共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
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
3.
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
4.
Martin Lara 《Celestial Mechanics and Dynamical Astronomy》2014,118(3):221-234
A simple rearrangement of the torque free motion Hamiltonian shapes it as a perturbation problem for bodies rotating close to the principal axis of maximum inertia, independently of their triaxiality. The complete reduction of the main part of this Hamiltonian via the Hamilton–Jacobi equation provides the action-angle variables that ease the construction of a perturbation solution by Lie transforms. The lowest orders of the transformation equations of the perturbation solution are checked to agree with Kinoshita’s corresponding expansions for the exact solution of the free rigid body problem. For approximately axisymmetric bodies rotating close to the principal axis of maximum inertia, the common case of major solar system bodies, the new approach is advantageous over classical expansions based on a small triaxiality parameter. 相似文献
5.
Gwenaël Boué Nicolas Rambaux Andy Richard 《Celestial Mechanics and Dynamical Astronomy》2017,129(4):449-485
We revisit the rotation dynamics of a rigid satellite with either a liquid core or a global subsurface ocean. In both problems, the flow of the fluid component is assumed inviscid. The study of a hollow satellite with a liquid core is based on the Poincaré–Hough model which provides exact equations of motion. We introduce an approximation when the ellipticity of the cavity is low. This simplification allows to model both types of satellite in the same manner. The analysis of their rotation is done in a non-canonical Hamiltonian formalism closely related to Poincaré’s “forme nouvelle des équations de la mécanique”. In the case of a satellite with a global ocean, we obtain a seven-degree-of-freedom system. Six of them account for the motion of the two rigid components, and the last one is associated with the fluid layer. We apply our model to Titan for which the origin of the obliquity is still a debated question. We show that the observed value is compatible with Titan slightly departing from the hydrostatic equilibrium and being in a Cassini equilibrium state. 相似文献
6.
R. A. Howland 《Celestial Mechanics and Dynamical Astronomy》1988,45(4):407-412
In the author's treatment of the ideal resonance problem (1988), a non-canonical transformation was employed to bring the original Hamiltonian to a form amenable to the use of standard action-angle variables. Though the strictly Hamiltonian form of equations of motion was thus compromised, their general form was maintained, allowing transformation of the system to arbitrary order and forestalling the introduction of elliptic functions until a final explicit integration required in this approach. The general theory of such transformations is presented, and some points regarding their application are discussed, leading to the conclusion that the approach is practically limited to systems with a single degree of freedom only. 相似文献
7.
T. S. Boronenko 《Celestial Mechanics and Dynamical Astronomy》2017,127(2):139-161
In this article, we present the Lie transformation algorithm for autonomous Birkhoff systems. Here, we are referring to Hamiltonian systems that obey a symplectic structure of the general form. The Birkhoff equations are derived from the linear first-order Pfaff–Birkhoff variational principle, which is more general than the Hamilton principle. The use of 1-form in formulating the equations of motion in dynamics makes the Birkhoff method more universal and flexible. Birkhoff’s equations have a tensorial character, so their form is independent of the coordinate system used. Two examples of normalization in the restricted three-body problem are given to illustrate the application of the algorithm in perturbation theory. The efficiency of this algorithm for problems of asymptotic integration in dynamics is discussed for the case where there is a need to use non-canonical variables in phase space. 相似文献
8.
Andrzej J. Maciejewski 《Celestial Mechanics and Dynamical Astronomy》1985,37(1):47-57
The Euler-Poinsot equations of rigid body motion are considered. The Euler parameters attitude parametrization is assumed. The Hamiltonian form of these equations is obtained. Two different solutions are shown. A short discussion of free motion in this new treatment is also given. 相似文献
9.
