The Euler-Poincaré equations and double bracket dissipation.(English)Zbl 0846.58048
Summary: This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincaré) system by a special dissipation term that has Brockett’s double bracket form. We show that a formally unstable equilibrium of the unperturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of this dissipation mechanism and its relation to Rayleigh dissipation functions. This work complements our earlier work [the authors, ‘Asymptotic stability, mistability, and stabilization of relative equilibria’, Proc. ACC, Boston IEEE, 1120-1125 (1991) and Ann. Inst. Henri Poincaré, Anal. Nonlineaire 11, No. 1, 37-90 (1994;Zbl 0834.58025)] in which we studied the corresponding problem for systems with symmetry with the dissipation added to the internal variables; here it is added directly to the group or Lie algebra variables. The mechanisms discussed here include a number of interesting examples of physical interest such as the Landau-Lifschitz equations for ferromagnetism, certain models for dissipative rigid body dynamics and geophysical fluids, and certain relative equilibria in plasma physics and stellar dynamics.
MSC:
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37N99 | Applications of dynamical systems |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
Citations:
Zbl 0834.58025Cite
References:
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