= Newtonian central configuration
{c}
{title2=$\nabla_iW=\Lambda m_i\mathbf a_i$}
A collision-free positive-mass configuration satisfies the displayed equation when every gravitational acceleration points proportionally toward a common centre. With $W=-\sum_{i<j}Gm_im_j/|\mathbf a_i-\mathbf a_j|$, 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.
Back to article page
