virial theorem
VirialTheorem
2004-11-19 01:01:34
2004-11-19 01:06:54
Definition
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We start with the moment of inertia about the origin for the system of particles, which is defined as
\begin{equation}\label{first}
I = \sum_i m_i {r_i}^2
\end{equation}
Differentiate using the chain rule. Note that a vector dotted into itself yields its magnitude square.
\begin{equation}
\vec{r_i} \cdot \vec{r_i} = r_i^2
\end{equation}
This lets us make the connection that
\begin{equation}
I = \sum_i m_i {r_i}^2 = \sum_i m_i (\vec{r_i} \cdot \vec{r_i})
\end{equation}
So after differentiating we get
\begin{equation}
\frac{dI}{ dt} = \sum_i m_i \frac{\vec{dr_i}}{dt} \cdot \vec{r_i} + \vec{r_i} \cdot \frac{\vec{dr_i}}{dt}
\end{equation}
Differentiating again yields
\begin{equation}
\frac{d^2I}{dt^2} = \sum_i m_i \frac{d^2\vec{r_i}}{dt^2} \cdot \vec{r_i} + \frac{\vec{dr_i}}{dt} \cdot \frac{\vec{dr_i}}{dt} + \frac{\vec{dr_i}}{dt} \cdot \frac{\vec{dr_i}}{dt} + \vec{r_i} \cdot \frac{d^2\vec{r_i}}{dt^2}
\end{equation}
In short form
\begin{equation}
\frac{d^2I}{dt^2} = 2\, \sum_i (m_i \dot{\vec{r_i}} \cdot \dot{\vec{r_i}} + m_i\vec{r_i} \cdot \ddot{\vec{r_i}})
\end{equation}