# zbMATH — the first resource for mathematics

Singularities of oscillations of a satellite on highly eccentric elliptic orbits. (English) Zbl 0898.34034
Singularities of an equation describing oscillations of a satellite with respect to its mass center are considered $(1+e\cos V) \;{{d^2\delta}\over{dV^2}}-2e\sin V\;{{d\delta}\over{dV}}+\mu \sin\delta =4e\sin V .$ Here $$\delta$$ is the doubled angle between the radius vector of the mass center of the satellite and one of its axis of inertia; $$\mu$$ is the inertial parameter of the satellite; $$e$$ is the eccentricity of the orbit; $$V$$ is the true anomaly of the position of the satellite on the orbit; $$e\leq 1$$, $$| \mu | \leq 3$$. It is regular if $$e<1$$. For $$e=1$$ and $$V=\pi$$, it is singular since the coefficient of the higher derivative vanishes. In this case the study of the motion as function of $$\mu$$ is reduced to a study of three limit equations, named basic, first and second. The dynamics of the system is completely studied.

##### MSC:
 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 70M20 Orbital mechanics
Full Text:
##### References:
  Bruno, A.D.; Bruno, A.D., First approximations of differential equations, Doklady akademii nauk, Russian acad. sci. doklady. mathem., 49, 2, 334-339, (1994), (Russian) · Zbl 0855.35027  Bruno, A.D.; Bruno, A.D., (), (Russian)  Bruno, A.D.; Petrovich, V.YU., Computation of periodic oscillations of a satellite. the singular case, (), (Russian) · Zbl 0847.70020  Bruno, A.D.; Varin, V.P.; Bruno, A.D.; Varin, V.P., First (second) limit problem for the equation of oscillations of a satellite, (), (Russian) · Zbl 0955.70019  Bruno, A.D., Algorithms of the local nonlinear analysis, () · Zbl 0676.34011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.