I was going through energy conservation for unconstrained multi-particle systems in Gregory's Classical Mechanics ("Energy Principle", p266).
In the section where he calculates the work done by conservative internal forces ($G_{ij}$), he writes that the rate of work done for a pair of forces:
$$ G_{ij} \cdot v_{i} + G_{ji} \cdot v_{j} = G_{ij} \cdot (v_i - v_j) = |G_{ij}|\hspace{.1cm}\hat{r_{ij}} \cdot \dot{r_{ij}} = |G_{ij}||\dot{r_{ij}}|. $$
I am confused with the last step, where he says that the velocity of $j$ wrt $i$ is parallel to the displacement between them, that $$r_{ij} \cdot \dot{r_{ij}} = |r_{ij}||\dot{r_{ij}}|.$$ For example, the system could be rotating in which case velocity is clearly not along displacement. Even if there are no external forces, I think other particles could drag the $j$ particle off of $r_{ij}$. It seems to me that this only holds in the case of 2 isolated particles. What am I missing?
