Hamiltonian Systems near Relative Equilibria

Hamiltonian Systems Near Relative Periodic Orbits

We give explicit differential equations for a symmetric Hamiltonian vector field near a relative periodic orbit. These decompose the dynamics into periodically forced motion in a Poincaré section transversal to the relative periodic orbit, which in turn forces motion along the group orbit. The structure of the differential equations inherited from the symplectic structure and symmetry propertie...

Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry

For Hamiltonian systems with spherical symmetry there is a marked difference between zero and non-zero momentum values, and amongst all relative equilibria with zero momentum there is a marked difference between those of zero and those of non-zero angular velocity. We use techniques from singularity theory to study the family of relative equilibria that arise as a symmetric Hamiltonian which ha...

Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation∗

This paper studies the destabilizing effects of dissipation on families of relative equilibria in Hamiltonian systems which are non-extremal constraint critical points in the energy-Casimir or the energy-momentum methods. The dissipation is allowed to destroy the conservation law associated with the symmetry group or Casimirs, as long as the family of relative equilibria stays on an invariant m...

Relative equilibria and conserved quantities in symmetric Hamiltonian systems

Families of Relative Equilibria in Hamiltonian Systems with Dissipation

In this note the influence of dissipation on families of relative equilibria in Hamiltonian systems will be considered. Relative equilibria can be described as critical points of an appropriate functional. This characterisation can be used to give sufficient conditions such that in finite dimensional systems with dissipation the extremal families of relative equilibria are stable under dissipat...