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1.
The plane, singly averaged, elliptical restricted three body problem is considered in the article. The first three terms are taken in the perturbing function. The equations of motion in terms of the canonical elements of Delaunay are obtained. And the change of the elements of motion of the satellite due to the perturbing function is calculated. An application is given in the case of a satellite in the earth-moon system. 相似文献
2.
In the present note we show how a simple canonical transformation in extended phase space makes the transition from classical Delaunay elements to the Scheifele variables. We do not use spherical coordinates or Hamilton-Jacobi theory. The meaning of the new variables is briefly explained, especially the use of the true anomaly as one of the variables. 相似文献
3.
Relations are established between the Delaunay variables defined over a phase space E in four dimensions and the Lissajous variables defined over a four-dimensional phase space F when the latter is mapped onto E by a parabolic canonical transformation. 相似文献
4.
Victor R. Bond 《Celestial Mechanics and Dynamical Astronomy》1982,27(2):203-210
The propagation of errors in the solutions of the differential equations for the orbital elements of perturbed two-body motion is investigated. It is shown that the error in the time-element grows linearly for differential equations for orbital elements when only perturbations are present on the right-hand side, cubically for formulations which have a two-body term on the right-hand side, and linearly for formulations based upon extended phase space Hamiltonians. 相似文献
5.
I. V. Tupikova 《Celestial Mechanics and Dynamical Astronomy》2009,104(1-2):129-144
A new mathematically correct approach to construct an averaging procedure for the motion of a massless body around the central body perturbed by fully interacting planets is developed and the errors of the standard solution are discussed. The new technique allows to combine the advantages of the Hamiltonian representation with the usage of standard osculating elements in combination with all the standard expansions of the perturbing functions. The main idea is to introduce new additional variables conjugate to all the standard elements and to work in a corresponding super phase space. In this way, the number of variables is doubled at first, but one has to deal with only one Hamiltonian. The artificially introduced variables disappear from the final averaged equations as well as from the transformation formulae connecting the osculating and the mean elements. 相似文献
6.
Luis Floría 《Celestial Mechanics and Dynamical Astronomy》1997,68(1):75-85
We study the possibilities and limitations of the application of generalized Delaunay-like transformations (in the 6-dimensional
phase space) and TR-like mappings (in the 8-dimensional, extended phase space) to perturbed two-body problems with a time-varying
Keplerian parameter μ(t), that is, to Gylden-type systems. For the sake of theoretical completeness, both negative- and positive-energy
motion (with nonstationary coupling parameter) are, in principle, considered. Our developments are intended to introduce canonical
variables parallelling the classical ones of Delaunay and the Delaunay-Similar variables of Scheifele.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
7.
Otávio de Oliveira Costa Filho Wagner Sessin 《Celestial Mechanics and Dynamical Astronomy》1999,74(1):1-17
This paper is concerned with the extended Delaunay method as well as the method of integration of the equations, applied to
first order resonance. The equations of the transformation of the extended Delaunay method are analyzed in the (p + 1)/p type resonance in order to build formal, analytical solutions for the resonant problem with more than one degree of freedom.
With this it is possible to gain a better insight into the method, opening the possibility for more generalized applications.
A first order resonance in the first approximation is carried out, giving a better comprehension of the method, including
showing how to eliminate the ‘Poincaré singularity’ in the higher orders.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
8.
