共查询到20条相似文献，搜索用时 46 毫秒
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David Hestenes 《Celestial Mechanics and Dynamical Astronomy》1983,30(2):151-170
Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.This work was partially supported by JPL under contract with the National Aeronautics and Space Administration. 相似文献
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Jörg Waldvogel 《Celestial Mechanics and Dynamical Astronomy》2008,102(1-3):149-162
Quaternions have been found to be the ideal tool for describing and developing the theory of spatial regularization in Celestial Mechanics. This article corroborates the above statement. Beginning with a summary of quaternion algebra, we will describe the regularization procedure and its consequences in an elegant way. Also, an alternative derivation of the theory of Kepler motion based on regularization will be given. Furthermore, we will consider the regularization of the spatial restricted three-body problem, i.e. the spatial generalization of the Birkhoff transformation. Finally, the perturbed Kepler motion will be described in terms of regularized variables. 相似文献
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Dieter J. Lelgemann 《Celestial Mechanics and Dynamical Astronomy》1983,30(3):309-321
Using Hill's variables, an analytical solution of a canonical system of six differential equations describing the motion of a satellite in the gravitational field of the earth is derived. The gravity field, expanded into spherical harmonics, has to be expressed as a function of the Hill variables. The intermediary is chosen to include the main secular terms. The first order solution retains the highly practical formal structure of Kaula's linear solution, but is valid for circular orbits and provides of course a spectral decomposition of radius vector and radial velocity. The resulting eccentricity functions are much simpler than the Hansen functions, since a series evaluation of the Kepler equation is avoided. The present solution may be extended to higher order solutions by Hori's perturbation method. 相似文献
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There exist many comets with near-parabolic orbits in the Solar System. Among various theories proposed to explain their origin, the Oort cloud hypothesis seems to be the most reasonable (Oort, 1950). The theory assumes that there is a cometary cloud at a distance 103 – 105 AU from the Sun and that perturbing forces from planets or stars make orbits of some of these comets become of near-parabolic type. Concerning the evolution of these orbits under planetary perturbations, we can raise the question: Will they stay in the Solar System forever or will they escape from it? This is an attractive dynamical problem. If we go ahead by directly solving the dynamical differential equations, we may encounter the difficulty of long-time computation. For the orbits of these comets are near-parabolic and their periods are too long to study on their long-term evolution. With mapping approaches the difficulty will be overcome. In another aspect, the study of this model has special meaning for chaotic dynamics. We know that in the neighbourhood of any separatrix i.e. the trajectory with zero frequency of the unperturbed motion of an Hamiltonian system, some chaotic motions have to be expected. Actually, the simplest example of separatrix is the parabolic trajectory of the two body problem which separates the bounded and unbounded motion. From this point of view, the dynamical study on near-parabolic motion is very important. Petrosky's elegant but more abstract deduction gives a Kepler mapping which describes the dynamics of the cometary motion (Petrosky, 1988). In this paper we derive a similar mapping directly and discuss its dynamical characters. 相似文献
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A two-point boundary value problem of the Kepler orbit similar to Lambert’s problem is proposed. The problem is to find a
Kepler orbit that will travel through the initial and final points in a specified flight time given the radial distances of
the two points and the flight-direction angle at the initial point. The Kepler orbits that meet the geometric constraints
are parameterized via the universal variable z introduced by Bate. The formula for flight time of the orbits is derived. The admissible interval of the universal variable
and the variation pattern of the flight time are explored intensively. A numerical iteration algorithm based on the analytical
results is presented to solve the problem. A large number of randomly generated examples are used to test the reliability
and efficiency of the algorithm. 相似文献
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In this paper we show that the Conditional Entropy of nearby orbits may be a useful tool to explore the phase space associated
to a given Hamiltonian. The arc length parameter along the orbits, instead of the time, is used as a random variable to compute
the entropy. In the first part of this work we summarise the main analytical results to support this tool while, in the second
part, we present numerical evidence that this technique is able to localise (stable) periodic and quasiperiodic orbits, 'aperiodic'
orbits (chaotic motion) and unstable periodic orbits (the 'source' of chaotic motion). Besides, we show that this technique
provides a measure of chaos which is similar to that given by the largest Lyapunov Characteristic Number. It is important
to remark that this method is very simple to compute and does not require long time integrations, just realistic physical
times.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
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We obtain thex - p
xPoincare phase plane for a two dimensional, resonant, galactic type Hamiltonian using conventional numerical integration,
a second order symplectic integrator and a map based on the averaged Hamiltonian. It is found that all three methods give
good results, for small values of the perturbation parameter, while the symplectic integrator does a better job than the mapping,
for large perturbations. The dynamical spectra are used to distinguish between regular and chaotic motion. 相似文献
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The Kepler problem including radiation pressure and drag is treated. The equation of the orbit is derived and the scalar and vector integrals of motion are obtained by direct operation on the vector form of the equation of motion. 相似文献
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Diarmuid O'Mathuna 《Celestial Mechanics and Dynamical Astronomy》1970,1(3-4):467-478
For Vinti's dynamical problem, there is proposed a new form of solution wherein all three coordinates are expressed in terms of one independent variable. The formulae for the three co-ordinates are clear generalizations of the corresponding formulae for the Kepler problem while the independent variable corresponds to the true anomaly. The solution is completed by the relation connecting the independent variable with time: the latter is a generalization of the well known Kepler time-angle relationship. From the form and method of solution the main qualitative features of the motion are readily derived. 相似文献
