= Gravitational binding energy from strip density
{title2=$W=-2G\int_0^\infty S(x)^2\,dx$}
For an isolated spherical distribution, shell assembly gives <gravitational potential energy> $W=-\int_0^\infty GM(r)r^{-1}\,dM(r)$. Use $S'=-2\pi r\rho$ and $M'=-2rS'$ to write $W=2G\int MS'\,dr=-2G\int SM'\,dr=4G\int rSS'\,dr$. A second <integration by parts> gives the displayed negative quadratic functional. The boundary terms $MS$ and $rS^2$ must vanish; finite-mass regular profiles with finite binding energy satisfy the usual sufficient decay conditions.
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