Dissipation Induced Instabilities
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 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.
© 1994 Gauthier-Villars. August, 1990; revised December 27, 1993, reprinted June 3, 1996. Research partially supported by the National Science Foundation grant DMS–90–02136,PYI grant DMS–91–57556,and AFOSR grant F49620–93–1–0037 Research partially supported by the AFOSR University Research Initiative Program under grants AFOSR-87-0073 and AFOSR-90-0105 and by the National Science Foundation's Engineering Research Centers Program NSFD CDR 8803012 Research partially supported by,DOE contract DE–FG03–92ER–25129,a Fairchild Fellowship at Caltech,and the Fields Institute for Research in the Mathematical Sciences Research partially supported by NSF Grant DMS 91–42613 and DOE contract DE–FG03–92ER–25129
Accepted Version - BlKrMaRa1994.pdf