共查询到20条相似文献，搜索用时 46 毫秒
1.
Madeleine Pascal 《Celestial Mechanics and Dynamical Astronomy》1985,36(4):319-347
We consider the motion of a dual spin satellite placed in the gravitational field of n material points, assuming that the satellite has no influence on the motion of these points. The main bodies are located at the libration points of the classical n bodies problem. We investigate the set of relative equilibria of the satellite. As in the elementary case of a gyrostat attracted by a single point, we show that this problem is equivalent to find the extremum of a quadratic function. We obtain all possible equilibria of the satellite by solving two algebraic equations. Sufficient conditions of stability of these relative equilibria are given. 相似文献
2.
In this paper the problem of two, and thus, after a generalization, of an arbitrary finite number, of rigid bodies is considered. We show that the Newton-Euler equations of motion are Hamiltonian with respect to a certain non-canonical structure. The system possesses natural symmetries. Using them we shown how to perform reduction of the number of degrees of freedom. We prove that on every stage of this process equations of motion are Hamiltonian and we give explicite form corresponding of non-canonical Poisson bracket. We also discuss practical consequences of the reduction. We prove the existence of 36 non-Lagrangean relative equilibria for two generic rigid bodies. Finally, we demonstrate that our approach allows to simplify the general form of the mutual potential of two rigid bodies. 相似文献
3.
The problem of two gyrostats in a central force field is considered. We prove that the Newton-Euler equations of motion are
Hamiltonian with respect to a certain non-canonical structure. The system posseses symmetries. Using them we perform the reduction
of the number of degrees of freedom. We show that at every stage of the reduction process, equations of motion are Hamiltonian
and give explicit forms corresponding to non-canonical Poisson brackets. Finally, we study the case where one of the gyrostats
has null gyrostatic momentum and we study the zero and the second order approximation, showing that all equilibria are unstable
in the zero order approximation.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
4.
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
5.
6.
H. M. Yehia 《Celestial Mechanics and Dynamical Astronomy》1987,41(1-4):275-288
The general problem of motion of a rigid body about a fixed point under the action of stationary non-symmetric potential and gyroscopic forces is considered. The equations of motion in the Euler-Poisson form are derived. An interpretation is given in terms of charged, magnetized gyrostat moving in a superposition of three classical fields. As an example, the problem of motion of a satellite — gyrostat on a circular orbit with respect to its orbital system is reduced to that of its motion in an inertial system under additional magnetic and Lorentz forces.When the body is completely symmetric about one of its axes passing through the fixed point, the above problem is found to be equivalent to another one, in which the body has three equal moments of inertia and the forces are symmetric around a space axis. The last problem is well-studied and the given analogy reveals a number of integrable cases of the original problem. A transformation is found, which gives from each of these cases a class of integrable cases depending on an arbitrary function. The equations of motion are also reduced to a single equation of the second order. 相似文献
7.
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
8.
François Farago Jacques Laskar Jocelyn Couetdic 《Celestial Mechanics and Dynamical Astronomy》2009,104(3):291-306
Exploring the global dynamics of a planetary system involves computing integrations for an entire subset of its parameter
space. This becomes time-consuming in presence of a planet close to the central star, and in practice this planet will be
very often omitted. We derive for this problem an averaged Hamiltonian and the associated equations of motion that allow us
to include the average interaction of the fast planet. We demonstrate the application of these equations in the case of the μ Arae system where the ratio of the two fastest periods exceeds 30. In this case, the effect of the inner planet is limited
because the planet’s mass is one order of magnitude below the other planetary masses. When the inner planet is massive, considering
its averaged interaction with the rest of the system becomes even more crucial. 相似文献
9.
