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1.
This paper describes an algorithm which brings a regularizable polynomial perturbation of a three degree of freedom Kepler problem into a normal form which Poisson commutes with the Kepler Hamiltonian. We illustrate the alogrithm with an example: the quadratic Zeeman effect. For other applications of this algorithm see [1],[4], and [5]. The authors have written a program in MAPLE which implements the constrained normal form. 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.
The planar isosceles three-body problem where the two symmetric bodies have small masses is considered as a perturbation of
the Kepler problem. We prove that the circular orbits can be continued to saddle orbits of the Isosceles problem. This continuation
is not possible in the elliptic case. Their perturbed orbits tend to a continued circular one or approach a triple collision.
The basic tool used is the study of the Poincaré maps associated with the periodic solutions.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
4.
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
5.
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
6.
Yu. S. Osipov 《Celestial Mechanics and Dynamical Astronomy》1977,16(2):191-208
The main result of this paper is a theorem on the trajectory equivalence of phase flows on isoenergetic surfaces with a positive energy level in the Kepler problem and perturbed kepler problem. The following two facts are crucial for proving it: firstly, an isomorphism of the phase flow on an isoenergetic surface in the Kepler problem and the geodesic flow in a constant curvature space. The isomorphism is studied in detail. In particular, all the integrals of the Kepler problem are obtained proceeding from the group-theory considerations. The second fact is a generalization of the theorem on structural stability of Anosov flows onto non-compact manifolds. 相似文献
7.
Claudio Vidal 《Celestial Mechanics and Dynamical Astronomy》2001,80(2):119-132
We consider perturbations of the Kepler problem that are symmetric with respect to the origin and admit a first integral of motion which is also symmetric with respect to the origin. It has been proved that each circular solution of the unperturbed problem gives rise to a periodic solution of the perturbed system. 相似文献
8.
Djordje S. Djukic 《Celestial Mechanics and Dynamical Astronomy》1993,56(4):523-540
This paper considers adiabatic invariants for the classical Kepler problem with resisting forces. The analysis is based on the theory of integrating factors and theory of adiabatic invariants in the Krylov-Bogoliubov-Mitropolski variables. The adiabatic invariants are series with respect to a small parameter. Also, for every particular case of nonconservative forces, it is shown that, with a complete set of adiabatic invariants, an approximate solution of the problem can be obtained. Four problems are analyzed in detail where approximate solutions are compared with numerical. 相似文献
9.
We propose the Ptolemaic transformation: a canonical change of variables reducing the Keplerian motion to the form of a perturbed Hamiltonian problem. As a solution of the unperturbed case, the Ptolemaic variables define an intermediary orbit, accurate up to the first power of eccentricity, like in the kinematic model of Claudius Ptolemy. In order to normalize the perturbed Hamiltonian we modify the recurrent Lie series algorithm of HoriuuMersman. The modified algorithm accounts for the loss of a term's order during the evaluation of a Poisson bracket, and thus can be also applied in resonance problems. The normalized Hamiltonian consists of a single Keplerian term; the mean Ptolemaic variables occur to be trivial, linear functions of the Delaunay actions and angles. The generator of the transformation may serve to expand various functions in Poisson series of eccentricity and mean anomaly. 相似文献
10.
Jan Vrbik 《Celestial Mechanics and Dynamical Astronomy》2001,80(2):111-118
A semi-analytical solution to the Kustaanheimo–Stiefel formulation of the perturbed Kepler problem is presented. 相似文献
11.
Bruno Cordani 《Celestial Mechanics and Dynamical Astronomy》2000,77(3):185-200
The usual action-angle variables for the Kepler Problem (the Delaunay variables) are not globally defined, leaving out some
orbits (circular orbits or those lying on the xy-plane). Moreover they are trascendental functions of the physical variables, making it quite difficult to write the perturbed
Hamiltonian. The way-out proposed here is to pass to a 8-dimensional rank-6 Poisson manifold, that is, to parametrize the
state of the Kepler Problem with two 4-dimensional vectors mutually orthogonal and of equal norm.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
12.
The accelerated Kepler problem (AKP) is obtained by adding a constant acceleration to the classical two-body Kepler problem.
