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Zbl 0834.58025
Bloch, A.M.; Krishnaprasad, P.S.; Marsden, J.E.; Ratiu, T.S.
Dissipation induced instabilities. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 11, No.1, 37-90 (1994). ISSN 0294-1449

Summary: 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.\par 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.

MSC 2000:: *37C75 Stability theory
70J25 Stability of linear oscillatory motions

Keywords: mechanics; stability; symmetry

Cited in: Zbl 0846.58048

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