5 papers
Stability by KAM confinement of certain wild, nongeneric relative equilibria of underwater vehicles with coincident centers of mass and bouyancy
George W. Patrick
Purely rotational relative equilibria of an ellipsoidal underwater vehicle occur at nongeneric momentum where the symplectic reduced spaces change dimension. The stability these re…
Stability of Poisson Equilibria and Hamiltonian Relative Equilibria by Energy Methods
George W. Patrick, Mark Roberts, Claudia Wulff
We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generali…
Reduction of the planar 4-vortex system at zero momentum
G. W. Patrick
The system of four point vortices in the plane has relative equilibria that behave as composite particles, in the case where three of the vortices have strength and one of t…
The Landau-Lifshitz Equation by Semidirect Product Reduction
G. W. Patrick
The Landau-Lifshitz equation is derived as the reduction of a geodesic flow on the group of maps into the rotation group. Passing the symmetries of spatial isotropy to the reduced…
Dynamics near Relative Equilibria: Nongeneric Momenta at a 1:1 Group-Reduced Resonance
G. W. Patrick
An interesting situation occurs when the linearized dynamics of the shape of a formally stable Hamiltonian relative equilibrium at nongeneric momentum 1:1 resonates with a frequenc…