In mechanics, the virial theorem provides a general equation relating the average over time of the total kinetic energy, $\backslash left\backslash langle\; T\; \backslash right\backslash rangle$, of a stable system, bound by potential forces, with that of the total potential energy, $\backslash left\backslash langle\; V\_\backslash text\; \backslash right\backslash rangle$, where angle brackets represent the average over time of the enclosed quantity. Mathematically, the theorem states :$2\; \backslash left\backslash langle\; T\; \backslash right\backslash rangle\; =\; -\backslash sum\_^N\; \backslash left\backslash langle\; \backslash mathbf\_k\; \backslash cdot\; \backslash mathbf\_k\; \backslash right\backslash rangle$ where F_k represents the force on the kth particle, which is located at position r_k. The word "virial" derives from vis, the Latin word for "force" or "energy", and was given its technical definition by Clausius in 1870. Fritz Zwicky was the first to use the virial theorem to deduce the existence of unseen matter, what is now called dark matter. 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 V(r) = αr that is proportional to some power n of the inter-particle distance r, the virial theorem adopts a simple form :$2\; \backslash langle\; T\; \backslash rangle\; =\; n\; \backslash langle\; V\_\backslash text\; \backslash rangle.$ Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step.

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