The Global Phase Structure of the Three Dimensional Isosceles Three Body Problem with Zero Energy

Meyer, Kenneth R.; Wang, Quidong

doi:10.1007/978-1-4613-8448-9_18

Cited by 4 publications

(2 citation statements)

References 3 publications

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“…He used two spatial isosceles 3-body problems to prove that five bodies can escape to infinity in a finite time without collision. Other works on the spatial isosceles 3-body problem are [16] and the references therein. If in the spatial isosceles 3-body problem the initial positions and velocities of the three bodies are contained in a plane, then the motion remains always in this plane, and we have the so-called planar isosceles 3-body problem.…”

Section: Ams Subject Classifications 70f15 37n05mentioning

confidence: 99%

Families of Periodic Orbits for the Spatial Isosceles 3-Body Problem

Corbera¹,

Llibre²

2004

SIAM J. Math. Anal.

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Abstract. 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 Poincaré's continuation method.Key words. periodic orbits, quasi-periodic orbits, 3-body problem, analytic continuation method AMS subject classifications. 70F15, 37N05DOI. 10.1137/S0036141002407880 1. Introduction. We consider a special case of the spatial 3-body problem, the spatial isosceles 3-body problem, or simply the isosceles problem. This problem consists of describing the motion of two equally massive bodies, m 1 = m 2 = 1/2, having initial conditions and velocities symmetric with respect to a straight line which passes through their center of mass, and a third body, with mass m 3 = µ, having initial position and velocity on this straight line. This problem is called the isosceles problem because the three bodies form an isosceles triangle at any time, eventually degenerated to a segment.The most interesting application of the spatial isosceles 3-body problem was given by Xia in [25]. He used two spatial isosceles 3-body problems to prove that five bodies can escape to infinity in a finite time without collision. Other works on the spatial isosceles 3-body problem are [16] and the references therein. If in the spatial isosceles 3-body problem the initial positions and velocities of the three bodies are contained in a plane, then the motion remains always in this plane, and we have the so-called planar isosceles 3-body problem. There are several papers about the planar isosceles 3-body problem, for instance, [9], [17], etc.When the third body of the isosceles 3-body problem has infinitesimal mass (i.e., µ = 0) then we obtain the restricted is...

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Section: Ams Subject Classifications 70f15 37n05mentioning

confidence: 99%

Families of Periodic Orbits for the Spatial Isosceles 3-Body Problem

Corbera¹,

Llibre²

2004

SIAM J. Math. Anal.

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“…The youtube video https://www.youtube.com/watch?v=fSmQyeKcj5k shows some images of a periodic spatial solution. Due to the importance and several applications that the spatial three body configuration offers, nowadays there is a large numbers of papers devoted to this problem, see for example [1,2,10,12,13,15,17,18] and specially [7] and the references therein. In all cited papers, the existence of periodic spatial isosceles solutions has been considered using variational methods, numerical methods or techniques of analytical continuation.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation of periodic orbits for the $N$-body problem, from a non geometrical family of solutions

Perdómo¹,

Rivera²,

Suárez³

2020

Preprint

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Given two positive real numbers M and m and an integer n > 2, it is well known that we can find a family of solutions of the (n + 1)-body problem where the body with mass M stays put at the origin and the other n bodies, all with the same mass m, move on the x-y plane following ellipses with eccentricity e. It is expected that this geometrical family that depends on e, has some bifurcations that produces solutions where the body in the center moves on the z-axis instead of staying put in the origin. By doing analytic continuation of a periodic numerical solution of the 4-body problem -the one displayed on the video http://youtu.be/2Wpv6vpOxXk -we surprisingly discovered that the origin of this periodic solution is not part of the geometrical family of elliptical solutions parametrized by the eccentricity e. It comes from a not so geometrical but easier to describe family. Having notice this new family, the authors find an exact formula for the bifurcation point in this new family and use it to show the existence of non planar periodic solution for any pair of masses M , m and any integer n. As a particular example we find a solution where three bodies with mass 3 move around a body with mass 7 that moves up and down.

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