Gravitational binding energy from strip density ($W=-2G{\int }_0^{\infty }S(x{)}^{2}dx$)

For an isolated spherical distribution, shell assembly gives gravitational potential energy $W=-{\int }_0^{\infty }GM(r)r^{-1}dM(r)$. Use $S^{\prime }=-2\pi r\rho$ and $M^{\prime }=-2r{S}^{\prime }$ to write $W=2G\int M{S}^{\prime }dr=-2G\int S{M}^{\prime }dr=4G\int rS{S}^{\prime }dr$. A second integration by parts gives the displayed negative quadratic functional. The boundary terms $MS$ and $r{S}^2$ must vanish; finite-mass regular profiles with finite binding energy satisfy the usual sufficient decay conditions.