(Palais, 1979) Many problems are set up as variational problems. That is, on a (possibly infinite-dimensional) manifold M with a group of symmetries G, we look for critical points of a G-invariant functional f. In order to do this, we might first restrict ourselves to looking at the set S of G-symmetric points of M (points p such that gp=p for all g). The principle asserts that if p is a critical point of f restricted to S, then p is in fact a critical point in M. In other words: “Critical symmetric points are symmetric critical points”.
For example, harmonic functions on a space X are critical points of an energy functional on a space of functions M = {X to R}. To find a harmonic map on X, we might start by considering only maps which are rotationally symmetric. The principle states that it suffices to consider only rotationally symmetric variations as well. This reduces the problem from a PDE to an ODE, how nice!
Anyway, the Principle does not always hold, but in some very general situations it does. Let’s find out about them.