A collection of $N$ particles, with masses $m_{i}$ and positions $\mathbf{x}_{i}$, interact through a gravitational potential

$$V=\sum_{i<j} V_{i j}=-\sum_{i<j} \frac{G m_{i} m_{j}}{\left|\mathbf{x}_{i}-\mathbf{x}_{j}\right|} .$$

Assume that the system is gravitationally bound, and that the positions $\mathbf{x}_{i}$ and velocities $\dot{\mathbf{x}}_{i}$ are bounded for all time. Further, define the time average of a quantity $X$ by

$$\bar{X}=\lim _{t \rightarrow \infty} \frac{1}{t} \int_{0}^{t} X\left(t^{\prime}\right) \mathrm{d} t^{\prime}$$

(a) Assuming that the time average of the kinetic energy $T$ and potential energy $V$ are well defined, show that

$$\bar{T}=-\frac{1}{2} \bar{V}$$

[You should consider the quantity $I=\frac{1}{2} \sum_{i=1}^{N} m_{i} \mathbf{x}_{i} \cdot \mathbf{x}_{i}$, with all $\mathbf{x}_{i}$ measured relative to the centre of mass.]

(b) Explain how part (a) can be used, together with observations, to provide evidence in favour of dark matter. [You may assume that time averaging may be replaced by an average over particles.]