Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation

Abstract

This paper studies the destabilizing effects of dissipation on familiesof relative equilibria in Hamiltonian systems which are non-extremalconstraint critical points in the energy-Casimir or theenergy-momentum methods. The dissipation is allowed to destroy theconservation law associated with the symmetry group or Casimirs, as longas the family of relative equilibria stays on an invariant manifold. Thisapproach complements earlier work in the literature, in which thedissipation did not affect the conservation law. Firstly, Chetaev's instability theorem is extended to invariantmanifolds and this extended theorem is used to prove the instability offamilies of relative equilibria for several examples. Secondly, it is shownthat families of non-extremal stationary solutions of the two-dimensionalincompressible homogeneous Euler equations are unstable for thecorresponding viscous perturbations of this system, i.e. for thetwo-dimensional Navier-Stokes equations. Also, the instability ofthe sleeping top relative equilibria under friction can easily be provedin this way, even before the Hamiltonian sleeping top becomes linearlyunstable. Finally, sufficient conditions are given for which friction destabilizes families of non-extremal relative equilibria in simple mechanical systems with Abelian symmetry.