共查询到20条相似文献，搜索用时 15 毫秒
1.
Paule Kustaanheimo 《Celestial Mechanics and Dynamical Astronomy》1972,6(1):52-59
The ordinary spinor differential Equation (20) of the unperturbed Kepler motion is obtained from the classical equation of motion (19) if one uses the spinor regularization (9) and postulates an essential subsidiary condition (10). A natural generalization for the Kepler motion follows by dropping this subsidiary conditions; it is the 8-parameter set of solutions of the spinor equation of motion (20). The sixteen natural extensive integrals (30)–(35) for this generalized Kepler motion are here deduced by means of the relativistic motors (2), (7) of the Spinor Ring Algebra. These integrals form, with respect to the Poisson bracket operation, a 15-dimensional Lie algebra (40)–(44), closely related to the Lie algebras in quantum mechanics.Dedicated to Professor G. Järnefelt on his 70th anniversary. 相似文献
2.
Liam M. Healy 《Celestial Mechanics and Dynamical Astronomy》2003,85(2):175-207
The Lie transfer map method may be applied to orbit propagation problems in celestial mechanics. This method, described in another paper, is a perturbation method applicable to Hamiltonian systems. In this paper, it is used to calculate orbits for zonal perturbations to the Kepler (two-body) problem, in both expansion in the eccentricity and closed form. In contrast with a normal form method like that of Deprit, the Lie transformations here are used to effect a propagation of phase space in time, and not to transform one Hamiltonian into another. 相似文献
3.
Maria Dina Vivarelli 《Celestial Mechanics and Dynamical Astronomy》1987,41(1-4):359-370
As an outcome of our previous notes [13, 14] on the quaternion regularization of the classical Kepler problem and pre-quantization of the Kepler manifold we show, first, that both the cross product of two quaternions and the cross product of their anti-involutes are susceptible of a simple geometrical representation in the ordinary 3-dimensional euclidean spaceR
3 and, secondly, that they satisfy anSO(4)-invariant relation that implies projection of curves from the quaternion space onto the spaceR
3. ThisSO(4)-invariance allows—in the particular case of orthogonal quaternions of equal norm—a straight derivation: (i) of the correspondence between the free motion on the surface of a sphereS
3 and the physical elliptical Kepler motion (collisions included) on a plane denoted by
w
; (ii) of the celebrated Kepler equation and (iii) of the Levi-Civita regularizing time transformation. With (i) and (ii) we recover some of Györgyi's [3] results. The aforesaid orbital plane
w
and the orbital plane *, arrived at independently by exploiting the Kustaanheimo-Stiefel regularizing transformation, are shown to be inclined exactly at an angle characterizing the ratio of the semi-axes of the elliptical orbits and intimately related to the cross product representation. Thus the eventual superimposition of the two planes confirms the intimate connection between the various regularization procedures—transforming the classical Kepler problem into the geodesic flow onS
3—and the Fock's procedure for the quantum theoretical Kepler problem of the hydrogen atom (accidental degeneracy).This research was supported by the Consiglio Nazionale delle Ricerche of Italy (C.N.R.-G.N.F.M.). 相似文献
4.
5.
Kyle T. Alfriend 《Celestial Mechanics and Dynamical Astronomy》1972,5(4):502-511
The question of whether or not there is a transfer of energy between the in-plane motion and out-of-plane motion in the neighborhood ofL
4 in the restricted problem of three bodies is investigated in this paper. The in-plane motion is assumed to be finite and the out-of-plane motion to be infinitesimal. The equation governing the out-of-plane motion becomes one with time varying coefficients. The stability of this equation is then investigated using Lie Series.Presented as a paper AAS No. 70-313, at the AAS/AIAA Astrodynamics Specialists Conference 1971 at Fort Lauderdale Fla., U.S.A. 相似文献
6.
Jan Vrbik 《Celestial Mechanics and Dynamical Astronomy》1998,71(4):273-287
The main purpose of this paper is to supply a proof of formulas for constructing a perturbative solution to the perturbed
Kepler problem by utilizing quaternion algebra of the Kustaanheimo–Stiefel formulation. The main advantage of this approach
is a removal, from the corresponding solution, of fast oscillations (in the case of conservative forces) and small divisors
(in the case of time-dependent forces).
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
7.
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
8.
J. Baumgarte 《Celestial Mechanics and Dynamical Astronomy》1980,21(1):43-43
It is shown that the Lie algebra so(4, 2) is characteristic for the three dimensional Keplerian motion provided the eccentric anomaly is used as the independent variable. This algebra generates all the integrals of motion and yields the guiding principle for reformulating all the Keplerian formulas.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978. 相似文献
9.
P. Deuflhard 《Celestial Mechanics and Dynamical Astronomy》1980,21(2):213-223
In this paper, a special extrapolation method for the numerical integration of perturbed Kepler problems (given in KS-formulation) is worked out and analyzed in detail. The underlying so-called Kepler discretization isexact for the pure (elliptic) Kepler motion. A numerically stable realization is presented together with a backward error analysis: this analysis shows that the effect of the arising rounding errors can be regarded as a small perturbation inferior to the physical perturbation. For test purposes, a well-known example describing the motion of an artificial Earth satellite in an equator plane subject to the oblateness perturbation is used to demonstrate the efficiency of the new extrapolation method.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978. 相似文献
10.
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
11.
