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# Families of periodic orbits for the spatial isosceles 3-body problem

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2004

Enllaç permanent: http://hdl.handle.net/10854/1900

Versió de l'editor: https://doi.org/10.1137/S0036141002407880

We study the families of periodic orbits of the spatial isosceles 3-body problem (for
small enough values of the mass lying on the symmetry axis) coming via the analytic continuation
method from periodic orbits of the circular Sitnikov problem. Using the first integral of the angular
momentum, we reduce the dimension of the ...»»»»

We study the families of periodic orbits of the spatial isosceles 3-body problem (for
small enough values of the mass lying on the symmetry axis) coming via the analytic continuation
method from periodic orbits of the circular Sitnikov problem. Using the first integral of the angular
momentum, we reduce the dimension of the phase space of the problem by two units. Since periodic
orbits of the reduced isosceles problem generate invariant two-dimensional tori of the nonreduced
problem, the analytic continuation of periodic orbits of the (reduced) circular Sitnikov problem
at this level becomes the continuation of invariant two-dimensional tori from the circular Sitnikov
problem to the nonreduced isosceles problem, each one filled with periodic or quasi-periodic orbits.
These tori are not KAM tori but just isotropic, since we are dealing with a three-degrees-of-freedom
system. The continuation of periodic orbits is done in two different ways, the first going directly from
the reduced circular Sitnikov problem to the reduced isosceles problem, and the second one using
two steps: first we continue the periodic orbits from the reduced circular Sitnikov problem to the
reduced elliptic Sitnikov problem, and then we continue those periodic orbits of the reduced elliptic
Sitnikov problem to the reduced isosceles problem. The continuation in one or two steps produces
different results. This work is merely analytic and uses the variational equations in order to apply
Poincar´e’s continuation method.^^^^

Article

(c) Society for Industrial and Applied Mathematics, 2003

CORBERA SUBIRANA, Montserrat; LLIBRE, Jaume. "Families of periodic orbits for the spatial isosceles 3-body problem". A: Siam Journal on Mathematical Analysis, 2003, vol. 35, núm. 5, pàg. 1311-1346. DOI. 10.1137/S0036141002407880