Two fully regular and universal solutions to the problem of spacecraft relative motion are derived from the Sperling–Burdet (SB) and the Kustaanheimo–Stiefel (KS) regularizations. There are no singularities in the resulting solutions, and their form is not affected by the type of reference orbit (circular, elliptic, parabolic, or hyperbolic). In addition, the solutions to the problem are given in compact tensorial expressions and directly referred to the initial state vector of the leader spacecraft. The SB and KS formulations introduce a fictitious time by means of the Sundman transformation. Because of using an alternative independent variable, the solutions are built based on the theory of asynchronous relative motion. This technique simplifies the required derivations. Closed-form expressions of the partial derivatives of orbital motion with respect to the initial state are provided explicitly. Numerical experiments show that the performance of a given representation of the dynamics depends strongly on the time transformation, whereas it is virtually independent from the choice of variables to parameterize orbital motion. In the circular and elliptic cases, the linear solutions coincide exactly with the results obtained with the Clohessy–Wiltshire and Yamanaka–Ankersen state-transition matrices. Examples of relative orbits about parabolic and hyperbolic reference orbits are also presented. Finally, the theory of asynchronous relative motion provides a simple mechanism to introduce nonlinearities in the solution, improving its accuracy. 相似文献
9.
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
10.
Within the framework of the Hamiltonian mechanics in the extended phase space, a set of canonical elements of the Delaunay type is developed in terms of an arbitrary independent angular variable. Application to the four classical anomalies-eccentric, true, elliptic, and mean-is presented. Particular attention is given to the generalized time equation and its conjugate energy equation. 相似文献
11.
Gabriel Mititelu 《Celestial Mechanics and Dynamical Astronomy》2009,103(4):327-342
A practical and important problem encountered during the atmospheric re-entry phase is to determine analytical solutions for
the space vehicle dynamical equations of motion. The author proposes new solutions for the equations of trajectory and flight-path
angle of the space vehicle during the re-entry phase in Earth’s atmosphere. Explicit analytical solutions for the aerodynamic
equations of motion can be effectively applied to investigate and control the rocket flight characteristics. Setting the initial
conditions for the speed, re-entering flight-path angle, altitude, atmosphere density, lift and drag coefficients, the nonlinear
differential equations of motion are linearized by a proper choice of the re-entry range angles. After integration, the solutions
are expressed with the Exponential Integral, and Generalized Exponential Integral functions. Theoretical frameworks for proposed
solutions as well as, several numerical examples, are presented. 相似文献
12.
Victor R. Bond 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):303-318
The Newtonian differential equations of motion for the two-body problem can be transformed into four, linear, harmonic oscillator equations by simultaneously applying the regularizing time transformation dt/ds=r and the Kustaanheimo-Stiefel (KS) coordinate transformation. The time transformation changes the independent variable from time to a new variables, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. This paper presents the derivation of uniform, regular equations for the perturbed twobody problem in the four-dimensional space. The variation of parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. These element differential equations are slightly modified by introducing two additional elements for the time to further improve long term stability of numerical integration.Originally presented at the AAS/AIAA Astrodynamics Specialists Conference, Vail, Colorado, July 1973 相似文献
13.
It is usually believed that we know everything to be known for any separable Hamiltonian system, i.e. an integrable system
in which we can separate the variables in some coordinate system (e.g. see Lichtenberg and Lieberman 1992, Regular and Chaotic Dynamics, Springer). However this is not always true, since through the separation the solutions may be found only up to quadratures,
a form that might not be particularly useful. A good example is the two-fixed-centers problem. Although its integrability
was discovered by Euler in the 18th century, the problem was far from being considered as completely understood. This apparent
contradiction stems from the fact that the solutions of the equations of motion in the confocal ellipsoidal coordinates, in
which the variables separate, are written in terms of elliptic integrals, so that their properties are not obvious at first
sight. In this paper we classify the trajectories according to an exhaustive scheme, comprising both periodic and quasi-periodic
ones. We identify the collision orbits (both direct and asymptotic) and find that collision orbits are of complete measure
in a 3-D submanifold of the phase space while asymptotically collision orbits are of complete measure in the 4-D phase space.
We use a transformation, which regularizes the close approaches and, therefore, enables the numerical integration of collision
trajectories (both direct and asymptotic). Finally we give the ratio of oscillation period along the two axes (the ‘rotation
number’) as a function of the two integrals of motion.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
14.
