稳定化方法与后稳定化方法的比较

对BaumgaLrte的稳定化和Chin的后稳定化进行了详尽讨论与数值比较．用经典数值方法并结合这两种稳定化方式都能提高数值精度和改善数值稳定性．在最佳稳定参数下稳定化精度一般不等价于后稳定化．两者精度优劣并无常定．考虑到Baumgarte的稳定化使得数值积分的右函数更复杂和增加计算耗费,尤其是存在稳定参数最佳选取的麻烦,故推荐后稳定化投入实算．但值得注意的是用后稳定化与没有经过稳定化处理的经典积分器来比不宜扩大积分步长．     

Numerical stabilization of Keplerian motion

The time transformation dt/ds=r is studied in detail and numerically stablized differential equations are obtained for =1,2, and 3/2. The case =1 corresponds to Baumgarte's results.     

Parabolic orbit determination. Comparison of the Olbers method and algebraic equations

In this paper, the Olbers method for the preliminary parabolic orbit determination (in the Lagrange–Subbotin modification) and the method based on systems of algebraic equations for two or three variables proposed by the author are compared. The maximum number of possible solutions is estimated. The problem of selection of the true solution from the set of solutions obtained both using additional equations and by the problem reduction to finding the objective function minimum is considered. The results of orbit determination of the comets 153P/Ikeya-Zhang and 2007 N3 Lulin are cited as examples.     

Analytical solutions for the equations of motion of a space vehicle during the atmospheric re-entry phase on a 2-D trajectory

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.     

An intermediate matching technique for solving two point boundary value problems using the perturbation method

The perturbation method, a numerical method for solving two point boundary value problems (TPBVP), is modified to attempt to improve inherent instability and sensitivity problems associated with the method. The desired solution to the TPBVP is divided into two time intervals. The differential equations required to define a solution to the two point boundary value problem are integrated independently over these shorter segments rather than consecutively over the entire trajectory. The independent integration of the differential equations over approximately half of the trajectory instead of the entire trajectory substantially decreases sensitivity and stability properties associated with the numerical integration. The equations for both time segments can be integrated simultaneously. By this procedure, a system of twice the dimension of the original problem is integrated for a period of time equal to half of the time interval for the original problem. To show the effectiveness of the method, two impulse trajectories which minimize the total velocity increment required to transfer a spacecraft from an Earth orbit into a lunar orbit are calculated.     

Asymptotic expansions in the perturbed two-body problem with application to systems with variable mass

The main theorems of the theory of averaging are formulated for slowly varying standard systems and we show that it is possible to extend the class of perturbation problems where averaging might be used. The application of the averaging method to the perturbed two-body problem is possible but involves many technical difficulties which in the case of the two-body problem with variable mass are avoided by deriving new and more suitable equations for these perturbation problems. Application of the averaging method to these perturbation problems yields asymptotic approximations which are valid on a long time-scale. It is shown by comparison with results obtained earlier that in the case of the two-body problem with slow decrease of mass the averaging method cannot be applied if the initial conditions are nearly parabolic. In studying the two-body problem with quick decrease of mass it is shown that the new formulation of the perturbation problem can be used to obtain matched asymptotic approximations.     

A stabilization procedure for the differential equations of the symmetric gyroscope including perturbing torques

This paper shows that for the free symmetric top a formulation of the equations of motion is possible, which is Liapunov stable. The formalism applied is equivalent to the conservative stabilization of the Keplerian problem. The perturbed problem appears in -stable form. This stabilization procedure could be useful in celestial mechanics, if the gyroscopic motion of a satellite is considered and one is interested in the exact position of the angles.     

Time transformations in the extended phase-space

Time transformations involving momenta in addition to the coordinates are studied from the points of view of stabilization and regularization of the equations of motion. The generalization of Sundman's transformation by using the potential function to transform the time is further generalized by using the Lagrangian function for the same purpose. The possibility of the stabilization of the equations of motion is investigated similarly to Stiefel's and Baumgarte's recent results but instead of a factorial, an additive control function is introduced in all equations of motion. The relation between the original and new independent variables is integrated by a modification of Ebert's theorem and it is shown that the new independent variable is Hamilton's principal function. Numerical examples illustrate the method and seem to indicate that the computation of close approach trajectories benefit especially by the transformations discussed. The Appendix offers an analytic treatment regarding the stabilization of the constant of energy.     

Radiative Transfer in Inhomogeneous Atmospheres. I

This series of papers is devoted to multiple scattering of light in plane parallel, inhomogeneous atmospheres. The approach proposed here is based on Ambartsumyan's method of adding layers. The main purpose is to show that one can avoid difficulties with solving various boundary value problems in the theory of radiative transfer, including some standard problems, by reducing them to initial value problems. In this paper the simplest one dimensional problem of diffuse reflection and transmission of radiation in inhomogeneous atmospheres with finite optical thicknesses is considered as an example. This approach essentially involves first determining the reflection and transmission coefficients of the atmosphere, which, as is known, are a solution of the Cauchy problem for a system of nonlinear differential equations. In particular, it is shown that this system can be replaced with a system of linear equations by introducing auxiliary functions P and S. After the reflectivity and transmissivity of the atmosphere are determined, the radiation field in it is found directly without solving any new equations. We note that this approach can be used to obtain the required intensities simultaneously for a family of atmospheres with different optical thicknesses. Two special cases of the functional dependence of the scattering coefficient on the optical thickness, for which the solutions of the corresponding equations can be expressed in terms of elementary functions, are examined in detail. Some numerical calculations are presented and interpreted physically to illustrate specific features of radiative transport in inhomogeneous atmospheres.     

