Restricted Three-Body Problem: An Approximation of its General Solution – Part One – the Manifold of Symmetric Periodic Solutions     

Kostantinos Papadakis  Constantine Goudas 《Astrophysics and Space Science》2006,305(2):99-124

This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x ₀ ∊ [−2,2],C ∊ [−2,5]) of the (x ₀, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x ₀ ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x ₀, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families.  相似文献   

贵港市矿业发展现状与对策     

李东昌 《南方国土资源》2002,15(1):57-60

文章从贵港市矿产资源情况出发 ,通过对开发现状及存在问题的分析 ,探讨保证资源的合理开发和永续利用 ,为经济建设提供物质支撑的理念 ,提出了矿业可持续发展的对策。  相似文献   

用遗传算法解算病态方程   总被引：7，自引：1，他引：6  

曾群意  欧吉坤 《大地测量与地球动力学》2003,23(3):93-97

对应用遗传算法解决病态方程问题进行了探讨。利用拟合法而不是通过法方程求解参数，从而避免了法方程系数求逆，使病态方程的解答有了较好的结果。通过模拟计算并和其他方法进行比较，证明该方法是可行的和有效的。  相似文献   

Non-Existence of New Meromorphic First Integrals in the Planar Three-Body Problem     

Alexei V. Tsygvintsev 《Celestial Mechanics and Dynamical Astronomy》2003,86(3):237-247

We consider the planar three-body problem and prove that, apart from some exceptional cases, there is no additional first integral meromorphic with respect to positions, mutual distances and momenta.  相似文献   

Preliminary Study on the Translunar Halo Orbits of the Real Earth–Moon System     

Miquel A. Andreu 《Celestial Mechanics and Dynamical Astronomy》2003,86(2):107-130

The computation of translunar Halo orbits of the real Earth–Moon system (REMS) has been an open problem for a long time, but now, it is possible to compute Halo orbits of the REMS in a systematic way. In this paper, we describe the method used for the numerical computation of Halo orbits for a time span longer than 41 years. Halo orbits of the REMS are computed from quasi-periodic Halo orbits of the quasi-bicircular problem (QBCP). The QBCP is a model for the dynamics of a spacecraft in the Earth–Moon–Sun system. It is a Hamiltonian system with three degrees of freedom and depending periodically on time. In this model, Earth, Moon and Sun are moving in a self-consistent motion close to bicircular. The computed Halo orbits of the REMS are compared with the family of Halo orbits of the QBCP. The results show that the QBCP is a good model to understand the main features of the Halo family of the REMS.  相似文献   

Virial oscillations of celestial bodies: V. The structure of the potential and kinetic energies of a celestial body as a record of its creation history     

V. I. Ferronsky  S. A. Denisik  S. V. Ferronsky 《Celestial Mechanics and Dynamical Astronomy》1996,64(3):167-183

The physical meaning of the terms of the potential and kinetic energy expressions, expanded by means of the density variation function for a nonuniform self-gravitating sphere, is discussed. The terms of the expansions represent the energy and the moment of inertia of the uniform sphere, the energy and the moment of inertia of the nonuniformities interacting with the uniform sphere, and the energy of the nonuniformities interacting with each other. It follows from the physical meaning of the above components of the energy structure, and also from the observational fact of the expansion of the Universe that the phase transition, notably, fusion of particles and nuclei and condensation of liquid and solid phases of the expanded matter accompanied by release of energy, must be the physical cause of initial thermal and gravitational instability of the matter. The released kinetic energy being constrained by the general motion of the expansion, develops regional and local turbulent (cyclonic) motion of the matter, which should be the second physical effect responsible for the creation of celestial bodies and their rotation.  相似文献   

Diffuse radiation,twilight, and photochemistry — I     

D. J. Lary  J. A. Pyle 《Journal of Atmospheric Chemistry》1991,13(4):373-392

A photochemical scheme which includes a detailed treatment of multiple scattering up to solar zenith angles of 96° (developed for use in a GCM) has been used to study partitioning within chemical families. Attention is drawn to the different zenith angle dependence of diffuse radiation for the two spectral regions <310 nm and >310 nm. The effect that this has on the so-called 40 km ozone problem is discussed. The importance of correctly including multiple scattering for polar ozone studies is emphasised.  相似文献   

Mass heterogeneities and convection in the earth's mantle inferred from gravity and core-mantle boundary irregularities   总被引：1，自引：0，他引：1  

Ondřej Čadek  Ctirad Matyska 《Pure and Applied Geophysics》1991,135(1):107-123

Mass heterogeneities in the earth's mantle are retrieved from the gravity data and the topography of the core-mantle boundary as well as the topography of the earth's surface. A mantle circulation induced by the heterogeneities is modelled by solving the Stokes problem for incompressible Newtonian fluid. The derived models of mantle motions correlate well with the plate tectonics and point at a close relation between the surface tectonic activity and the processes in the vicinity of the core-mantle boundary.  相似文献   

中国南部二叠、三叠纪“混生生物群”及二叠、三叠系界线     

何锡麟  沈树忠 《吉林大学学报(地球科学版)》1991,(2)

本文系统描述了中国南部二叠、三叠纪“混生生物群”。根据笔者的统计,目前巳经发现有“二叠纪型”分子36属53种与三叠纪菊石、双壳类、牙形剌共生。 笔者认为中国南部可以以牙形剌Anchignathodus parvus 的出现作为三叠系的底界,双壳类Pteria ussurica variabilis,菊石 Hypophiceras的出现可以作为三叠系开始的辅助标志。文中还将这一界线与世界上其它地区的二叠、三叠系界线进行了对比。  相似文献   

Palaeoenvironmental significance of Late Permian palaeosols in the South‐Eastern Iberian Ranges,Spain     

RAÚL DE LA HORRA  Ma ISABEL BENITO  JOSE LÓPEZ‐GÓMEZ  ALFREDO ARCHE  JOSÉ F. BARRENECHEA  JAVIER LUQUE 《Sedimentology》2008,55(6):1849-1873

The Late Permian (Wuchiapingian) Alcotas Formation in the SE Iberian Ranges consists of one red alluvial succession where abundant soil profiles developed. Detailed petrographical and sedimentological studies in seven sections of the Alcotas Formation allow six different types of palaeosols, with distinctive characteristics and different palaeogeographical distribution, to be distinguished throughout the South‐eastern Iberian Basin. These characteristics are, in turn, related to topographic, climatic and tectonic controls. The vertical distribution of the palaeosols is used to differentiate the formation in three parts from bottom to top showing both drastic and gradual vertical upwards palaeoenvironmental changes in the sections. Reconstruction of palaeoenvironmental conditions based on palaeosols provides evidence for understanding the events that occurred during the Late Permian, some few millions of years before the well‐known Permian‐Triassic global crisis.  相似文献