Bharat Mahajan Srinivas R. Vadali Kyle T. Alfriend 《Celestial Mechanics and Dynamical Astronomy》2018,130(3):25
A novel approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral and sectorial spherical harmonics is presented in this work. It is shown that the exact solution for the Delaunay normalization can be reduced to quadratures by the application of Deprit’s Lie-transform-based perturbation method. Two different series representations of the quadratures, one in powers of the eccentricity and the other in powers of the ratio of the Earth’s angular velocity to the satellite’s mean motion, are derived. The latter series representation produces expressions for the short-period variations that are similar to those obtained from the conventional method of relegation. Alternatively, the quadratures can be evaluated numerically, resulting in more compact expressions for the short-period variations that are valid for an elliptic orbit with an arbitrary value of the eccentricity. Using the proposed methodology for the Delaunay normalization, generalized expressions for the short-period variations of the equinoctial orbital elements, valid for an arbitrary tesseral or sectorial harmonic, are derived. The result is a compact unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes, which is nonsingular for the resonant orbits, and is closed-form in the eccentricity as well. The accuracy of the proposed theory is validated by comparison with numerical orbit propagations. 相似文献
15.
16.
For use in numerical studies of rotational motion, a set of elements is introduced for the torque-free rotational motion of a rigid body around its barycenter. The elements are defined as the initial values of a modification of the Andoyer canonical variables. A computational procedure is obtained for determining these elements from the combination of the spin angular momentum vector and a triad defining the orientation of the rigid body. A numerical experiment shows that the errors of transformation between the elements and variables are sufficiently small. The errors increase linearly with time for some elements and quadratically for some others. 相似文献
17.
E. Leimanis 《Astronomische Nachrichten》1976,297(3):141-143
The requirement that near a singular point of the equations of motion the power series expansions of the old variables in terms of the new ones start with second order terms leads to the transformation z = sin21/2w related to that of THIELE -BURRAU . Using this new transformation, a derivation of the regularized equations of motion is given. The original as well as the regularized equations of motion are of interest, for example, for calculating the initial values of the orbital elements for SCHWARZSCHILD's periodic solutions (LEIMANIS and OLUND 1972). 相似文献
18.
The solution to the motion of a satellite in an eccentric orbit and in resonance with the second-degree sectorial harmonic of the potential field is developed. The method of solution used parallels the well known von Zeipel method of general perturbations. The solution consists of expressions for the variations of the Delaunay variables. These expressions are composed of the perturbations developed by Brouwer in 1959 for the motion of an artificial satellite plus first-order perturbations due to the second-degree sectorial harmonic (in terms of the Legendre normal elliptic integrals of the first and second kind).This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration. 相似文献
19.
Simple Model of a Stochastically Excited Solar Dynamo 总被引：2，自引：0，他引：2
The aim of this paper is to investigate the dynamical nature of the complexity observed in the time evolution of the sunspot
number. We report a detailed analysis of the sunspot number time series, and use the daily records to build the phase space
of the underlying dynamical system. The observed features of the phase space prompted us to describe the global behavior of
the solar cycle in terms of a noise-driven relaxation oscillator. We find the equations whose solutions best fit the observed
series, which adequately describe the shape of the peaks and the oscillations of the system. The system of equations obtained
from this fitting procedure is shown to be equivalent to a truncation of the dynamo equations. A linear transformation maps
the phase space of these equations into the phase space reconstructed from the observations. The irregularities of the solar
cycle were modeled through the introduction of a stochastic parameter in the equations to simulate the randomness arising
in the process of eruption of magnetic flow to the solar surface. The mean values and deviations obtained for the periods,
rise times and peak values, are in good agreement with the values obtained from the sunspot time series. 相似文献
20.
The problem of the critical inclination is treated in the Hamiltonian framework taking into consideration post-Newtonian corrections as well as the main correction term of sectorial harmonics for an earth-like planet. The Hamiltonian is expressed in terms of Delaunay canonical variables. A canonical transformation is applied to eliminate short period terms. A modified critical inclination is obtained due to relativistic and the first sectorial harmonics corrections. 相似文献