Dissipation Induced Instabilities
Bloch, A., P. S. Krishnaprasad, J. E. Marsden, T. S. Ratiu
Analyse Nonlineaire, Annuales Institute H. Poincaré, 11, 37-90
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. This
also includes the special case of systems with no symmetry and ordinary equilibria. The
theorem is proved by combining the techniques of Chetaev, who proved instability theorems
using a special Chetaev-Lyapunov function, with those of Hahn, which enable one to
strengthen the Chetaev results from Lyapunov instability to spectral instability. The
main achievement is to strengthen Chetaev's methods 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.