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.     

Motions in the field of two rotating magnetic dipoles. III: Zero-velocity curves and surfaces

The zero-velocity curves and surfaces created by two parallel or antiparallel magnetic dipoles moving on concentric circles are computed. It is found that for all combinations of values of the mass parameter µ and the ratio of the magnitudes of the magnetic moments of the two dipoles, there exist closed regions inside of which charged particles are permanently trapped.     

Hydromagnetic Rayleigh problem for a porous wall in slip flow regime

An analysis of the Rayleigh problem in MHD for a porous wall in a slip flow regime is considered. The normal velocity of suction/injection at the wall is assumed to be time dependent. The solution of the problem is obtained in the form of a power series, in terms of known functions. The variations of the velocity field and the skin friction are shown graphically and are followed by a quantitative discussion.Science Update, The Pioneer, India, 21 July 1995 edition     

Sliding element simulating the response of slip surfaces and its application for the prediction of earthquake-induced landslide movement using one-dimensional dynamic analyses

Landslides - Earthquake-induced landslides involve excessive movement of slopes, usually along slip surfaces. This seismic movement of slopes may depend crucially on (a) the soil response along the...     

Seismic Displacement of Gravity Walls by a Two-body Model

Gravity walls retaining dry soil are modeled as a system of two bodies: (a) the gravity wall that slides along the wall-foundation soil boundary and (b) the critical soil wedge in the soil behind the wall. The strength of the system is defined by both the frictional and the cohesional components of resistance. The angle of the prism of the critical soil wedge behind the wall is obtained using the limit equilibrium method. The model accounts for changes in the geometry of the backfill soil behind the wall by considering the displacements at the end of each time step under limit equilibrium. The model shows that the standard (single) block model is over-conservative for the extreme case of critical-to-applied-seismic acceleration ratios less than about 0.30, but works well for cases where this ratio ranges between 0.5 and 0.8. The model is applied to predict the seismic displacement of gravity walls (a) tested in the shaking-table and (b) studied numerically by elaborate elasto–plastic analyses.     

Aegean Sea ridge barrier-and-basin sedimentation patterns

An isopach map of the Aegean Sea, based on a regionally comprehensive seismic-profile network, reveals the highly irregular distribution of unconsolidated sediments of post-Miocene age. Geologically recent structural activity has considerably modified the seafloor configuration of the north and south Aegean, and depositional patterns are to varying degree related to the complex Aegean physiography. From north to south, a series of ridges, islands, and plateaus have acted as effective barriers behind which sediments are trapped, primarily in depressions. Sources of sediment in the north Aegean troughs and basins include rivers and suspensate-rich water masses; material was also provided by the erosion of plateaus and ridges during phases of Pliocene uplift and regressions and Quaternary eustatic low sea-level stands when the Aegean became virtually land-locked and isolated from the Black Sea. Volcanic as well as terrigenous material has accumulated in the central and south Aegean. However, the rapid depositional rates in the south Aegean are more closely related to the flow exchange with the Levantine Basin and the consequent ponding of material behind the Peloponnesus-Crete and Crete-Rhodes ridges.     

Spatio-temporal network analysis for studying climate patterns

A fast, robust and scalable methodology to examine, quantify, and visualize climate patterns and their relationships is proposed. It is based on a set of notions, algorithms and metrics used in the study of graphs, referred to as complex network analysis. The goals of this approach are to explain known climate phenomena in terms of an underlying network structure and to uncover regional and global linkages in the climate system, while comparing general circulation models outputs with observations. The proposed method is based on a two-layer network representation. At the first layer, gridded climate data are used to identify “areas”, i.e., geographical regions that are highly homogeneous in terms of the given climate variable. At the second layer, the identified areas are interconnected with links of varying strength, forming a global climate network. This paper describes the climate network inference and related network metrics, and compares network properties for different sea surface temperature reanalyses and precipitation data sets, and for a small sample of CMIP5 outputs.     

Classification and detection of dominant factors in geospatial patterns of traditional settlements in China

Journal of Geographical Sciences - The geospatial distribution pattern in traditional Chinese settlements (TCSs) reflects the traditional harmony between humans and nature, which has been learned...     

The general solution of the HENON–HEILES problem

The general solution of the Henon–Heiles system is approximated inside a domain of the (x, C) of initial conditions (C is the energy constant). The method applied is that described by Poincaré as ‘the only “crack” permitting penetration into the non-integrable problems’ and involves calculation of a dense set of families of periodic solutions that covers the solution space of the problem. In the case of the Henon–Heiles potential we calculated the families of periodic solutions that re-enter after 1–108 oscillations. The density of the set of such families is defined by a pre-assigned parameter ε (Poincaré parameter), which ascertains that at least one periodic solution is computed and available within a distance ε from any point of the domain (x, C) for which the approximate general solution computed. The approximate general solution presented here corresponds to ε = 0.07. The same solution is further improved by “zooming” into four square sub-domain of (x, C), i.e. by computing sufficient number of families that reduce the density parameter to ε = 0.003. Further zooming to reduce the density parameter, say to ε = 10⁻⁶, or even smaller, although easily performable in both areas occupied by stable as well as unstable solutions, was found unnecessary. The stability of all members of each and all families computed was calculated and presented in this paper for both the large solution domain and for the sub-domains. The correspondence between areas of the approximate general solution occupied by stable periodic solutions and Poincaré sections with well-aligned section points and also correspondence between areas occupied by unstable solutions and Poincaré sections with randomly scattered section points is shown by calculating such sections. All calculations were performed using the Runge-Kutta (R-K) 8th order direct integration method and the large output received, consisting of many thousands of families is saved as “Atlas of the General Solution of the Henon–Heiles Problem,” including their stability and is available at request. It is concluded that approximation of the general solution of this system is straightforward and that the chaotic character of its Poincaré sections imposes no limitations or difficulties.     

New types of motion in Störmer's problem

Application of boundary value techniques in the case of an electron performing relativistic motions within a magnetic dipole such as that of the Earth supplemented by a scanning process by means of which the entire phase space of the problem can be investigated, six new types of periodic motion have been discovered and computed. The stability of these motions is investigated and their direct bearing on formation and shape of the Van Allen zones of the Earth is discussed.