Editing Equipartition theorem (section)

===Relation to the virial theorem===
{{See also|Virial theorem|Generalized coordinates|Hamiltonian mechanics}}

The general equipartition theorem is an extension of the [[virial theorem]] (proposed in 1870<ref>{{cite journal | last = Clausius | first = R | author-link = Rudolf Clausius | year = 1870 | title = Ueber einen auf die Wärme anwendbaren mechanischen Satz | journal = Annalen der Physik | volume = 141 | issue = 9 | pages = 124–130 | url = http://gallica.bnf.fr/ark:/12148/bpt6k152258 | doi=10.1002/andp.18702170911|language=de|bibcode = 1870AnP...217..124C }}<br />{{cite journal | last = Clausius | first = RJE | author-link = Rudolf Clausius | year = 1870 | title = On a Mechanical Theorem Applicable to Heat | journal = [[Philosophical Magazine]] |series=Series 4 | volume = 40 | pages = 122–127}}</ref>), which states that
<math display="block">
\left\langle \sum_k q_k \frac{\partial H}{\partial q_{k}} \right\rangle = 
\left\langle \sum_k p_k \frac{\partial H}{\partial p_{k}} \right\rangle = 
\left\langle \sum_k p_k \frac{dq_k}{dt} \right\rangle = -\left\langle \sum_k q_k \frac{dp_k}{dt} \right\rangle,
</math>
where ''t'' denotes [[time]].<ref name="goldstein_1980" /> Two key differences are that the virial theorem relates ''summed'' rather than ''individual'' averages to each other, and it does not connect them to the [[temperature]] ''T''. Another difference is that traditional derivations of the virial theorem use averages over time, whereas those of the equipartition theorem use averages over [[phase space]].