 Newtonian central configuration

A collision-free positive-mass configuration satisfies the displayed equation when every gravitational acceleration points proportionally toward a common centre. With $W=-{\sum }_{i<j}G{m}_i{m}_j/|{\mathbf{a}}_i-{\mathbf{a}}_j|$, homogeneity gives $\Lambda I=-W>0$, $I={\sum }_i{m}_i|{\mathbf{a}}_i{|}^2$. Pairwise force cancellation implies ${\sum }_i{m}_i{\mathbf{a}}_i=0$. Such a configuration generates a nonrotating homothetic gravitational motion by multiplying all position vectors by one positive scale factor.