Abstract. Hip-Hop solutions of the 2N -body problem with equal masses are shown to exist using an analytic continuation argument. These solutions are close to planar regular 2N -gon relative equilibria with small vertical oscillations. For fixed N , an infinity of these solutions are three-dimensional choreographies, with all the bodies moving along the same closed curve in the inertial frame.
In this paper we analyse the ejection-collision (EC) orbits of the planar restricted three body problem. Being µ ∈ (0, 0.5] the mass parameter, and taking the big (small) primary with mass 1 − µ (µ), an EC orbit will be an orbit that ejects from the big primary, does an excursion and collides with it. As it is well known, for any value of the mass parameter µ ∈ (0, 0.5] and sufficiently restricted Hill regions (that is, for big enough values of the Jacobi constant C), there are exactly four EC orbits. We check their existence and extend numerically these four orbits for µ ∈ (0, 0.5] and for smaller values of the Jacobi constant. We introduce the concept of n-ejection-collision orbits (n-EC orbits) and we explore them numerically for µ ∈ (0, 0.5] and values of the Jacobi constant such that the Hill bounded possible region of motion contains the big primary and does not contain the small one. We study the cases 1 ≤ n ≤ 10 and we analyse the continuation of families of such n-EC orbits, varying the energy, as well as the bifurcations that appear.
We consider a non-autonomous system of ordinary differential equations. Assume that the time dependence is periodic with a very high frequency 1/ε, where ε is a small parameter and differentiability with respect to the parameter is lost when ε equals zero. We derive from Arenstorf's implicit function theorem a set of conditions to show the existence of periodic solutions. These conditions look formally like the standard analytic continuation method, namely, checking that a certain minor does not vanish. We apply this result to show the existence of a new class of periodic orbits of very large radii in the three-dimensional elliptic restricted three-body problem for arbitrary values of the masses of the primaries.
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