Orbit closures

Let $G$ be a connected linear algebraic group, let $V$ be a finite dimensional algebraic $G$-module, and let $O_1$ and $O_2$ be two $G$-orbits in $V$. I shall describe a constructive way to find out whether or not $O_1$ lies in the closure of $O_2$. This yields a constructive way to find out whether given two points of $V$ lie in the same orbit or not. Several classical problems in algebra and algebraic geometry are reduced to this problem.