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 $V$. The talk is aimed at a discussion of the constructive ways of finding out whether or not $O_1$ lies in the Zariski closure of $O_2$. This yields the constructive ways of finding 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.