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

Abstract

We prove that the stability problem of a vertical uniform rotation of a heavy top is completely solved by using the linearization method and the conserved quantities of the differential system which describe the rotation of the heavy top

The stability problem and special solutions for the 5-components Maxwell–Bloch equations

“…Indeed, any linear combination αDX H| O 0 (0, 0, 0, 0, 0) + βDX I| O 0 (0, 0, 0, 0, 0) has the characteristic polynomial (t 2 + β 2 ) 2 , which has non-distinct eigenvalues. Its stability property can be established using an algebraic method (see [1], [5], [6], [7]). More precisely, the system of algebraic equations H(x 1 , y 1 , x 2 , y 2 , z) = H(0, 0, 0, 0, 0), I(x 1 , y 1 , x 2 , y 2 , z) = I(0, 0, 0, 0, 0), C(x 1 , y 1 , x 2 , y 2 , z) = C(0, 0, 0, 0, 0) has as unique solution the equilibrium (0, 0, 0, 0, 0), leading to the following stability result.…”

Energy Methods in the Stability Problem for the 𝔰𝔬(4) Free Rigid Body

“…We prove that Lyapunov functions are positive or negative definite studying the non-degeneracy of the Hessian matrix associated to the Lyapunov function at the equilibrium point. If the Hessian is degenerate at an equilibrium point, one can still study the Lyapunov stability using algebraic methods, see [6], [7], [8]. 2 The geometry and stability problem for the so(4) free rigid body…”

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 .…”