5.3: The Virial Theorem

For a potential energy homogeneous in the coordinates, of degree \(\begin{equation}k\end{equation}\), say, and spatially bounded motion, there is a simple relation between the time averages of the kinetic energy, \(\begin{equation}\bar{T}, \text { and potential energy, } \bar{U} \end{equation}\). It’s called the virial theorem.

Theorem \(\PageIndex{1}\): Virial Theorem

\[ 2 \bar{T}=k \bar{U}\]

Proof

Since

\[ T=\sum_{i} \frac{1}{2} m_{i} \vec{v}_{i}^{2}, \quad \vec{p}_{i}=m_{i} \vec{v}_{i}=\partial T / \partial \vec{v}_{i}\]

we have

\[ 2 T=\sum_{i} \vec{p}_{i} \cdot \vec{v}_{i}=\frac{d}{d t}\left(\sum_{i} \vec{p}_{i} \cdot \vec{r}_{i}\right)-\sum_{i} \vec{r}_{i} \cdot \dot{\vec{p}}_{i}\]

We now average the terms in this equation over a very long time, that is, take

\[ \bar{f}=\lim _{\tau \rightarrow \infty} \frac{1}{\tau} \int_{0}^{\tau} f(t) d t\]

Since we’ve said the orbits are bounded in space, and we assume also in momentum, the exact differential term contributes

\[ \frac{1}{\tau}\left[\left(\sum_{i} \vec{p}_{i} \cdot \vec{r}_{i}\right)_{\text {at final }}-\left(\sum_{i} \vec{p}_{i} \cdot \vec{r}_{i}\right)_{\text {at initial }}\right] \rightarrow 0 \]

in the limit of infinite time.

So we have the time averaged

\[2 \bar{T}=\sum_{i} \vec{r}_{i} \cdot \partial U / \partial \vec{r}_{i} \]

and for a potential energy a homogeneous function of degree \(k\) in the coordinates, from Euler’s theorem:

\[ 2 \bar{T}=k \bar{U}\]

So, for example, in a simple harmonic oscillator the average kinetic energy equals the average potential energy, and for an inverse-square system, the average kinetic energy is half the average potential energy in magnitude, and of opposite sign (being of course positive).