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It is shown that each lobe of the figure-eight orbit is star-shaped, which indicates that the corresponding polar angle is monotone. In general, it is not clear under which assumptions a trajectory in a minimizer is star-shaped. In this paper, we study minimizers connecting two fixed-ends (i.e. the Bolza problem) in the planar three-body problem with two equal masses. We show that if the Jacobi coordinates of the two fixed-ends are both orthogonal, then for any minimizer, both Jacobi vectors describe a star-shaped curve. If the Jacobi coordinates are orthogonal only on one of the fixed-ends and they are in adjacent closed quadrants, then their polar angles have at most one critical point. As an application, we show the existence and prove some geometric properties of two families of periodic orbits. [ABSTRACT FROM AUTHOR]

This work studies periodic orbits as action minimizers in the spatial isosceles three-body problem with mass . In each period, the body with mass moves up and down on a vertical line, while the other two bodies have the same mass , and rotate about this vertical line symmetrically. For given , such periodic orbits form a one-parameter set with a rotation angle as the parameter.Two new phenomena are found for this set. First, for each , this set of periodic orbits bifurcate from a circular Euler (central configuration) orbit to a Broucke (collision) orbit as increases from to . There exists a critical rotation angle , where the orbit is a circular Euler orbit if ; a spatial orbit if ; and a Broucke (collision) orbit if . The exact formula of is numerically proved to be . Second, oscillating behaviors occur at rotation angle for all . Actually, the orbit with runs on its initial periodic shape for only a few periods. It breaks the first periodic shape and becomes irregular in a moment. However, it runs close to a different periodic shape after a while. In a short time, it falls apart from the second periodic shape and runs irregularly again. Such oscillation continues as time increases. Up to , the orbit is bounded and keeps oscillating between periodic shapes and irregular motions. Further study implies that, for each , the angle between any two consecutive periodic shapes is a constant. When , similar oscillating behaviors are expected. [ABSTRACT FROM AUTHOR]

In the three-body problem, it is known that there exists a special set of periodic orbits: spatial isosceles periodic orbits. In each period, one body moves up and down along a straight line, and the other two bodies rotate around this line. In this work, we revisit this set of orbits by applying variational method. Two unexpected phenomena are discovered. First, this set is not always spatial. It actually bifurcates from the circular Euler (central configuration) orbit to the Broucke (collision) orbit. Second, one of the orbits in this set encounters an oscillating behavior. By running its initial condition, the orbit stays periodic for only a few periods before it becomes irregular. However, it moves close to another periodic shape in a while. Shortly it falls apart again and starts running close to a third periodic shape after a moment. This oscillation continues as t increases. Actually, up to t = 1.2 × 105, the orbit is bounded and keeps oscillating between periodic shapes and irregular motions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the existence and linear stability of the Broucke periodic orbit in the planar three-body problem. In each period of this orbit, there are two binary collisions (or BC for short) between the outer bodies, while the inner body reaches its minimum or maximum at the time of each BC. A surprising simple existence proof of this orbit is given. The initial condition of this orbit is shown to be a supremum of some well-chosen set. The linear stability is then analyzed by Robertsʼ symmetry reduction method. It is shown that the Broucke periodic orbit with equal masses is linearly stable. [Copyright &y& Elsevier]

This paper gives an analytic existence proof of the Schubart periodic orbit with arbitrary masses, a periodic orbit with singularities in the collinear three-body problem. A 'turning point' technique is introduced to exclude the possibility of extra collisions and the existence of this orbit follows by a continuity argument on differential equations generated by the regularized Hamiltonian. [ABSTRACT FROM AUTHOR]