The problem of the stability of the equilibrium position of a Hamiltonian system at 3:1 resonance

✍ Scribed by A.P. Markeyev

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 476 KB
• Volume: 65
• Category: Article
• ISSN: 0021-8928
• DOI: 10.1016/s0021-8928(01)00068-5

✦ Synopsis

The stability of the equilibrium position of an autonomous Hamiltonian system with two degrees of freedom is investigated. It is assumed that the equilibrium is stable in the linear approximation, the frequencies 0)1 and 0)2 of small oscillations are connected by the resonance relation 0)1 --30)2, and the Hamilton function is not sign-definite in the neighbourhood of the equilibrium position. The critical case when it is necessary to take into account terms higher than the fourth power in the expansion of the Hamilton function in series in order to obtain strictly valid conclusions on the stability of the equilibrium position is investigated. Sufficient conditions for stability and instability, which are expressed in terms of the coefficients of the expansion up to the sixth power inclusive, are obtained. The results are used in the problem of the stability of steady rotation of a dynamically symmetrical artificial satellite -of a rigid body around the normal to the plane of the circular orbit of its centre of mass.