Homothetic gravitational motion
= Homothetic gravitational motion
{title2=$\ddot a=-\Lambda/a^2$}
For a <Newtonian central configuration>, set $\mathbf x_i(t)=a(t)\mathbf a_i$ with $a>0$. The gravitational force scales as $a^{-2}$, reducing all equations to the displayed scalar <ordinary differential equation>. Its first integral is $\dot a^2/2-\Lambda/a=k/2$. The total <energy> is $kI/2$, so the scalar energy constant exactly tracks the many-body energy for the fixed shape.