Stability of Some Equilibrium Solutions of the Problem of Motion of a Gyrostat in an Incompressible Ideal Fluid

Article type: Research Article

Authors: Vera, J.A.^{a; 1} | Vigueras, A.^{b; 2}

Affiliations: [a] Departamento de Matemática Aplicada y Estadística Universidad Politécnica de Cartagena, Spain | [b] C/ Paseo Alfonso XIII, 52. 30203 Cartagena (Murcia), Spain

Note: [1] E-mail: [email protected].

Note: [2] E-mail: [email protected].

Abstract: Kirchhoff's equations for a gyrostat in a incompressible ideal fluid, have been written as a Lie-Poisson system, using a non-canonical Hamiltonian formulation. Next, we provide some equilibria of the problem, when the gyrostatic momentum is constant and adopts the form l = (0, 0, l). By means of the energy-Casimir method, we obtain sufficient conditions for the stability of some of these equilibria. Besides, by studying the linearized equations of motion, we offer the necessary conditions for the stability.

Keywords: Kirchhoff equations, Lie-Poisson systems, relative equilibria, stability, linearization, energy-Casimir

Keywords: 70H14, 70G45, 70E99

DOI: 10.3233/JCM-2003-3306

Journal: Journal of Computational Methods in Sciences and Engineering, vol. 3, no. 3, pp. 425-432, 2003