Relative equilibria of molecules

We describe a method for finding the families of relative equilibria of molecules which bifurcate from an equilibrium point as the angular momentum is increased from 0. Relative equilibria are steady rotations about a stationary axis during which the shape of the molecule remains constant. We show that the bifurcating families correspond bijectively to the critical points of a function h on the 2-sphere which is invariant under an action of the symmetry group of the equilibrium point. From this it follows that for each rotation axis of the equilibrium configuration there is a bifurcating family of relative equilibria for which the molecule rotates about that axis. In addition, for each reflection plane there is a family of relative equilibria for which the molecule rotates about an axis perpendicular to the plane. We also show that if the equilibrium is non-degenerate and stable then the minima, maxima and saddle points of h correspond respectively to relative equilibria which are (orbitally) Liapounov stable, linearly stable and linearly unstable. The stabilities of the bifurcating branches of relative equilibria are computed explicitly for XY2, X3 and XY4 molecules. These existence and stability results are corollaries of more general theorems on relative equi