and let $U\in C^\infty((0,\infty);\mathbb R)$ satisfy $U\circ\rho\in L^1(\mathbb R^n,\mathcal L^n)$. Let $V\in C^\infty(\mathbb R^n;\mathbb R)$ have at most quadratic growth, and let $W\in C^\infty(\mathbb R^n;\mathbb R)$ be even and have at most quadratic growth. Define functionals on such densities by