Hamiltonian Dynamics of a Gyrostat in the N-Body Problem: Relative Equilibria

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.     

Numerical stabilization of the differential equations of Keplerian motion

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.     

Symplectic adaptive algorithm for solving nonlinear two-point boundary value problems in Astrodynamics

In this paper, from a Hamiltonian point of view, the nonlinear optimal control problems are transformed into nonlinear two-point boundary value problems, and a symplectic adaptive algorithm based on the dual variational principle is proposed for solving the nonlinear two-point boundary value problem. The state and the costate variables within a time interval are approximated by using the Lagrange polynomial and the costate variables at two ends of the time interval are taken as independent variables. Then, based on the dual variational principle, the nonlinear two-point boundary value problems are replaced by a system of nonlinear equations which can preserve the symplectic structure of the nonlinear optimal control problem. Furthermore, the computational efficiency of the proposed symplectic algorithm is improved by using the adaptive multi-level iteration idea. The performance of the proposed algorithm is tested by the problems of Astrodynamics, such as the optimal orbital rendezvous problem and the optimal orbit transfer between halo orbits.     

The critical layers and other singular regions in ideal hydrodynamics and magnetohydrodynamics

The governing singular differential equations are stated for certain systems in ideal hydrodynamics and magnetohydrodynamics. The solutions of these equations are studied in the neighborhood of the singular regions, and can be conveniently characterised by the roots of the corresponding indicial equation. The nature of the solution determines the nature of the dispersion relation (relating frequency and wavenumberk) and of the Green's function for the problem. The differences between the various classes of problems are discussed, and exemplified by considering the initial value problem for three cases: (a) unstratified shear flow, (b) stratified shear flow, and (c) a static magnetofluid. The latter case is typical of a number of problems of astrophysical interest, and possesses a rich mathematical and physical structure.     

Accurate computation of highly eccentric satellite orbits

For computing highly eccentric (e0.9) Earth satellite orbits with special perturbation methods, a comparison is made between different schemes, namely the direct integration of the equations of motion in Cartesian coordinates, changes of the independent variable, use of a time element, stabilization and use of regular elements. A one-step and a multi-step integration are also compared.It is shown that stabilization and regularization procedures are very helpful for non or smoothly perturbed orbits. In practical cases for space research where all perturbations are considered, these procedures are no longer so efficient. The recommended method in these cases is a multi-step integration of the Cartesian coordinates with a change of the independent variable defining an analytical step size regulation. However, the use of a time element and a stabilization procedure for the equations of motion improves the accuracy, except when a small step size is chosen.     

Stablization and time transformation

We suggest a simple stabilization technique to reduce the along-track error in the numerical Integration of the Lagrange's equations of motion. We also investigate the equations of motion of the two-body problem after applying the Sundman transformation dt = r^αds, both with and without introducing the energy integral. In both cases, we show how the stability of the equations varies with α and in the case with the energy integral, we show that every solution is a quasi-periodic function of s with two frequencies.     

Orbitally stable multistep methods

Lambert and Watson (1976) examine the family of symmetric linear multistep methods for the special second-order initial value problem, and connect the property of symmetry with a property of periodicity. The problems of celestial mechanics may be formulated as second-order initial value problems, but these frequently incorporate the first derivative explicity. It is common for such equations to be reduced to a system of first-order equations. Thus motivated, we utilize ideas from the aforementioned paper to determine the family of linear multistep methods for first-order initial value problems that possess an analogous property of periodicity. This family of orbitally stable methods is illustrated by examining the regularized equations of motion of an artificial earth satellite in an oblate atmosphere.     

Exact linearization of non-planar intermediary orbits in the satellite theory

In this paper we consider the reduction of the equations of motion for non-planar perturbed two body problems into linear form. It is seen that this can be easily accomplished for any element of the class of radial intermediaries to the satellite problem proposed by Deprit in 1981, since they have a functional dependence suitable for linearization. The transformation is worked out by using an adequate set of redundant variables. Four harmonic oscillators are obtained, of which two are coupled through gyroscopic terms. Their constant frequencies contain the secular contribution of the main problem of artificial satellite theory up to the order of the considered intermediary. Therefore, this result may well be interesting in relation to the study and prediction of accurate long-term solutions to satellite problems.     

On the use of the autonomous Birkhoff equations in Lie series perturbation theory

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.     

Analysis of some degenerate quadruple collisions

We consider the trapezoidal problem of four bodies. This is a special problem where only three degrees of freedom are involved. The blow up method of McGehee can be used to deal with the quadruple collision. Two degenerate cases are studied in this paper: the rectangular and the collinear problems. They have only two degrees of freedom and the analysis of total collapse can be done in a way similar to the one used for the collinear and isosceles problems of three bodies. We fully analyze the flow on the total collision manifold, reducing the problem of finding heteroclinic connections to the study of a single ordinary differential equation. For the collinear case, from which arises a one parameter family of equations, the analysis for extreme values of the parameter is done and numerical computations fill up the gap for the intermediate values. Dynamical consequences for possible motions near total collision as well as for regularization are obtained.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.Dedicated to Prof. Szebehely on the occasion of his sixtieth birthday.     

Equivalent problems in rigid body dynamics—II

The problem of motion of a dynamically symmetric gyrostat acted upon by non-symmetric potential forces admitting a cyclic integral is considered. This problem is brought into equivalence with another one concerning the motion of a similar gyrostat under the action of axisymmetric potential forces. Using this analogy, new integrable cases of each of the two problems are obtained from the known cases of the other. The equations of motion are reduced to a single equation of the second order.