On the Coleman's principle concerning the stationary points of invariant functionals

Let $G$ be a set of transformations of a topological space $X$ and $X_0$ be a set of all $G$-invariant points of $X$. Let $f$ be a functional defined on $X$ and invariant under the transformations from $G$. We find some weak conditions on $X$, $G$ and $f$ under which the following fact is true: if $x_0\in X_0$ is a stationary point of the restriction of $f$ to $X_0$, then $x_0$ is a stationary point of $f$ on the whole $X$. We also give some applications of our abstract theorems, obteined in this article to the different multidimensional variational problems, Skyrme's non-linear field model being in their number.