Horizontal generator of contact stability ($({\iota }_{Y_t}d{\alpha }_t){|}_{{\xi }_t}=-{\dot{\alpha }}_t{|}_{{\xi }_t}$)

For a family of contact forms, require $Y_t\in \ker {\alpha }_t$ and solve $({\iota }_{Y_t}d{\alpha }_t){|}_{\ker {\alpha }_t}=-{\dot{\alpha }}_t{|}_{\ker {\alpha }_t}$. The contact distribution symplectic form gives a unique smooth solution. Then ${\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t=h_t{\alpha }_t$, with $h_t={\dot{\alpha }}_t(R_t)$ and $R_t$ the Reeb vector field.