Hartree mass and energy conservation
= Hartree mass and energy conservation
{c}
The gravitational Hartree flow conserves
$$
M(u)=\int|u|^2,
\qquad
E(u)=\frac12\int|\nabla u|^2-\frac14\int|\nabla\phi|^2.
$$
The self-adjointness of the Newtonian convolution turns the time derivative of the potential energy into $\operatorname{Re}\int\phi u\overline{u_t}$, which cancels the derivative of the kinetic energy by the equation.
= Hartree mass conservation
{synonym}
= Hartree energy conservation
{synonym}