Consider a classical particle subject to Newton’s second law \(\dot{\textbf p}=\textbf F\). If one takes the dot product of both sides with the particle’s velocity \(\dot{\textbf x}\) so that \(\dot{\textbf p}\cdot\dot{\textbf x}=\textbf F\cdot\dot{\textbf x}\), the left hand side is an exact total derivative \(\dot T=\dot{\textbf p}\cdot\dot{\textbf x}\) of the kinetic energy \(T=\frac{\textbf p\cdot\dot{\textbf x}}{2}\) while the right hand side is by definition the total power \(P\) developed, thus obtaining the familiar \(\dot T=P\). Of course we all know how useful this is, especially when \(\textbf F=-\frac{\partial V}{\partial\textbf x}\) is conservative so that \(P=-\dot V\) and integration yields a conserved energy \(E=T+V\).
But what if, instead of dotting the equation of motion \(\dot{\textbf p}=\textbf F\) with the velocity \(\dot{\textbf x}\) as we did above, we were instead curious about what fruitful insights might arise if we simply dot the equation of motion \(\dot{\textbf p}=\textbf F\) with the position \(\textbf x\) itself rather than the velocity \(\dot{\textbf x}\). This leads to:
\[\dot{\textbf p}\cdot\textbf x=\textbf F\cdot\textbf x\]
But we notice that the right-hand side of the equation looks like the work done by the force \(\textbf F\) if the particle were moved from the origin \(\textbf 0\) to its actual position \(\textbf x\) (strictly speaking, this interpretation is only exactly correct is \(\textbf F\) is a constant force). This suggests that the left-hand side of the equation should be amenable to some kind of energy interpretation. One can notice the “integration by parts” combination \(\dot{\textbf p}\cdot\textbf x=\frac{d}{dt}(\textbf x\cdot\textbf p)-\textbf p\cdot\dot{\textbf x}\) and moreover the last term is just \(2T=\textbf p\cdot\dot{\textbf x}\) twice the kinetic energy, confirming the energy interpretation mentioned above. The remaining term inside the time derivative is a bit more mysterious, consisting of the dot product of the position \(\textbf x\) with its conjugate momentum \(\textbf p\) (feels like a combination you would see in Hamiltonian mechanics, but nothing immediately comes to mind). Let’s call it the virial \(\tilde L:=\textbf x\cdot\textbf p\) because it resembles angular momentum \(\textbf L=\textbf x\times\textbf p\) except with the cross product \(\times\) replaced by a dot product \(\cdot\).
The key insight of the virial theorem is that, in many physical situations of interest in the real world (e.g. bound orbits in a Coulomb potential), the average rate of change of the virial \(\tilde L\) on sufficiently long timescales \(\Delta t\to\infty\) is zero. To see this, recall that the average rate of change over a time interval \(\Delta t\) is just \(\frac{\Delta\tilde L|_{\Delta t}}{\Delta t}\). Under what conditions does this quotient vanish? Of course, it could just happen to sporadically vanish at random specific values of \(\Delta t\) but that’s not a particularly reliable phenomenon. Instead, the more robust way it would (approximately) vanish is if the denominator \(\Delta t\to\infty\) is large. But there is a catch here, which is that the numerator \(\Delta\tilde L|_{\Delta t}\) could also blow up to \(\infty\) at the same time. So really, the requirement we’re after is that the numerator \(\Delta\tilde L|_{\Delta t}\) does not grow faster than the denominator \(\Delta t\) as \(\Delta t\to\infty\). In other words, we require sublinear numerator growth \(\Delta\tilde L|_{\Delta t}=o_{\Delta t\to\infty}(\Delta t)\). Of course, it could be that the numerator actually decays as \(\Delta t\to\infty\), or remains bounded \(\Delta\tilde L|_{\Delta t}=\Theta_{\Delta t\to\infty}(1)\).
Assuming therefore that the physical situation of interest satisfies the \( \frac{\Delta\tilde L|_{\Delta t}}{\Delta t}=0\) hypothesis of the virial theorem, it follows that, on suitably long time scales \(\Delta t\):
\[2\langle T\rangle_{\Delta t}=-\langle\textbf F\cdot\textbf x\rangle_{\Delta t}\]
In the specific case where the force \(\textbf F\) is not only conservative but specifically arises from a central power-law potential energy \(V(\textbf x)=k|\textbf x|^n\), then the corresponding central force is \(\textbf F=-\frac{\partial V}{\partial\textbf x}=-nk|\textbf x|^{n-2}\textbf x\) and so happily \(\textbf F\cdot\textbf x=-nV(\textbf x)\) yielding the useful corollary of the virial theorem \(2\langle T\rangle_{\Delta t}=n\langle V\rangle_{\Delta t}\) for central power-law forces. For the isotropic harmonic oscillator \(n=2\), this gives the familiar equipartition theorem \(\langle T\rangle_{\Delta t}=\langle V\rangle_{\Delta t}\) while for the Coulomb potential \(n=-1\) this gives \(2\langle T\rangle_{\Delta t}=-\langle V\rangle_{\Delta t}\). Finally, it is not difficult to generalize the virial theorem to multi-particle systems, where the generalization is essentially the obvious one.