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

$$T=\sum_{i} \dfrac{1}{2} m_{i} \vec{v}_{i}^{2}, \quad \vec{p}_{i}=m_{i} \vec{v}_{i}=\partial T / \partial \vec{v}_{i}$$

we have

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

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

$$\bar{f}=\lim _{\tau \rightarrow \infty} \dfrac{1}{\tau} \int_{0}^{\tau} f(t) d t$$

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

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

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).