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

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

Oct 2006

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006ap%26ss.305...99p&link_type=abstract

Astrophysics and Space Science, Volume 305, Issue 2, pp.99-124

Astronomy and Astrophysics

Astrophysics

3

Planar Restricted Problem, Periodic Solutions, Family Curves, Basic Families, Escape Solutions, Asymptotic Solutions, Heteroclinic Solutions, Completeness Criterion, Partition Theorem, Approximate General Solution

Scientific paper

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 ɛ [-2,2], C ɛ [-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 ɛ [-6,2], C ɛ [-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.

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