ISSN:
1432-1467
Keywords:
Key words. relative equilibria, molecules, symmetry, symplectic reduction, bifurcation, stability AMS Classification. 58F05, 58F14, 70H33, 70K20
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Summary. We describe a method for finding the families of relative equilibria of molecules that 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 two-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 nondegenerate 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 XY 2 , X 3 , and XY 4 molecules. These existence and stability results are corollaries of more general theorems on relative equilibria of G -invariant Hamiltonian systems that bifurcate from equilibria with finite isotropy subgroups as the momentum is varied. In the general case, the function h is defined on the Lie algebra dual {\frak g} * and the bifurcating relative equilibria correspond to critical points of the restrictions of h to the coadjoint orbits in {\frak g} * .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s003329900064
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