Abstract
Abstract The nonlinear stability and the existence of periodic orbits of the equilibrium states of the Clebsch’s system are discussed.. Numerical integration using the Lie-Trotter integrator and the analytic approximate solutions using Multistage Optimal Homotopy Asymptotic Method are presented, too.
Highlights
The Clebsch’s system was proposed in 1870 and it represents a speci c famous case of the Kircho equations which describes the motion of a rigid body in an ideal uid
The equilibrium states eM,N,P are nonlinearly stable for any M, N, P ∈ R
It is a constant of motion of the dynamics (3)
Summary
The Clebsch’s system was proposed in 1870 (see [1] for details) and it represents a speci c famous case of the Kircho equations which describes the motion of a rigid body in an ideal uid. The Clebsch’s case was obtained from the equations: x. − a )x x where a , a , a are di erent and nonzero constants It is well-known that its rst integrals are:. 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.
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