In the n-body problem a central configuration is formed if the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration vector. We consider the problem: given a collinear configuration of four bodies, under what conditions is it possible to choose positive masses which make it central. We know it is always possible to choose three positive masses such that the given three positions with the masses form a central configuration. However for an arbitrary configuration of four bodies, it is not always possible to find positive masses forming a central configuration. In this paper, we establish an expression of four masses depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically we show that there is a compact region in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the compact region, it is always possible to choose positive masses to make the configuration central.

h i g h l i g h t s • Develop variational method with structural prescribed boundary conditions (SPBC). • This method numerically finds solutions and periodic solutions for N-body problem. • This method analytically proves the existence of such solutions numerically found. • Stable star pentagon choreography are analytically proved and numerically found. • Many stable choreographic and other periodic solutions are discovered and proved. Communicated by B. Sandstede Keywords: Variational method Choreographic periodic solutions Variational method with structural prescribed boundary conditions (SPBC) Stability Central configurations n-body problem a b s t r a c tIn this paper, we give a rigorous proof of the existence of infinitely many simple choreographic solutions in the classical Newtonian 4-body problem. These orbits are discovered by a variational method with structural prescribed boundary conditions (SPBC). This method provides an initial path that is obtained by minimizing the Lagrangian action functional over the SPBC. We prove that the initial path can be extended to a periodic or quasi-periodic solution. With computer-assistance, a family of choreographic orbits of this type is shown to be linearly stable. Among the many linearly stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution. We also prove the existence of infinitely many double choreographic periodic solutions, infinitely many non-choreographic periodic solutions and uncountably many quasi-periodic solutions. Each type of periodic solutions has many stable solutions and possibly infinitely many stable solutions. Our results with SPBC largely complement the current results by minimizing the action on a loop space.

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In this paper, we study the variational properties of two special orbits: the Schubart orbit and the Broucke-Hénon orbit. We show that under an appropriate topological constraint, the action minimizer must be either the Schubart orbit or the Broucke-Hénon orbit. One of the main challenges is to prove that the Schubart orbit coincides with the action minimizer connecting a collinear configuration with a binary collision and an isosceles configuration. A new geometric argument is introduced to overcome this challenge.

It is known that a central configuration of the planar four body problem consisting of three particles of equal mass possesses a symmetry if the configuration is convex or is concave with the unequal mass in the interior. We use analytic methods to show that besides the family of equilateral triangle configurations, there are exactly one family of concave and one family of convex central configurations, which completely classifies such central configurations.