Hamiltonian dynamics of a rigid body in a central gravitational field

This paper explores the dynamic properties of the planar system of an ellipsoidal satellite in an equatorial orbit about an oblate primary. In particular, we investigate the conditions for which the satellite is bound in librational motion or when the satellite will circulate with respect to the primary. We find the existence of stable equilibrium points about which the satellite can librate, and explore both the linearized and non-linear dynamics around these points. Absolute bounds are placed on the phase space of the libration-orbit coupling through the use of zero-velocity curves that exist in the system. These zerovelocity curves are used to derive a sufficient condition for when the satellite's libration is bound to less than 90 • . When this condition is not satisfied so that circulation of the satellite is possible, the initial conditions at zero libration angle are determined which lead to circulation of the satellite. Exact analytical conditions for circulation and the maximum libration angle are derived for the case of a small satellite in orbits of any eccentricity.

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“…(21). The plus terms correspond to the case where φ 2 = 0 or π, and the minus terms correspond the the other cases where φ 2 = π/2 or 3π/2.…”

Section: Appendixmentioning

confidence: 99%

Dust motions in quasi-statically charged binary asteroid systems

Maruskin

Bellerose

Wong

et al. 2013

Celest Mech Dyn Astr

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“…The coupled dynamics of a rigid body in a central gravity field has been investigated in several earlier works [17][18][19]. Relative equilibria and stability of coupled orbitattitude dynamics have been studied in a J 2 gravity field [20,21] and in a second degree and order gravity field as well [22,23].…”

Section: Introductionmentioning

confidence: 99%

Stabilization of coupled orbit–attitude dynamics about an asteroid utilizing Hamiltonian structure

2017

Astrodyn

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The gravitationally coupled orbit-attitude dynamics, also called the full dynamics, in which the spacecraft is modeled as a rigid body, is a high-precision model for the motion in the close proximity of an asteroid. A feedback control law is proposed to stabilize relative equilibria of the coupled orbit-attitude motion in a uniformly rotating second degree and order gravity field by utilizing the Hamiltonian structure. The feedback control law is consisted of potential shaping and energy dissipation. The potential shaping makes the relative equilibrium a minimum of the modified Hamiltonian by modifying the potential artificially. With the energy-Casimir method, it is theoretically proved that an unstable relative equilibrium can always be stabilized in the Lyapunov sense by the potential shaping with sufficiently large feedback gains. Then, the energy dissipation leads the motion to converge to the relative equilibrium. The proposed stabilization control law has a simple form and is easy to implement autonomously, which can be attributed to the utilization of natural dynamical behaviors in the controller design.

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“…Notable studies in this regard include Wang, et al [9], concerning the problem of a rigid body in a central Newtonian field; El-Gohary [10], regarding the problem of a gyrostat in a Newtonian force field; and Maciejewski [11], examining the problem of two rigid bodies in mutual Newtonian attraction. These papers have been generalized to the case of a gyrostat by Mondéjar, et al [12] to the case of two gyrostats in mutual Newtonian attraction.…”

Section: Introductionmentioning

confidence: 99%

“…In Vera [17] and a series of recent papers of Vera and Vigueras [9,10], we studied the noncanonical Hamiltonian dynamics of + 1 bodies in Newtonian attraction, where of them are rigid bodies with spherical distribution of mass or material points while the other one is a triaxial gyrostat. Using the symmetries of the system, we made two reductions, at each step giving the Poisson structure of the reduced space.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian relative equilibria for a gyrostat in the three-body problem

López

2009

Open Physics

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Abstract:In this paper we consider the noncanonical Hamiltonian dynamics of a gyrostat in the three-body problem. By means of geometric mechanics methods, we study the approximate Poisson dynamics that arise when we develop the potential of the system in Legendre series and truncate this to an arbitrary order . After reduction of the dynamics by means of the two symmetries of the system, we consider the existence and number of equilibria which we denominate of Lagrangian type, in analogy with classic results on the topic. Necessary and sufficient conditions are established for their existence in an approximate dynamics of order , and explicit expressions for these equilibria are given, this being useful for the subsequent study of their stability. The number of Lagrangian equilibria is thoroughly studied in approximate dynamics of orders zero and one. The main result of this work indicates that the number of Lagrangian equilibria in an approximate dynamics of order for ≥ 1 is independent of the order of truncation of the potential, if the gyrostat S 0 is almost spherical. In relation to the stability of these equilibria, necessary and sufficient conditions are given for linear stability of Lagrangian equilibria when the gyrostat is almost spherical. In this way, we generalize the classical results on equilibria of the three-body problem and many results provided by other authors using more classical techniques for the case of rigid bodies. 45.20.dc, 45.20.Jj, 45.50.Pk, 47.10.Df PACS (2008): Keywords:hamiltonian system • three-body problem • gyrostat • lagrangian equilibria • stability © Versita Warsaw and Springer-Verlag Berlin Heidelberg.

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Hamiltonian dynamics of a rigid body in a central gravitational field

Cited by 86 publications

References 28 publications

Dust motions in quasi-statically charged binary asteroid systems

Dust motions in quasi-statically charged binary asteroid systems

Stabilization of coupled orbit–attitude dynamics about an asteroid utilizing Hamiltonian structure

Lagrangian relative equilibria for a gyrostat in the three-body problem

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