Accession Number:

ADA454960

Title:

Dissipation Induced Instabilities

Descriptive Note:

Research rept.

Corporate Author:

MARYLAND UNIV COLLEGE PARK SYSTEMS RESEARCH CENTER

Report Date:

1993-03-09

Pagination or Media Count:

55.0

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 Chetaevs 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, Machay 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.

Subject Categories:

  • Mechanics
  • Numerical Mathematics
  • Theoretical Mathematics

Distribution Statement:

APPROVED FOR PUBLIC RELEASE