Two Orbits: When Is One in the Closure of the Other?

Let $G$ be a connected linear algebraic group, let $V$ be a finite dimensional algebraic $G$-module, and let $\mathcal O_1$ and $\mathcal O_2$ be two $G$-orbits in $V$. We describe a constructive way to find out whether or not $\mathcal O_1$ lies in the closure of $\mathcal O_2$.