On stability of resonance rotation of a dynamically symmetric satellite in a plane of elliptical orbit

Theoretical mechanics


Bardin B. S.*, Chekina E. A.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: bardin@yandex.ru
**e-mail: chekina_ev@mail.ru


We deal with the stability problem of a resonant rotation of a symmetric rigid body about its center of mass in an elliptical orbit. The motion of the body can be described by canonical system of differential equations with Hamiltonian

where , and are the satellite's equatorial and polar moments of inertia respectively; is the eccentricity of the orbit; ν is the true anomaly.

If the parameters α, satisfy the relation then the canonical equations with Hamiltonian has particular solution [2].

The resonant rotation is a planar motion such that the body completes three rotations in absolute space during two orbital revolutions of its center of mass. In this paper we study the stability of the resonant rotation in an extended formulation taking into account both planar and spatial perturbations.

Let us introduce perturbations by means of the formulas

The equations of perturbed motion have canonical form, that is the variables satisfy the equations

By analyzing linearized equations of perturbed motion there was found eccentricity interval , where the resonant rotation is unstable. Outside this interval a nonlinear stability study has been performed and subintervals of formal stability and stability for the most of initial data have been found. The conclusions on stability were obtained by an analysis of coefficients of the Hamiltonian normal form normalized up to terms of fourth order. A nonlinear normalization of the Hamiltonian was performed by using an approach based on construction of symplectic map generated by the Hamiltonian system.

Table 1. Results of the stability study in nonresonant case


Conclusion on stability

stability for the most of initial data

formal stability

stability for the most of initial data


Hamiltonian system, symplectic map, normal form, resonance, satellite, stability


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