Some remarks on the Clebsch's system

Stability Problems and Analytical Integration for the Clebsch’s System

“…In this paper we analyze the nonlinear stability of the equilibrium states of Clebsch's system. Some results obtained in [2] -like the stability of the equilibrium states e M,N , e M,N , e M,N -are improved and some other new, like the stability of the equilibrium states e M,N,P which are developed. Then, using Moser-Weinstein theorem with zero eigenvalue, we were able to prove the existence of the periodic orbits around the nonlinear stable equilibria.…”

“…We shall discuss now the numerical integration of the equations (3) via the Lie-Trotter formula ( [7]). Following [2] or [7], the Lie-Trotter integrator can be written as:…”

“…where a , a , a are di erent and nonzero constants. It is well-known that its rst integrals are: During the time since its publication, a lot of problems about the Clebsch's system have been studied like its almost Lie-Poisson structure ( [2]), the Lax formulation ( [3]) or its Hirota-Kimura type discretization ( [4]).…”

“…The paper's structure is as follows: rst, the nonlinear stability of the equilibrium states of Clebsch's dynamics is discussed. About this problem, only partial results were found in [2] due to the fact that the existence of a Hamilton-Poisson structure is still an open problem, and for the almost Hamilton-Poisson structure proposed only one Casimir function was found instead of two. We use here the Arnold's method in order to obtain some new results, which does not require a Hamilton-Poisson structure.…”