RELATIVE EQUILIBRIA OF MOLECULES

We describe a method for finding the families of relative equilibria o f molecules that bifurcate from an equilibrium point as the angular mo mentum is increased from 0. Relative equilibria are steady rotations a bout 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 two-sphere which is inv ariant under an action of the symmetry group of the equilibrium point. From this it follows that for each rotation axis of the equilibrium c onfiguration there is a bifurcating family of relative equilibria for which the molecule rotates about that axis. In addition, for each refl ection plane there is a family of relative equilibria for which the mo lecule rotates about an axis perpendicular to the plane. We also show that if the equilibrium is nondegenerate and stable, then the minima, maxima, and saddle points of h correspond respectively to relative equ ilibria which are (orbitally) Liapounov stable, linearly stable, and l inearly unstable. The stabilities of the bifurcating branches of relat ive equilibria are computed explicitly for XY2, X-3, and XY4 molecules . These existence and stability results are corollaries of more genera l theorems on relative equilibria of G-invariant Hamiltonian systems t hat bifurcate from equilibria with finite isotropy subgroups as the mo mentum is varied. In the general case, the function h is defined on th e Lie algebra dual g and the bifurcating relative equilibria correspo nd to critical points of the restrictions of h to the coadjoint orbits in g.