On the stability of certain motions of a rigid body-gyrostat in an incompressible ideal fluid

Abstract

This work is devoted to investigating the stability of motion of a rigid body-gyrostat in an incompressible ideal fluid. This motion is assumed to happen under the action of neutral forces (buoyancy and gravity) whose centers are not coincide. The equations of motion are introduced, and they are expressed by means of the Hamiltonian function in the framework of the Lie-Poisson system. We prove that the problem of motion of a gyrostat in an incompressible ideal fluid is equivalent to the general problem of the motion of a rigid body under the action of a combination of potential and gyroscopic forces. We endeavor for studying the stability of two possible types of stationary solutions which describe the spin motion and non-spin motion of a gyrostat. For the non-spin motion, we consider two stationary solutions for which the translation motion is either in direction of the gravity or in the perpendicular direction on it. For the spin motion, we consider an stationary solution describing physically translation along and rotation about the same axis, say, the third one. The linear approximation method is applied to get the sufficient conditions for instability or in other words, the necessary conditions for stability. The energy-Casimir method is utilized to provide sufficient conditions for stability.