[2]	A.S. Davydov, "Quantum mechanics" , Pergamon (1965) (Translated from Russian)
[3]	J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, "Molecular theory of gases and liquids" , Wiley (1954)
[4]	I.I. Ol'khovskii, "A course in theoretical mechanics for physicists" , Moscow (1974) (In Russian)