Dissipation Induced Instabilities for Euler Equations and the Brockett Double Bracket Equations

Jerry Marsden, UC Berkeley

This talk will focus on the perturbation of conservative Euler-Poincare type equations by a special dissipation term that has Brockett's double bracket form. We show that a formally unstable equilibrium of the unperturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of this dissipation mechanism and its relation to Rayleigh dissipation functions. This work complements earlier work of Bloch, Krishnaprasad, Marsden and Ratiu in which the corresponding problem for systems with symmetry with the dissipation added to the internal variables is studied; here it is added directly to the Lie algebra variables. A number of examples, including the Landau-Lifschitz equations for ferromagnetism and the equations for pseudo rigid bodies with viscoelastic dissipation are discussed. We will also point out some relations with feedback stabilization for rigid bodies with internal rotors and for the Landau Lifschitz equations.