0} \begin{split} 0 & = \frac{d}{dt} \left ( \int {\mathcal{F}}\int f v \ dv\,dx + \int {\mathcal{G}}\int g v \ dv\,dx \right ) \\ & \quad + \int \left [ \left (\int f v \ dv \right ) \left (\int f \hat{v}_0 \ dv \right) 