Why does the mean derivative of the scalar virial equal zero over one period?

[Karol]

Homework Statement

In the Virial theorem The scalar virial G is defined by the equation:
$$G=\vec{p}\cdot \vec{r}$$
Where $\vec{p}$ is the momentum vector and $\vec{r}$ the location vector.
When i take the mean of the derivative $\bar{\dot{G}}$ over a whole period T it equals 0. why?

Homework Equations

$$\vec{p}\cdot \vec{r}=(mv)\cdot \cos \theta \cdot r$$

The Attempt at a Solution

I understand this scalar product is zeroed during one period, but why?

 

[ehild]

You said the function G is periodic. You want the mean of its time-derivative.

How do you calculate the mean of a function?

What is the integral of the derivative?

ehild

 

[Karol]

I think i understand.
$$\bar{\dot{G}}=\frac{1}{T}\int_{0}^{T}\frac{dG}{dt}dt=\frac{1}{T}(G(T)-G(0))$$
Because the end point and the start point are identical G(T)=G(0)

 

[ehild]

I think i understand.
$$\bar{\dot{G}}=\frac{1}{T}\int_{0}^{T}\frac{dG}{dt}dt=\frac{1}{T}(G(T)-G(0))$$
Because the end point and the start point are identical G(T)=G(0)

Correct

ehild