DISSIPATION INDUCED INSTABILITIES

The main goal of this paper is to prove that if the energy-momentum (o r energy-Casimir) method predicts formal instability of a relative equ ilibrium in a Hamiltonian system with symmetry, then with the addition of dissipation, the relative equilibrium becomes spectrally and hence linearly and nonlinearly unstable. The energy-momentum method assumes that one is in the context of a mechanical system with a given symmet ry group. Our result assumes that the dissipation chosen does not dest roy the conservation law associated with the given symmetry group - th us, we consider internal dissipation. This also includes the special c ase of systems with no symmetry and ordinary equilibria. The theorem i s proved by combining the techniques of Chetaev, who proved instabilit y theorems using a special Chetaev-Lyapunov function, with those of Ha hn, which enable one to strengthen the Chetaev results from Lyapunov i nstability to spectral instability. The main achievement is to strengt hen Chetaev's methods to the context of the block diagonalization vers ion of the energy momentum method given by Lewis, Marsden, Posbergh, a nd Simo. However, we also give the eigenvalue movement formulae of Kre in, MacKay and others both in general and adapted to the context of th e normal form of the linearized equations given by the block diagonal form, as provided by the energy-momentum method. A number of specific examples, such as the rigid body with internal rotors, are provided to illustrate the results.