This setting models the dynamics of a jet-sustaining accretion disk and its content of forming planets as the disk loses linear
momentum through the asymmetric jet-counterjet system it powers. The dynamics of the accelerated Kepler problem is analyzed
using physical as well as parabolic coordinates. The latter naturally separate the problem’s Hamiltonian into two unidimensional
Hamiltonians. In particular, we identify the origin of the secular resonance in the AKP and determine analytically the radius
of stability boundary of initially circular orbits that are of particular interest to the problem of radial migration in binary
systems as well as to the truncation of accretion disks through stellar jet acceleration. 相似文献
13.
M. A. Vashkov’yak 《Solar System Research》2018,52(4):359-370
The well-known problem of motion in a central field integrable in quadratures is considered. The force function of the problem depends only on the particle distance to the chosen coordinate origin. In the general case of an arbitrary central force, a rigorous analytical solution of the problem cannot be obtained due to the complexity of the integrals. In this paper we propose a semi-analytical method of constructing an approximate solution for the case where the distance varies in a limited range that allows the time dependences of the polar coordinates to be obtained using elliptic functions and integrals. As an example, we consider the model problems of the perturbed motion of hypothetical Jovian and lunar equatorial satellites as well as the problem of the motion of a single star in the principal plane of a galaxy. The methodical accuracy has been estimated by a comparison with the numerical solution. 相似文献
14.
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
15.
Radial,transverse and normal satellite position perturbations due to the geopotential 总被引：1，自引：0，他引：1
Perturbations in the position of a satellite due to the Earth's gravitational effects are presented. The perturbations are given in the radial, transverse (or alongtrack) and normal (or cross-track) components. The solution is obtained by projecting the Kepler element perturbations obtained by Kaula [Kaula, 1966] into each of the three components. The resulting perturbations are presented in a form analogous to the form of Kaula's solution which facilitates implementation and interpretation. 相似文献
16.
When the elimination of the parallax and the elimination of the perigee is applied to the zonal problem of the artificial satellite, a one-degree of freedom Hamiltonian is obtained. The classical way to integrate this Hamiltonian is by applying the Delaunay normalization, however, changing the time to the perturbed true anomaly and the variable to the inverse of the distance, the Hamilton equations become a perturbed harmonic oscillator. In this paper we apply the Krylov—Bogoliubov—Mitropolsky (KBM) method to integrate the perturbed harmonic oscillator as an alternative method to the Delaunay normalization. This method has no problem with small eccentricities and inclinations, and shows good numerical results in the evaluation of ephemeris of satellites.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
17.
John G. Bryant 《Celestial Mechanics and Dynamical Astronomy》1999,73(1-4):269-280
We introduce a new kind of canonical variables that prove very useful when the order of a Hamiltonian system can be reduced
by one, as in the case of isoenergetic reduction, and of what we call homogeneous reduction. The Kepler Problem, Geometrical
Optics and McGehee Blow-up are discussed as examples. Finally we carry out the isoenergetic reduction of the general N-Body
Problem using the new variables, and briefly discuss its application to the problem of collision.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
18.
19.
Jan Vrbik 《Celestial Mechanics and Dynamical Astronomy》2005,91(3-4):217-237
A recently developed iterative technique for solving the perturbed Kepler problem is explained, in a step-by-step detail,
using the simple example of oblateness perturbations. The results are then extended to deal with higher-degree zonal harmonics. 相似文献
20.
Martín Lara Jesús F. Palacián Ryan P. Russell 《Celestial Mechanics and Dynamical Astronomy》2010,108(1):1-22
Preliminary mission design for planetary satellite orbiters requires a deep knowledge of the long term dynamics that is typically
obtained through averaging techniques. The problem is usually formulated in the Hamiltonian setting as a sum of the principal
part, which is given through the Kepler problem, plus a small perturbation that depends on the specific features of the mission.
It is usually derived from a scaling procedure of the restricted three body problem, since the two main bodies are the Sun
and the planet whereas the satellite is considered as a massless particle. Sometimes, instead of the restricted three body
problem, the spatial Hill problem is used. In some cases the validity of the averaging is limited to prohibitively small regions,
thus, depriving the analysis of significance. We find this paradigm at Enceladus, where the validity of a first order averaging
based on the Hill problem lies inside the body. However, this fact does not invalidate the technique as perturbation methods
are used to reach higher orders in the averaging process. Proceeding this way, we average the Hill problem up to the sixth
order obtaining valuable information on the dynamics close to Enceladus. The averaging is performed through Lie transformations
and two different transformations are applied. Firstly, the mean motion is normalized whereas the goal of the second transformation
is to remove the appearance of the argument of the node. The resulting Hamiltonian defines a system of one degree of freedom
whose dynamics is analyzed. 相似文献