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1.
Li-Sheng Wang P. S. Krishnaprasad J. H. Maddocks 《Celestial Mechanics and Dynamical Astronomy》1990,50(4):349-386
This paper concerns the dynamics of a rigid body of finite extent moving under the influence of a central gravitational field.
A principal motivation behind this paper is to reveal the hamiltonian structure of the n-body problem for masses of finite
extent and to understand the approximation inherent to modeling the system as the motion of point masses. To this end, explicit
account is taken of effects arising because of the finite extent of the moving body. In the spirit of Arnold and Smale, exact
models of spin-orbit coupling are formulated, with particular attention given to the underlying Lie group framework. Hamiltonian
structures associated with such models are carefully constructed and shown to benon-canonical. Special motions, namely relative equilibria, are investigated in detail and the notion of anon-great circle relative equilibrium is introduced. Non-great circle motions cannot arise in the point mass model. In our analysis,
a variational characterization of relative equilibria is found to be very useful.
Thereduced hamiltonian formulation introduced in this paper suggests a systematic approach to approximation of the underlying dynamics
based on series expansion of the reduced hamiltonian. The latter part of the paper is concerned with rigorous derivations
of nonlinear stability results for certain families of relative equilibria. Here Arnold's energy-Casimir method and Lagrange
multiplier methods prove useful.
This work was supported in part by the AFOSR University Research Initiative Program under grant AFOSR-87-0073, by AFOSR grant
89-0376, and by the National Science Foundation's Engineering Research Centers Program: NSFD CDR 8803012. The work of P.S.
Krishnaprasad was also supported by the Army Research Office through the Mathematical Sciences Institute of Cornell University. 相似文献
2.
The motion of a point mass in the J 2 problem is generalized to that of a rigid body in a J 2 gravity field. The linear and nonlinear stability of the classical type of relative equilibria of the rigid body, which have been obtained in our previous paper, are studied in the framework of geometric mechanics with the second-order gravitational potential. Non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are obtained through a Poisson reduction process by means of the symmetry of the problem. The linear system matrix at the relative equilibria is given through the multiplication of the Poisson tensor and Hessian matrix of the variational Lagrangian. Based on the characteristic equation of the linear system matrix, the conditions of linear stability of the relative equilibria are obtained. The conditions of nonlinear stability of the relative equilibria are derived with the energy-Casimir method through the projected Hessian matrix of the variational Lagrangian. With the stability conditions obtained, both the linear and nonlinear stability of the relative equilibria are investigated in details in a wide range of the parameters of the gravity field and the rigid body. We find that both the zonal harmonic J 2 and the characteristic dimension of the rigid body have significant effects on the linear and nonlinear stability. Similar to the classical attitude stability in a central gravity field, the linear stability region is also consisted of two regions that are analogues of the Lagrange region and the DeBra-Delp region respectively. The nonlinear stability region is the subset of the linear stability region in the first quadrant that is the analogue of the Lagrange region. Our results are very useful for the studies on the motion of natural satellites in our solar system. 相似文献
3.
Equilibrium Points and Central Configurations for the Lennard-Jones 2- and 3-Body Problems 总被引：1，自引：1，他引：0
Montserrat Corbera Jaume Llibre Ernesto Pérez-Chavela 《Celestial Mechanics and Dynamical Astronomy》2004,89(3):235-266
In this paper we study the relative equilibria and their stability for a system of three point particles moving under the
action of a Lennard-Jones potential. A central configuration is a special position of the particles where the position and
acceleration vectors of each particle are proportional, and the constant of proportionality is the same for all particles.
Since the Lennard-Jones potential depends only on the mutual distances among the particles, it is invariant under rotations.
In a rotating frame the orbits coming from central configurations become equilibrium points, the relative equilibria. Due
to the form of the potential, the relative equilibria depend on the size of the system, that is, depend strongly of the momentum
of inertia I. In this work we characterize the relative equilibria, we find the bifurcation values of I for which the number of relative equilibria is changing, we also analyze the stability of the relative equilibria.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
4.
