Stability of equilibrium states in the Zhukovski case of heavy gyrostat using algebraic methods

Abstract

Communicated by M. BrokateWe study the stability of the equilibrium points of a skew product system and analyze the possibility to construct a Lyapunov function by using a set of conserved quantities and solving an algebraic system. We apply the theoretical results to study the stability of an equilibrium state of a heavy gyrostat in the Zhukovski case.

A note on stability of spacecrafts and underwater vehicles

“…Proof We use the algebraic method, see Comanescu (Theorem ()), which is presented in the Hamilton‐Poisson context in Ortega et al (Corollary 4.3) and used in the papers . If ( Π e , 0 ) is a strict local extremum of , then the algebraic system has no root besides z e in some neighborhood of z e .…”

“…We use the algebraic method, see Comanescu 3 (Theorem (2.3)), which is presented in the Hamilton-Poisson context in Ortega et al 11 (Corollary 4.3) and used in the papers. 12,1314 If ( e , 0) is a strict local extremum of H z e , then the algebraic system…”

Hessian Operators on Constraint Manifolds

“…Usually the above theorem is applied backwards, where the vector field X S is the restriction of a vector field X ∈ X(M ) to an invariant submanifold S under the dynamics generated by the vector field X. In the case when F 1 , ..., F k , G are conserved quantities for the vector field X and the conditions (ii) and (iii) of the above theorem are satisfied, then the equilibrium point x e is also stable for the dynamics generated by the vector field X according to the algebraic method, see [6], [7], [8].…”

A note on stability of the vertical uniform rotations of the heavy top

“…Theorem 2.2 (iii) offer an algebraic method to prove the Lyapunov stability of an equilibrium point. This method have been used in [3] and [4] for studying the stability problem of the uniform rotations of a torque-free gyrostat and also for studying the stability problem of the equilibrium states of a heavy gyrostat (Zhukovskii case).…”