Paolo Lanzano 《Astrophysics and Space Science》1969,5(3):300-322
We consider two spheroidal rigid bodies of comparable size constituting the components of an isolated binary system. We assume that (1) the bodies are homogeneous oblate ellipsoids of revolution, and (2) the meridional eccentricities of both components are small parameters.We obtain seven nonlinear differential equations governing simultaneously the relative motion of the two centroids and the rotational motion of each set of body axes. We seek solutions to these equations in the form of infinite series in the two meridional eccentricities.In the zero-order approximation (i. e., when the meridional eccentricities are neglected), the equations of motion separate into two independent subsystems. In this instance, the relative motion of the centroids is taken as a Kepler elliptic orbit of small eccentricity, whereas for each set of body axes we choose a composite motion consisting of a regular precession about an inertial axis and a uniform rotation about a body axis.The first part of the paper deals with the representation of the total potential energy of the binary system as an infinite series of the meridional eccentricities. For this purpose, we had to (1) derive a formula for representing the directional derivative of a solid harmonic as a combination of lower order harmonics, and (2) obtain the general term of a biaxial harmonic as a polynomial in the angular variables.In the second part, we expound a recurrent procedure whereby the approximations of various orders can be determined in terms of lower-order approximations. The rotational motion gives rise to linear differential equations with constant coefficients. In dealing with the translational motion, differential equations of the Hill type are encountered and are solved by means of power series in the orbital eccentricity. We give explicit solutions for the first-order approximation alone and identify critical values of the parameters which cause the motion to become unstable.The generality of the approach is tantamount to studying the evolution and asymptotic stability of the motion.Research performed under NASA Contract NAS5-11123. 相似文献
10.
Radu Balan 《Celestial Mechanics and Dynamical Astronomy》1995,63(1):59-79
The purpose of this paper is to study the motion of a spinless axisymmetric rigid body in a Newtonian field when we suppose the motion of the center of mass of the rigid body is on a Keplerian orbit. In this case the system can be reduced to a Hamiltonian system with configuration space of a two-dimensional sphere. We prove that the restricted planar motion is analytical nonintegrable and we find horseshoes due to the eccentricity of the orbit. In the caseI
3/I
1>4/3, we prove that the system on the sphere is also analytical nonintegrable.On leave from the Polytechnic Institute of Bucharest, Romania. 相似文献
11.
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
12.
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
13.
The restricted 2+2 body problem is considered. The infinitesimal masses are replaced by triaxial rigid bodies and the equations of motion are derived in Lagrange form. Subsequently, the equilibrium solutions for the rotational and translational motion of the bodies are detected. These solutions are conveniently classified in groups according to the several combinations which are possible between the translational equilibria and the constant orientations of the bodies. 相似文献
14.
In this paper, the translational-rotational motions of an axisymmetric rigid body and two spherical rigid bodies under the influence of their mutual gravitational attraction are considered. The equations of motion in the canonical elements of Delaunay-Andoyer are obtained. The elements of motion in the zero and first approximations can be determined. 相似文献
15.
Simulation of the full two rigid body problem using polyhedral mutual potential and potential derivatives approach 总被引:3,自引:0,他引:3
Eugene G. Fahnestock Daniel J. Scheeres 《Celestial Mechanics and Dynamical Astronomy》2006,96(3-4):317-339
Herein we investigate the coupled orbital and rotational dynamics of two rigid bodies modelled as polyhedra, under the influence of their mutual gravitational potential. The bodies may possess any arbitrary shape and mass distribution. A method of calculating the mutual potential’s derivatives with respect to relative position and attitude is derived. Relative equations of motion for the two body system are presented and an implementation of the equations of motion with the potential gradients approach is described. Results obtained with this dynamic simulation software package are presented for multiple cases to validate the approach and illustrate its utility. This simulation capability is useful both for addressing questions in dynamical astronomy and for enabling spacecraft missions to binary asteroid systems. 相似文献
16.
David L. Richardson 《Celestial Mechanics and Dynamical Astronomy》1980,22(3):231-236
A lagrangian formulation for the three-dimensional motion of a satellite in the vicinity of the collinear points of the circular-restricted problem is reconsidered. It is shown that the influence of the primaries can be expressed in the form of two third-body disturbing functions. By use of this approach, the equations for the Lagrangian and for the motion itself are readily developed into highly compact expressions. All orders of the non-linear developments are shown to be easily obtainable using well-known recursive relationships. The resulting forms for these equations are well suited for use in the initial phase of canonical or non-canonical investigations. 相似文献
17.