K. Uldall Kristiansen M. Vereshchagin K. Goździewski P. L. Palmer R. M. Roberts 《Celestial Mechanics and Dynamical Astronomy》2012,112(2):169-190
In this paper we consider the two-body problem of a spherical pseudo-rigid body and a rigid sphere. Due to the rotational
and “re-labelling” symmetries, the system is shown to possess conservation of angular momentum and circulation. We follow
a reduction procedure similar to that undertaken in the study of the two-body problem of a rigid body and a sphere so that
the computed reduced non-canonical Hamiltonian takes a similar form. We then consider relative equilibria and show that the
notions of locally central and planar equilibria coincide. Finally, we show that Riemann’s theorem on pseudo-rigid bodies
has an extension to this system for planar relative equilibria. 相似文献
10.
The problem of the attitude dynamics of a triaxial gyrostat under no external torques and one constant internal rotor, is a three degrees-of-freedom system, although thanks to the existence of integrals of motion it can be reduced to only one degree-of-freedom problem. We introduce coordinates to represent the orbits of constant angular momentum as a flow on a sphere. This representation shows that the problem is equivalent to a quadratic Hamiltonian depending on two parameters. We find the exact solution of the orbits in terms of elliptic functions. By making use of properties of elliptic functions we find the solution at each region of the parametric partition from the solution of one region. We also prove that heteroclinic orbits are planar curves. 相似文献
11.
We deal with the study of the spatial restricted three-body problem in the case where the small particle is far from the primaries, that is, the so-called comet case. We consider the circular problem, apply double averaging and compute the relative equilibria of the reduced system. It appears that, in the circular problem, we find not only part of the equilibria existing in the elliptic case, but also new ones. These critical points are in correspondence with periodic and quasiperiodic orbits and invariant tori of the non-averaged Hamiltonian. We explain carefully the transition between the circular and the elliptic problems. Moreover, from the relative equilibria of elliptic type, we obtain invariant 3-tori of the original system. 相似文献
12.
Vasile Mioc Ernesto Pérez-Chavela Magda Stavinschi 《Celestial Mechanics and Dynamical Astronomy》2003,86(1):81-106
The two-body problem associated to an anisotropic Schwarzschild-type field is being tackled. Both the motion equations and the energy integral are regularized via McGehee-type transformations. The regular vector field exhibits nice symmetries that form a commutative group endowed with an idempotent structure. The physically fictitious flows on the collision and infinity manifolds, as well as the local flows in the neighbourhood of these manifolds, are fully described. Homothetic, spiral, and oscillatory orbits are pointed out. Some features of the global flow are depicted for all possible levels of energy. For the negative-energy case, few things have been done. The positive-energy global flow does not have zero-velocity curves; every orbit is of the type ejection – escape or capture – collision. In the zero-energy case, the collision and infinity manifolds have a very similar structure. The existence of eight trajectories that connect the equilibria on these manifolds is proved. The projectability of the zero-energy global flow completes the full understanding of the problem in this case. 相似文献
13.
Aaron J. Rosengren Daniel J. Scheeres 《Celestial Mechanics and Dynamical Astronomy》2014,118(3):197-220
We consider sets of natural vectorial orbital elements of the Milankovitch type for perturbed Keplerian motion. These elements are closely related to the two vectorial first integrals of the unperturbed two-body problem; namely, the angular momentum vector and the Laplace–Runge–Lenz vector. After a detailed historical discussion of the origin and development of such elements, nonsingular equations for the time variations of these sets of elements under perturbations are established, both in Lagrangian and Gaussian form. After averaging, a compact, elegant, and symmetrical form of secular Milankovitch-like equations is obtained, which reminds of the structure of canonical systems of equations in Hamiltonian mechanics. As an application of this vectorial formulation, we analyze the motion of an object orbiting about a planet (idealized as a point mass moving in a heliocentric elliptical orbit) and subject to solar radiation pressure acceleration (obeying an inverse-square law). We show that the corresponding secular problem is integrable and we give an explicit closed-form solution. 相似文献
14.
H. M. Yehia 《Celestial Mechanics and Dynamical Astronomy》1987,41(1-4):289-295
The problem of motion of a dynamically symmetric gyrostat acted upon by non-symmetric potential forces admitting a cyclic integral is considered. This problem is brought into equivalence with another one concerning the motion of a similar gyrostat under the action of axisymmetric potential forces. Using this analogy, new integrable cases of each of the two problems are obtained from the known cases of the other. The equations of motion are reduced to a single equation of the second order. 相似文献
15.
