 Solution

Work in an inertial centre of mass frame and write $I={\sum }_i{m}_i|{\mathbf{r}}_i{|}^2$. Differentiation and Newton's second law giveThe pairwise Newtonian gravitational potential energy isFor each pair, combining the two force contributions gives $-G{m}_i{m}_j/|{\mathbf{r}}_i-{\mathbf{r}}_j|$. Therefore ${\sum }_i{\mathbf{r}}_i\cdot {\mathbf{F}}_i=\Omega$, andFor a stationary configuration, or after a long-time average of a bounded system whose endpoint term vanishes, the virial theorem givesThe brackets may be suppressed when $T,\Omega$ denote equilibrium averages. A collapsing cloud or an eccentric Kepler orbit need not obey $2T+\Omega =0$ instantaneously. Confining surface pressure, magnetic stress or other external forces add their own virial terms; the displayed result concerns an isolated gravitational particle system.