William W. Hooker 《Celestial Mechanics and Dynamical Astronomy》1975,11(3):337-359
Equations of motion are derived for systems of rotationally interconnected bodies in which the terminal bodies may be flexible and the remaining bodies are rigid. The bodies may have an arbitrary topological tree arrangement; that is, there are no closed loops of bodies. This derivation extends earlier results for systems of interconnected rigid bodies only, and is much simpler than several other recent works on terminal flexible bodies. The model for a flexible body assumes that the elastic deformation is representable as a time-varying linear combination of given mode shapes.The paper also derives the appropriate form for gravitational terms, so that the equations can be used for flexible satellites. Also included are expressions for kinetic energy and angular momentum so that in case these are theoretically constant, they can be used to monitor the accuracy of the numerical integration. The paper concludes with a section showing how interbody constraint forces and torques (which do not appear in the equations of motion) can be recovered from quantities available in this formulation, and also how to treat state variables which are prescribed functions of time.A digital computer program based on the equations derived here has been used to simulate a spinning Skylab (with flexible booms) and also the interplanetary Viking (with flexible solar panels and thrust vector control).We announce with regret that Bill Hooker died in an avalanche while on a mountain-climbing expedition in Peru, July 1974. 相似文献
18.
We consider the Sitnikov problem; from the equations of motion we derive the approximate Hamiltonian flow. Then, we introduce
suitable action–angle variables in order to construct a high order normal form of the Hamiltonian. We introduce Birkhoff Cartesian
coordinates near the elliptic orbit and we analyze the behavior of the remainder of the normal form. Finally, we derive a
kind of local stability estimate in the vicinity of the periodic orbit for exponentially long times using the normal form
up to 40th order in Cartesian coordinates. 相似文献
19.
R. A. Howland 《Celestial Mechanics and Dynamical Astronomy》1988,45(1-3):207-211
A new approach to the librational solution of the Ideal Resonance Problem has been devised--one in which a non-canonical transformation is applied to the classical Hamiltonian to bring it to the form of the simple harmonic oscillator. Although the traditional form of the canonical equations of motion no longer holds, a quasi-canonical form is retained in this single-degree-of-freedom system, with the customary equations being multiplied by a non-constant factor. While this makes the resulting system amenable to traditional transformation techniques, it must then be integrated directly. Singularities of the transformation in the circulation region limit application of the method to the librational region of motion.Computer-assisted algebra has been used in all three stages of the solution to fourth order of this problem: using a general-purpose FORTRAN program for the quadratic analytical solution of Hamiltonians in action-angle variables, the initial transformation is carried out by direct substitution and the resulting Hamiltonian transformed to eliminate angular variables. The resulting system of differential equations, requiring the expected elliptic functions as part of their solution, is currently in the process of being integrated using the LISP-based REDUCE software, by programming the required recursive rules for elliptic integration.Basic theory of this approach and the computer implementation of all these techniques is described. Extension to higher order of the solution is also discussed. 相似文献
20.
Taeyoung Lee Melvin Leok N. Harris McClamroch 《Celestial Mechanics and Dynamical Astronomy》2007,98(2):121-144
Equations of motion, referred to as full body models, are developed to describe the dynamics of rigid bodies acting under
their mutual gravitational potential. Continuous equations of motion and discrete equations of motion are derived using Hamilton’s
principle. These equations are expressed in an inertial frame and in relative coordinates. The discrete equations of motion,
referred to as a Lie group variational integrator, provide a geometrically exact and numerically efficient computational method
for simulating full body dynamics in orbital mechanics; they are symplectic and momentum preserving, and they exhibit good
energy behavior for exponentially long time periods. They are also efficient in only requiring a single evaluation of the
gravity forces and moments per time step. The Lie group variational integrator also preserves the group structure without
the use of local charts, reprojection, or constraints. Computational results are given for the dynamics of two rigid dumbbell
bodies acting under their mutual gravity; these computational results demonstrate the superiority of the Lie group variational
integrator compared with integrators that are not symplectic or do not preserve the Lie group structure. 相似文献