J. A. Vera 《Astrophysics and Space Science》2009,323(4):375-382
The non-canonical Hamiltonian dynamics of a triaxial gyrostat in Newtonian interaction with two punctual masses is considered.
This serves as a model for the study of the attitude dynamics of a spacecraft located at a Lagrangian equilibrium point of
the system formed by a binary asteroid and a spacecraft. Using geometric-mechanics methods, the approximated dynamics that
arises when developing the potential in series of Legendre functions and truncating the series to the second harmonics is
studied. Working in the reduced problem, the existence of equilibria in Lagrangian form are studied, in analogy with classic
results on the topic. In this way, the classical results on equilibria of the three-body problem, as well as other results
by different authors that use more conventional techniques for the case of rigid bodies, are generalized. The rotational Poisson
dynamics of a spacecraft located at a Lagrangian equilibrium and the study of the nonlinear stability of some important equilibria
are considered. The analysis is done in vectorial form avoiding the use of canonical variables and the tedious expressions
associated with them. 相似文献
16.
J. F. Palacián 《Celestial Mechanics and Dynamical Astronomy》2007,98(4):219-249
We study the dynamics of a satellite (artificial or natural) orbiting an Earth-like planet at low altitude from an analytical
point of view. The perturbation considered takes into account the gravity attraction of the planet and in particular it is
caused by its inhomogeneous potential. We begin by truncating the equations of motion at second order, that is, incorporating
the zonal and the tesseral harmonics up to order two. The system is formulated as an autonomous Hamiltonian and has three
degrees of freedom. After three successive Lie transformations, the system is normalised with respect to two angular co-ordinates
up to order five in a suitable small parameter given by the quotient between the angular velocity of the planet and the mean
motion of the satellite. Our treatment is free of power expansions of the eccentricity and of truncated Fourier series in
the anomalies. Once these transformations are performed, the truncated Hamiltonian defines a system of one degree of freedom
which is rewritten as a function of two variables which generate a phase space which takes into account all of the symmetries
of the problem. Next an analysis of the system is achieved obtaining up to six relative equilibria and three types of bifurcations.
The connection with the original system is established concluding the existence of various families of invariant 3-tori of
it, as well as quasiperiodic and periodic trajectories. This is achieved by using KAM theory techniques. 相似文献
17.
D. Viswanath 《Celestial Mechanics and Dynamical Astronomy》2006,94(2):213-235
The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses
1− μ and μ that circle each other with period equal to 2π. For small μ, a resonant periodic motion of the massless particle
in the rotating frame can be described by relatively prime integers p and q, if its period around the heavier primary is approximately 2π p/q, and by its approximate eccentricity e. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions.
We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study
of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem
are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations
to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with q/p equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances.
We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3,
and 1/2 resonances for any μ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist
for μ small enough in the unaveraged restricted three-body problem. 相似文献
18.
19.
An appropriate generalization of the Jacobi equation of motion for the polar moment of inertia I is considered in order to study the N-body problem with variable masses. Two coupled ordinary differential equations governing the evolution of I and the total energy E are obtained. A regularization scheme for this system of differential equations is provided. We compute some illustrative
numerical examples, and discuss an average method for obtaining approximate analytical solutions to this pair of equations.
For a particular law of mass loss we also obtain exact analytical solutions. The application of these ideas to other kind
of perturbed gravitational N-body systems involving drag forces or a different type of mass variation is also considered.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
20.
The restricted three-body problem is reconsidered by replacing the point-like primaries of the classical problem by a pair of axisymmetric rigid bodies which have a plane of symmetry perpendicular to their axes, and the infinitesimal mass by a gyrostat. The conditions for the circular motion of the primaries around their center of mass are stated and they yield the classification of all possible orientations of these bodies into four groups according to the value of their angular velocity. Then the equations of motion of the gyrostat are derived and solved for the equilibrium configurations of the system. 相似文献