Dissipation Induced Instabilities

Abstract

The main goal of this paper is to prove that if the energy- momentum (or energy-Casimir) method predicts formal instability of a relative equilibrium 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 symmetry group. Our result assumes that the dissipation chosen does not destroy the conservation law associated with the given symmetry group -- thus, we consider internal dissipation. Our result also includes the special case of systems with no symmetry and ordinary equilibria. Our result is proved by combining the techniques of Chetaev, who proved instability theorems using a special Chetaev- Lyapunov function, those of Hahn, which enable one to strengthen the Chetaev results from Lyapunov instability to spectral instability. Our main achievement is to strengthen these results to the context of the block diagonalization version of the energy momentum method given by Lewis. Marsden, Posbergh, and Simo. However, we also give the eigenvalue movement formulae of Krein, MacKay and others both in general and adapted to the context of the normal form of the linearized equations given by the block diagoanl 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.