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The Lie-Poisson structure of the LAE- equation
François Gay-Balmaz, Tudor S. Ratiu
This paper shows that the time map of the averaged Euler equations, with Dirichlet, Neumann, and mixed boundary conditions is canonical relative to a Lie-Poisson bracket constr…
The dynamics around stable and unstable Hamiltonian relative equilibria
Juan-Pablo Ortega, Tudor S. Ratiu
For a symmetric Hamiltonian system, lower bounds for the number of relative equilibria surrounding stable and formally unstable relative equilibria on nearby energy levels are give…
Bifurcation of relative equilibria in mechanical systems with symmetry
Pascal Chossat, Debra Lewis, Juan-Pablo Ortega +1
The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system…
Reduction in principal fiber bundles: covariant Euler-Poincare equations
Marco Castrillon Lopez, Tudor S. Ratiu, Steve Shkoller
Let be a principal G-bundle, and let be a G-invariant Lagrangian density. We obtain the Euler-Poincare equations for the reduced Lagran…
Curvature of the Virasoro-Bott group
Peter W. Michor, Tudor Ratiu
We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the c…