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

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.     

Crustal and upper mantle structures of the brazilian coast

The Rayleigh wave phase and group velocities in the period range of 24–39 sec, obtained from two earthquakes which occurred in northeastern brazil and which were recorded by the Brazilian seismological station RDJ (Rio de Janeiro), have been used to study crustal and upper mantle structures of the Brazilian coastal region. Three crustal and upper mantle models have been tried out to explain crustal and upper mantle structures of the region. The upper crust has not been resolved, due basically to the narrow period range of the phase and group velocities data. The phase velocity inversions have exhibited good resolutions for both lower crust and upper mantle, with shear wave velocities characteristic of these regions. The group velocity data inversions for these models have showed good results only for the lower crust. The shear wave velocities of the lower crust (3.86 and 3.89 km/sec), obtained with phase velocity inversions, are similar to that (=3.89 km/sec) found byHwang (1985) to the eastern South American region, while group velocity inversions have presented shear velocity (=3.75 km/sec) similar to that (=3.78 km/sec) found byLazcano (1972) to the Brazilian shield. It was not possible to define sharply the crust-mantle transition, but an analysis of the phase and group velocity inversions results has indicated that the total thickness of the crust should be between 30 and 39 km. The crustal and upper mantle model, obtained with phase velocity inversion, can be used as a preliminary model for the Brazilian coast.     

基于Poulos弹性理论的被动桩改进算法

传统的Poulos弹性理论仅适合于均质土中土体侧向位移时桩的性状分析,无法考虑土的层状特性。通过引入层状地基中作用一水平集中力的广义Mindlin解和地面作用有竖向荷载时的应力和位移通解,对Poulos方法进行了改进,使之扩展到多层土中,还用于研究堆载条件下的被动桩变形和受力响应。算例分析表明,改进弹性理论要比Poulos方法更为严密、合理,提高了计算精度,应用范围也更广。     

Modelling of periodic wave transformation in the inner surf zone

In this paper, we analyse the ability of the nonlinear shallow-water (NSW) equations to predict wave distortion and energy dissipation of periodic broken waves in the inner surf zone. This analysis is based on the weak-solution theory for conservative equations. We derive a new one-way model, which applies to the transformation of non-reflective periodic broken waves on gently sloping beaches. This model can be useful to develop breaking-wave parameterizations (in particular broken-wave celerity expression) in both time-averaged wave models and time-dependent Boussinesq-type models. We also derive a new wave set-up equation which provides a simple and explicit relation between wave set-up and energy dissipation. Finally, we compare numerical simulations of both, the NSW model and the simplified one-way model, with spilling wave breaking experiments and we find a good agreement.     

水溶液中聚合物在粘土上的吸附

水溶液中聚合物在粘土上的吸附对于聚合物污染防治和污水处理等都有重要意义。Scheutjens-Fleer理论是应用范围最广的聚合物吸附理论,适应于从单体到长链高聚物,从链无相互作用的稀溶液到纯聚合物的吸附研究。本文简要介绍了有关聚合物吸附的重要参数,包括吸附量、耦合分数或直接表面覆盖、平均层厚度和链段密度,以及相应的测定方法。文中分析了影响聚合物吸附研究发展的主要问题,指出水溶液中聚合物在粘土上的吸附研究的发展方向是综合研究吸附等温线和A/V效应、耦合分数、吸附层厚度、吸附分级现象和对粘土层间的影响,并与Scheutjens-Fleer理论计算结果进行比较。     

沃克环流上升区赤道低空强西风带的分析

采用ＴＯＧＡ／ＣＯＡＲＥ国际合作考察期间（１９９２年１０月─１９９３年２月）获得的２６７次定点（２°Ｓ，１５５°Ｅ）定时高空大气探测资料，进行计算分析，发现１９９２年１２月─１９９３年２月低空存在一支西风急流，有两次分别持续半月之久，而１９９２年１１月的两次赤道西风急流，因无赤道高空急流配合，仅持续２ｄ便消失。本文还指出：（１）赤道低空西风急流是各种海－气指数产生ＥＮＳＯ异常的重要信息；（２）赤道高空急流有滞后赤道低空急流２ｄ左右的响应关系；（３）赤道低空急流是纬向水汽输送的狭窄通道。     

四阶奇异边值问题无穷多个正解存在性的一个注记

本文考虑四阶奇异边值问题x( 4 ) =λa(t) f (t,x(t) ) ,0 相似文献   

Hammerstein型非线性积分方程解的存在性

该文采用化为积分方程组的方法 ,利用锥上不动点指数计算 ,在不要求非线性项 f (x,u)非负的情况下 ,证明 Hammerstein型非线性积分方程 φ(x) =∫Gκ(x,y) f (y,φ(y) ) dy非平凡解和多解存在性的一些新的结果。此结果可用来证明非线性常微分方程两点边值问题解的存在性。     

On the radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas

The radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas of finite depth is investigated. The analytical expressions for the radiated and diffracted potentials are derived as infinite series by use of the method of separation of variables. The unknown coefficients in the series are determined by the eigenfunction expansion matching method. The expressions for wave forces, hydrodynamic coefficients and reflection and transmission coefficients are given and verified by the boundary element method. Using the present analytical solution, the hydrodynamic influences of the angle of incidence, the submergence, the width and the thickness of the structure on the wave forces, hydrodynamic coefficients, and reflection and transmission coefficients are discussed in detail.