Alexander A. Burov Anna D. Guerman Vasily I. Nikonov 《Celestial Mechanics and Dynamical Astronomy》2018,130(9):58
The existence, stability and bifurcation analysis is performed for equilibria of a material point in the gravitational field of three homogeneous penetrable balls fixed in absolute frame. The radii of the balls are assumed finite. In the case when the mass distribution admits a symmetry axis, analytic expressions are written out, allowing one to investigate the properties of equilibrium positions located both on the symmetry axis and outside it. The stability of solutions is studied; domains with different instability degree are described. 相似文献
5.
In this paper, a model is developed for the dynamics of a system of two bodies whose material points are under the influence of a central gravitational force. One of the bodies is assumed to be rigid and spherically symmetric, while the other is assumed to be deformable. To develop a tractable model for the system, the deformable body is modeled using Cohen and Muncaster's theory of a pseudo-rigid body. The resulting model of the system has several of the features, such as angular momentum conservation, exhibited by more restrictive models. We also show how the self-gravitation of the deformable body can be accommodated using appropriate constitutive equations for a force tensor. This enables our model to subsume many existing models of ellipsoidal figures of equilibrium. After the model and its conservations have been discussed, attention is restricted to steady motions of the system. Several results, which generalize recent works on rigid satellites, are established for these motions. For a specific choice of constitutive equations for the pseudo-rigid body, we determine the steady motions with the aid of a numerical continuation method. These results can also be considered as generalizations of earlier works on Roche's ellipsoids of equilibrium. 相似文献
6.
We consider the solid-solid interactions in the two body problem. The relative equilibria have been previously studied analytically and general motions were numerically analyzed using some expansion of the gravitational potential up to the second order, but only when there are no direct interactions between the orientation of the bodies. Here we expand the potential up to the fourth order and we show that the secular problem obtained after averaging over fast angles, as for the precession model of Boué and Laskar [Boué, G., Laskar, J., 2006. Icarus 185, 312-330], is integrable, but not trivially. We describe the general features of the motions and we provide explicit analytical approximations for the solutions. We demonstrate that the general solution of the secular system can be decomposed as a uniform precession around the total angular momentum and a periodic symmetric orbit in the precessing frame. More generally, we show that for a general n-body system of rigid bodies in gravitational interaction, the regular quasiperiodic solutions can be decomposed into a uniform precession around the total angular momentum, and a quasiperiodic motion with one frequency less in the precessing frame. 相似文献
7.
The problem of three bodies when one of them is a gyrostat is considered. Using the symmetries of the system we carry out two reductions. Global considerations about the conditions for relative equilibria are made. Finally, we restrict to an approximated model of the dynamics and a complete study of the relative equilibria is made. 相似文献
8.
We consider the non-canonical Hamiltonian dynamics of a gyrostat in Newtonian interaction with n spherical rigid bodies. Using the symmetries of the system we carry out two reductions. Then, working in the reduced problem,
we obtain the equations of motion, a Casimir function of the system and the equations that determine the relative equilibria.
Global conditions for existence of relative equilibria are given. Besides, we give the variational characterization of these
equilibria and three invariant manifolds of the problem; being calculated the equations of motion in these manifolds, which
are described by means of a canonical Hamiltonian system. We give some Eulerian and Lagrangian equilibria for the four body
problem with a gyrostat. Finally, certain classical problems of Celestial Mechanics are generalized. 相似文献
9.
A. P. Markeev 《Astronomy Letters》2005,31(9):627-633
We investigate the stability of the periodic motion of a satellite, a rigid body, relative to the center of mass in a central Newtonian gravitational field in an elliptical orbit. The orbital eccentricity is assumed to be low. In a circular orbit, this periodic motion transforms into the well-known motion called hyperboloidal precession (the symmetry axis of the satellite occupies a fixed position in the plane perpendicular to the radius vector of the center of mass relative to the attractive center and describes a hyperboloidal surface in absolute space, with the satellite rotating around the symmetry axis at a constant angular velocity). We consider the case where the parameters of the problem are close to their values at which a multiple parametric resonance takes place (the frequencies of the small oscillations of the satellite’s symmetry axis are related by several second-order resonance relations). We have found the instability and stability regions in the first (linear) approximation at low eccentricities. 相似文献
10.
