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Relative equilibria of molecules

Montaldi, JA and Roberts, RM (1999) Relative equilibria of molecules JOURNAL OF NONLINEAR SCIENCE, 9 (1). 53 - 88. ISSN 0938-8974

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Abstract

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 XY2 , X3, and XY4 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}* .

Item Type: Article
Uncontrolled Keywords: Science & Technology, Physical Sciences, Technology, Mathematics, Applied, Mechanics, Physics, Mathematical, Mathematics, Physics, relative equilibria, molecules, symmetry, symplectic reduction, bifurcation, stability, POLYATOMIC-MOLECULES, HAMILTONIAN-SYSTEMS, SURFACES, ROTATION
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Divisions: Faculty of Engineering and Physical Sciences > Mathematics
Depositing User: Mr Adam Field
Date Deposited: 27 May 2010 14:41
Last Modified: 23 Sep 2013 18:33
URI: http://epubs.surrey.ac.uk/id/eprint/1462

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