Exploring the Virial Theorem: Understanding the Derivative of G Over a Period T

[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

 

[HalfLostAndSearching]

The Virial theorem is a fundamental concept in classical mechanics that relates the average kinetic and potential energies of a system. In this case, the scalar virial G is defined as the dot product of the momentum vector and the location vector. This means that G is a measure of the average momentum and position of the system.

When we take the mean of the derivative of G over a period T, we are essentially looking at the average change in G over that period. Since G is a measure of average momentum and position, this derivative represents the average rate of change of momentum and position over time.

Now, according to the Virial theorem, the average kinetic energy (K) of a system is equal to half of the average potential energy (U). Mathematically, this can be expressed as K = 1/2U. This means that the average rate of change of momentum (which is proportional to kinetic energy) is equal to half of the average rate of change of position (which is proportional to potential energy).

Since these two rates of change are equal but opposite in direction, their sum will always be zero. This means that the derivative of G, which is a combination of these two rates of change, will also be zero over a period T. In other words, the average change in G over a period T will equal zero because the average changes in momentum and position will cancel each other out.

In conclusion, the reason why the mean of the derivative of G over a period T is zero is because of the fundamental relationship between kinetic and potential energy described by the Virial theorem. This theorem tells us that the average changes in momentum and position of a system will always balance each other out, resulting in a net change of zero over a period of time.