Christoph Lhotka 《Celestial Mechanics and Dynamical Astronomy》2017,127(4):397-428
In this paper, we consider the elliptic collinear solutions of the classical n-body problem, where the n bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler–Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler–Moulton collinear solution of n-bodies splits into \((n-1)\) independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler 2-body problem at Kepler elliptic orbit, and each of the other \((n-2)\) systems is the essential part of the linearized Hamiltonian system at an elliptic Euler collinear solution of a 3-body problem whose mass parameter is modified. Then the linear stability of such a solution in the n-body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the 3-body problems, which for example then can be further understood using numerical results of Martínez et al. on 3-body Euler solutions in 2004–2006. As an example, we carry out the detailed derivation of the linear stability for an elliptic Euler–Moulton solution of the 4-body problem with two small masses in the middle. 相似文献
12.
A coordinate system is defined on the phase space of a perturbed Keplerian system after the mean anomaly has been averaged out, for the purpose of explaining how eliminating the longitude of the ascending node reduces the orbital space to a two-dimensional sphere in case the system admits an axial symmetry. Concomitantly, on the submanifold of direct osculating ellipses, the CDM variables are replaced by functions which form the basis of a Poisson algebra isomorphic to the Lie algebra so(3) of the rotation group SO(3); furthermore, in these variables, the doubly reduced phase flow appears like a rotation of the reduced phase space. 相似文献
13.
R. Broucke 《Astrophysics and Space Science》1980,72(1):33-53
In this article we collect several results related to the classical problem of two-dimensional motion of a particle in the field of a central force proportional to a real power of the distancer. At first we generalize Whittaker's result of the fourteen powers ofr which lead to intergrability with elliptic functions. We enumerate six more general potentials, including Whittaker's fourteen potentials as particular cases (Sections 2 and 3).Next, we study the stability of the circular solutions, which are the singular solutions of the problem, in Whittaker's terminology. The stability index is computed as a function of the exponentn and its properties are explained, especially in terms of bifurcations with other families of ordinary periodic solutions (Sections 4, 5 and 7). In Section 6, the detailed solution of the inverse cube force problem is given in terms of an auxiliary variable which is similar to the eccentric anomaly of the Kepler problem.Finally, it is shown that the stable singular circular solutions of the central force problem generalize to stable singular elliptic solutions of the two-fixed-center problem. The stability and the bifurcations with other families of periodic solutions of the two-fixed-center problem are also described. 相似文献
14.
The Kepler problem for the resistive force r/r
2 is known to have a conserved vector which is the analogue to Hamilton's vector for the standar Kepler problem. In this note it is shown in a very elementary way that many similar force laws display the same property. The orbit equation can be obtained easily in such cases. 相似文献
15.
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
16.
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
17.
Haruo Yoshida 《Celestial Mechanics and Dynamical Astronomy》1993,56(1-2):27-43
In this paper various aspect of symplectic integrators are reviewed. Symplectic integrators are numerical integration methods for Hamiltonian systems which are designed to conserve the symplectic structure exactly as the original flow. There are explicit symplectic schemes for systems of the formH=T(p)+V(q), and implicit schemes for general Hamiltonian systems. As a general property, symplectic integrators conserve the energy quite well and therefore an artificial damping (excitation) caused by the accumulation of the local truncation error cannot occur. Symplectic integrators have been applied to the Kepler problem, the motion of minor bodies in the solar system and the long-term evolution of outer planets. 相似文献
18.
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
19.
Ashraf Hamdy Mostafa K. Ahmed Mohamad Radwan Fawzy Ahmed Abd El-Salam 《Earth, Moon, and Planets》2003,93(4):261-273
The objective of the present work is to develope explicit analytical expressions for the small amplitude orbits of the infinitesimal
mass about the equilibrium points in the elliptic restricted three body problem. To handle this dynamical problem, the Hamiltonian
for the elliptic problem is formed with the true anomaly and then with the eccentric anomaly as independent variables. The
origin is then transformed to a fixed point and the Hamiltonian is developed up to O(4) in the eccentricity, e, (which plays the role of the small parameter of the problem) of the primaries. The integration of the model problem under
consideration is undertaken by means of a perturbation technique based on Lie series developments, which leads to the solution
of the canonical equations of motion. 相似文献
20.
D. Bancelin D. Hestroffer W. Thuillot 《Celestial Mechanics and Dynamical Astronomy》2012,112(2):221-234
The integration of the equations of motion in gravitational dynamical systems—either in our Solar System or for extra-solar
planetary systems—being non integrable in the global case, is usually performed by means of numerical integration. Among the
different numerical techniques available for solving ordinary differential equations, the numerical integration using Lie
series has shown some advantages. In its original form (Hanslmeier and Dvorak, Astron Astrophys 132, 203 1984), it was limited to the N-body problem where only gravitational interactions are taken into account. We present in this paper a generalisation of the
method by deriving an expression of the Lie terms when other major forces are considered. As a matter of fact, previous studies
have been done but only for objects moving under gravitational attraction. If other perturbations are added, the Lie integrator
has to be re-built. In the present work we consider two cases involving position and position-velocity dependent perturbations:
relativistic acceleration in the framework of General Relativity and a simplified force for the Yarkovsky effect. A general
iteration procedure is applied to derive the Lie series to any order and precision. We then give an application to the integration
of the equation of motions for typical Near-Earth objects and planet Mercury. 相似文献