The elliptic isosceles restricted three-body problem with collision, is a restricted three-body problem where the primaries move having consecutive elliptic collisions and the infinitesimal mass is moving in the plane perpendicular to the primaries motion that passes through the center of mass of the primary system. Our purpose in this paper is to prove the existence of many families of periodic solutions using Continuation’s method, where the perturbing parameter is related with the energy of the primaries. This work is merely analytic and uses symmetry conditions and appropriate coordinates.
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References
Alvarez M., Llibre J. (1998). Restricted three-body problems and the non-regularization of the eliptic collision restricted isosceles three-body problem. Extracta Mathematicae 13(1): 73–94
Alvarez M., Llibre J. (2001). Heteroclinic orbits and Bernoulli shift for the elliptic collision restricted three-body problem. Arch. Rational Mech. Anal. 156(4): 317–357
Arenstorf R. (1966). A new method of perturbation theory and its application to the satellite problem of celestial mechanics. J. Reine Angew. Math. 221: 113–145
Cors J., Pinyol C., Soler J. (2001). Periodic solutions in the spatial elliptic restricted three-body problem. Physica D 154: 195–206
Cors J., Pinyol C., Soler J. (2005). Analytic continuation in the case of non-regular dependency on a small parameter with an application to celestial mechanics. Physica D 219: 1–19
Llibre J., Pasca D. (2006). Periodic orbits of the planar collision restricted 3-body problem. Celest. Mech. 96(1): 19–29
Martínez J., Orellana R. (1997). Orbit’s structure in the isosceles rectilinear restricted three-body problem. Celes. Mech. Dyn. Astron. 67: 275–291
Meyer, K., and Hall, G. (1992). Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Springer-Verlag.
Meyer K. (1999). Periodic Solutions of the N-body Problem, Lecture Notes in Math., 1719. Springer-Verlag, Berlin
Meyer K., Howison Clarissa R. (2000). Doubly-symmetric periodic solutions in the spatial restricted three-body problem. J. Diff. Eqs. 163: 174–197
Puel F. (1979). Numerical study of the rectilinear isosceles restricted problem. Celest. Mech. 20(2): 105–123
Schmidt D. (1972). Families of periodic orbits in the restricted problem of three bodies connecting families of direct and retrograde orbits. SIAM, J. Appl. Math. 1: 27–37
Stiefel, E. L., and Scheifele, G. (1971). Linear and Regular Celestial Mechanics, Springer-Verlag.
Szebehely V. (1967). Theory of Orbits. Academic Press, New York
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Partially supported by Dirección de Investigación UBB, 064608 3/RS.
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de Fatima Brandão, L., Vidal, C. Periodic Solutions of the Elliptic Isosceles Restricted Three-body Problem with Collision. J Dyn Diff Equat 20, 377–423 (2008). https://doi.org/10.1007/s10884-007-9080-3
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DOI: https://doi.org/10.1007/s10884-007-9080-3
Key words
- Restricted three-body problem
- isosceles restricted three-body problem
- periodic solutions
- symmetries
- continuation’s method