The motion of a point mass in the J 2 problem has been generalized to that of a rigid body in a J 2 gravity field for new high-precision applications in the celestial mechanics and astrodynamics. Unlike the original J 2 problem, the gravitational orbit-rotation coupling of the rigid body is considered in the generalized problem. The existence and properties of both the classical and non-classical relative equilibria of the rigid body are investigated in more details in the present paper based on our previous results. We nondimensionalize the system by the characteristic time and length to make the study more general. Through the study, it is found that the classical relative equilibria can always exist in the real physical situation. Numerical results suggest that the non-classical relative equilibria only can exist in the case of a negative J 2, i.e., the central body is elongated; they cannot exist in the case of a positive J 2 when the central body is oblate. In the case of a negative J 2, the effect of the orbit-rotation coupling of the rigid body on the existence of the non-classical relative equilibria can be positive or negative, which depends on the values of J 2 and the angular velocity Ω e . The bifurcation from the classical relative equilibria, at which the non-classical relative equilibria appear, has been shown with different parameters of the system. Our results here have given more details of the relative equilibria than our previous paper, in which the existence conditions of the relative equilibria are derived and primarily studied. Our results have also extended the previous results on the relative equilibria of a rigid body in a central gravity field by taking into account the oblateness of the central body. 相似文献
11.
The motion of two mutually-attracting gyrostats in the Newtonian field of forces of a major body is considered. The equilibrium positions of the translational motions are found out and grouped, relative to the attractive body, in collinear and equilateral equilibria. The stationary orientations at these points are also detected and the stability of all equilibrium-configurations is investigated by means of the Routh-Hurwitz criterion 相似文献
12.
The paper deals with a simple photo-gravitational model of N+1 bodies. The motion of a small particle which subjects both
the gravitational attraction and the radiation pressure is studied in a regular polygon configuration of N bodies. The dynamical
features of this model are investigated for a wide range of values of the radiation parameters by mapping its equilibrium
points and periodic orbits. The results show that for these values, radiation merely affects quantitatively the characteristics
of the system, while it leaves unaffected the stability of the particle periodic motions and equilibria.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
13.
Esther Barrabés Josep M. Cors Claudio Vidal 《Celestial Mechanics and Dynamical Astronomy》2017,129(1-2):153-176
We outline some aspects of the dynamics of an infinitesimal mass under the Newtonian attraction of three point masses in a symmetric collinear relative equilibria configuration when a repulsive Manev potential (\(-1/r +e/r^{2}\)), \(e>0\), is applied to the central mass. We investigate the relative equilibria of the infinitesimal mass and their linear stability as a function of the mass parameter \(\beta \), the ratio of mass of the central body to the mass of one of two remaining bodies, and e. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass. 相似文献
14.
Jonathan Braithwaite 《Monthly notices of the Royal Astronomical Society》2009,397(2):763-774
In this third paper in a series on stable magnetic equilibria in stars, I look at the stability of axisymmetric field configurations and, in particular, the relative strengths of the toroidal and poloidal components. Both toroidal and poloidal fields are unstable on their own, and stability is achieved by adding the two together in some ratio. I use Tayler's stability conditions for toroidal fields and other analytic tools to predict the range of stable ratios and then check these predictions by running numerical simulations. If the energy in the poloidal component as a fraction of the total magnetic energy is written as Ep / E , it is found that the stability condition is a ( E / U ) < Ep / E ≲ 0.8 where E /U is the ratio of magnetic to gravitational energy in the star and a is some dimensionless factor whose value is of order 10 in a main-sequence star and of order 103 in a neutron star. In other words, whilst the poloidal component cannot be significantly stronger than the toroidal, the toroidal field can be very much stronger than the poloidal–given that in realistic stars we expect E / U < 10−6 . The implications of this result are discussed in various contexts such as the emission of gravitational waves by neutron stars, free precession and a 'hidden' energy source for magnetars. 相似文献
15.