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The aim of the paper is to study the long term evolution of comet Halleys orbit taking into account small errors in the initial conditions. Recent papers deal with mapping methods to model cometary dynamics; (e.g. Petrosky and Broucke, 1987 and Chirikov and Vecheslavov, 1986). They will be discussed critically and compared with our own results. We then tested the model using numerical integration methods. For the moment we limited our calculation to 2.105 years, but a 106 year integration is still in progress. We show the expected dynamical evolution of Hallyes orbit taking into account also smaller and larger errors of the initial conditions (nongravitational effects are only roughly estimated). Finally we discuss alsothe controversal opinions concerning the role of the planets (especially the earth). 相似文献
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V. I. Ferronsky S. A. Denisik S. V. Ferronsky 《Celestial Mechanics and Dynamical Astronomy》1982,27(3):285-304
A general approach to the solution of the perturbed oscillation problem for celestial bodies is considered. The solution sought describes unperturbed virial oscillations (zero approximation) affected by external perturbing effects. In the general case, these perturbations can be expressed by an arbitrary given function of time, Jacobi's function and its first derivative. Standard methods and modes of perturbation theory are used for solution of the problem.It is shown that while studying the evolution of a celestial body as a dissipative system in the framework of perturbed virial oscillations, the analytical expression for perturbing function can be derived, assuming the celestial body to be an oscillating electrical dipole emitting electromagnetic energy.The general covariant form of Jacobi's equation is derived and its spur is examined. It is shown that the scalar form of Jacobi's equation appears to be more universal than Newton's laws of motion from which it is derived. 相似文献
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J. Baumgarte 《Celestial Mechanics and Dynamical Astronomy》1972,5(4):490-501
A stabilization of the classical equations of two-body motion is offered. It is characterized by the use of the regularizing independent variable (eccentric anomaly) and by the addition of a control-term to the differential equations. This method is related to the KS-theory (Stiefel, 1970) which performed for the first time a stabilization of the Kepler motion. But in contrast to the KS-theory our method does not transform the coordinates of the particle. As far as the theory of stability and the numerical experiments are concerned we restrict ourselves to thepure Kepler motion. But, of course, the stabilizing devices will also improve the accuracy of the computation of perturbed orbits. We list, therefore, also the equations of the perturbed motion. 相似文献
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The equations of motion of the planar three-body problem split into two parts, called an external part and an internal part. When the third mass approaches zero, the first part tends to the equations of the Kepler motion of the primaries and the second part to the equations of motion of the restricted problem.We discuss the Hill stability from these equations of motion and the energy integral. In particular, the Jacobi integral for the circular restricted problem is seen as an infinitesimal-mass-order term of the Sundman function in this context. 相似文献
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Action-angle variables for the Levi-Civita regularized planar Kepler problem were introduced independently first by Chenciner and then by Deprit and Williams. The latter used explicitly the so-called Lissajous variables. When applied to the transformed Keplerian Hamiltonian, the Lissajous transformation encounters the difficulty of being defined in terms of the constant frequency parameter, whereas the Kepler problem transformed into a harmonic oscillator involves the frequency as a function of an energy-related canonical variable. A simple canonical transformation is proposed as a remedy for this inconvenience. The problem is circumvented by adding to the physical time a correcting term, which occurs to be a generalized Kepler’s equation. Unlike previous versions, the transformation is symplectic in the extended phase space and allows the treatment of time-dependent perturbations. The relation of the extended Lissajous–Levi-Civita variables to the classical Delaunay angles and actions is given, and it turns out to be a straightforward generalization of the results published by Deprit and Williams. 相似文献
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Harold S. Morton Jr. John L. Junkins Jeffrey N. Blanton 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):287-301
Two new analytical solutions for Poinsot motion in terms of Euler parameters are derived. The first solution is a straightforward ‘universal’ (no branches) time series practical for short time motion calculations or as a basis for analytical continuation. The second, more involved solution is also universal but is not restricted to short times; it is in terms of circular, hyperbolic, and elliptic functions and elliptic integrals. 相似文献
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Zachary E. Wardell 《Monthly notices of the Royal Astronomical Society》2003,341(2):423-433
In a previous investigation, a model of three-body motion was developed which included the effects of gravitational radiation reaction. The aim was to describe the motion of a relativistic binary pulsar that is perturbed by a third mass and look for resonances between the binary and third-mass orbits. Numerical integration of an equation of relative motion that approximates the binary gives evidence of such resonances. These ( m : n ) resonances are defined for the present purposes by the resonance condition, m ω= 2 n Ω , where m and n are relatively prime integers and ω and Ω are the angular frequencies of the binary orbit and third-mass orbit (around the centre of mass of the binary), respectively. The resonance condition consequently fixes a value for the semimajor axis a of the binary orbit for the duration of the resonance because of the Kepler relationship ω= a −3/2 . This paper outlines a method of averaging developed by Chicone, Mashhoon and Retzloff, which renders a non-linear system that undergoes resonance capture into a mathematically amenable form. This method is applied to the present system and one arrives at an analytical solution that describes the average motion during resonance. Furthermore, prominent features of the full non-linear system, such as the frequency of oscillation and antidamping, accord with their analytically derived formulae. 相似文献