Mohammed Adel Sharaf Mervat El-Sayed Awad Samiha Al-Sayed Abdullah Najmuldeen 《Earth, Moon, and Planets》1992,56(2):141-164
In this paper, the classical and generalized Sundman time transformations are used to establish new generating set of differential equations of motion in terms of the Eulerian redundant parameters. The implementation of this set on digital computers for the commonly used independent variables is developed once and for all. Motion prediction algorithms based on these equations are developed in a recursive manner for the motions in the Earth's gravitational field with axial symmetry whatever the number of the zonal harmonic terms may be. Applications for the two types of short and long term predictions are considered for the perturbed motion in the Earth's gravitational field with axial symmetry with zonal harmonic terms up to J
36
. Numerical results proved the very high efficiency and flexibility of the developed equations. 相似文献
16.
Energy and stability in the Full Two Body Problem 总被引：1，自引：0，他引：1
The conditions for relative equilibria and their stability in the Full Two Body Problem are derived for an ellipsoid–sphere
system. Under constant angular momentum it is found that at most two solutions exist for the long-axis solutions with the
closer solution being unstable while the other one is stable. As the non-equilibrium problem is more common in nature, we
look at periodic orbits in the F2BP close to the relative equilibrium conditions. Families of periodic orbits can be computed
where the minimum energy state of one family is the relative equilibrium state. We give results on the relative equilibria,
periodic orbits and dynamics that may allow transition from the unstable configuration to a stable one via energy dissipation.
相似文献
17.
In the present work we consider asymmetric gyrostat which has a homogeneous viscoelastic disc and two bars attached to it. Furthermore, the gyrostat has a rotor oriented inside it such that the rotor is statically and dynamically balanced. This sytem has a rotational motion around its center of mass in a circular orbit under a central gravitational field. Bending vibrations of the bars and the disc are accompanied by dissipation of energy, which is the cause of the evolution of the system's rotational motion. Using the method of separation of motion and averaging, the approximate equations describing the evolution of rotational motion in terms of Andoyer canonical variables are obtained. The stationary motions for the system are deduced, together with the conditions of its stability. 相似文献
18.
V. A. Sarychev S. A. Mirer A. A. Degtyarev 《Celestial Mechanics and Dynamical Astronomy》2008,100(4):301-318
Attitude motion of a satellite subjected to gravitational and aerodynamic torques in a circular orbit is considered. In special
case, when the center of pressure of aerodynamic forces is located in one of the principal central planes of inertia of the
satellite, all equilibrium orientations are determined. Existence conditions of all equilibria are obtained and evolution
of domains with a fixed number of equilibria is investigated in detail. All bifurcation values of the system’s parameters
corresponding to the qualitative change of these domains are determined. Sufficient conditions of stability are obtained for
each equilibrium orientation using generalized integral of energy. 相似文献
19.
Roland Sergysels 《Celestial Mechanics and Dynamical Astronomy》1988,44(1-2):155-166
As is well known, the orbital and rotational motions of a solid are coupled, and the integrals of energy and angular momentum (in a gravitational field with spherical symmetry) impose restrictions on them. We study the regions allowed to the motion in configurational space. It turns out that even in the crudest model (planar motion of a triple rod) the restrictions on the libration angle and the orbital radius of the center of mass are coupled, so that excessive ellipticity of the orbit excludes stabilization in the neighbourhood of the spoke equilibrium position by gravitational forces only.Chargé de Cours. 相似文献
20.
Mervat El-Sayed Awad 《Earth, Moon, and Planets》1988,43(1):7-20
In this paper of the series, a special perturbation technique of Encke-type associated with the KS regularized variables will be developed for satellite motions in the Earth's gravitational field with axial symmetry. Its computational algorithm is of recursive nature and could be applied for any perturbed conic motion whatever the number of the zonal harmonic coefficients may be. Applications of the algorithm are also included.Now at the Department of Mathematics, Girls College of Education, Jeddah, Saudi Arabia. 相似文献