On the circular Sitnikov problem: the alternation of stability and instability in the family of vertical motions

This paper is devoted to the special case of the restricted circular three-body problem, when the two primaries are of equal mass, while the third body of negligible mass performs oscillations along a straight line perpendicular to the plane of the primaries (so called periodic vertical motions). The main goal of the paper is to study the stability of these periodic motions in the linear approximation. A special attention is given to the alternation of stability and instability within the family of periodic vertical motions, whenever their amplitude is varied in a continuous monotone manner.     

Stability of motion in the Sitnikov 3-body problem

We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m ₁ = m ₂ = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 ≤ m ₃ < 10⁻³, placed initially on the z-axis. We begin by finding for the restricted problem (with m ₃ = 0) an apparently infinite sequence of stability intervals on the z-axis, whose width grows and tends to a fixed non-zero value, as we move away from z = 0. We then estimate the extent of “islands” of bounded motion in x, y, z space about these intervals and show that it also increases as |z| grows. Turning to the so-called extended Sitnikov problem, where the third particle moves only along the z-axis, we find that, as m ₃ increases, the domain of allowed motion grows significantly and chaotic regions in phase space appear through a series of saddle-node bifurcations. Finally, we concentrate on the general 3-body problem and demonstrate that, for very small masses, m ₃ ≈ 10⁻⁶, the “islands” of bounded motion about the z-axis stability intervals are larger than the ones for m ₃ = 0. Furthermore, as m ₃ increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away “disperse” at larger m ₃ values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries, as observed in the m ₃ = 0 case.     

On the Sitnikov problem

A mapping which reflects the properties of the Sitnikov problem is derived. We study the mapping instead of the original differential equations and discover that there exists a hyperbolic invariant set. The theoretical prediction of the disorder region agrees remarkably with numerical results. We also discuss the LCEs and KS-entropy of the dynamical system.This project is supported by the National Science Foundation of China.     

Transient chaos in the Sitnikov problem

In this paper, we show the important role of chaotic transients in Celestial Mechanics through the Sitnikov problem. We compare the two kinds of chaos, permanent and transient, and provide the chaotic saddle of the Sitnikov problem giving also some important quantitative properties of this fractal set. Additionally, we present a link between the stickiness effect of tori and chaotic scattering.     

Region of motion in the Sitnikov four-body problem when the fourth mass is finite

This article deals with the region of motion in the Sitnikov four-body problem where three bodies (called primaries) of equal masses fixed at the vertices of an equilateral triangle. Fourth mass which is finite confined to moves only along a line perpendicular to the instantaneous plane of the motions of the primaries. Contrary to the Sitnikov problem with one massless body the primaries are moving in non-Keplerian orbits about their centre of mass. It is investigated that for very small range of energy h the motion is possible only in small region of phase space. Condition of bounded motions has been derived. We have explored the structure of phase space with the help of properly chosen surfaces of section. Poincarè surfaces of section for the energy range ?0.480≤h≤?0.345 have been computed. We have chosen the plane (q ₁,p ₁) as surface of section, with q ₁ is the distance of a primary from the centre of mass. We plot the respective points when the fourth body crosses the plane q ₂=0. For low energy the central fixed point is stable but for higher value of energy splits in to an unstable and two stable fixed points. The central unstable fixed point once again splits for higher energy into a stable and three unstable fixed points. It is found that at h=?0.345 the whole phase space is filled with chaotic orbits.     

The global solution of the N-body problem

The problem of finding a global solution for systems in celestial mechanics was proposed by Weierstrass during the last century. More precisely, the goal is to find a solution of the n-body problem in series expansion which is valid for all time. Sundman solved this problem for the case of n = 3 with non-zero angular momentum a long time ago. Unfortunately, it is impossible to directly generalize this beautiful theory to the case of n > 3 or to n = 3 with zero-angular momentum.A new blowing up transformation, which is a modification of McGehee's transformation, is introduced in this paper. By means of this transformation, a complete answer is given for the global solution problem in the case of n > 3 and n = 3 with zero angular momentum.The main result in this paper has appeared in Chinese in Acta Astro. Sinica. 26 (4), 313–322. In this version some mistakes have been rectified and the problems we solved are now expressed in a much clearer fashion.     

An analytical approach to small amplitude solutions of the extended nearly circular Sitnikov problem

The model of extended Sitnikov Problem contains two equally heavy bodies of mass m moving on two symmetrical orbits w.r.t the centre of gravity. A third body of equal mass m moves along a line z perpendicular to the primaries plane, intersecting it at the centre of gravity. For sufficiently small distance from the primaries plane the third body describes an oscillatory motion around it. The motion of the three bodies is described by a coupled system of second order differential equations for the radial distance of the primaries r and the third mass oscillation z. This problem which is dealt with for zero initial eccentricity of the primaries motion, is generally non integrable and therefore represents an interesting dynamical system for advanced perturbative methods. In the present paper we use an original method of rewriting the coupled system of equations as a function iteration in such a way as to decouple the two equations at any iteration step. The decoupled equations are then solved by classical perturbation methods. A prove of local convergence of the function iteration method is given and the iterations are carried out to order 1 in r and to order 2 in z. For small values of the initial oscillation amplitude of the third mass we obtain results in very good agreement to numerically obtained solutions.     

The Sitnikov problem for several primary bodies configurations

In this paper we address an \(n+1\)-body gravitational problem governed by the Newton’s laws, where n primary bodies orbit on a plane \(\varPi \) and an additional massless particle moves on the perpendicular line to \(\varPi \) passing through the center of mass of the primary bodies. We find a condition for the described configuration to be possible. In the case when the primaries are in a rigid motion, we classify all the motions of the massless particle. We study the situation when the massless particle has a periodic motion with the same minimal period as the primary bodies. We show that this fact is related to the existence of a certain pyramidal central configuration.     

The structure of the extended phase space of the Sitnikov problem

The extended phase space of the Sitnikov problem is studied by using a stroboscopic map and computing escape times. Comparisons of phase portraits and plots of escape times reveal the intrinsic connection between the geometry of the phase space and the dynamical behaviour of the system. Properties of the phase space are analysed both in the central regular region and far from it. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The main mean motion commensurabilities in the planar circular and elliptic problem

We study the motion of asteroids in the main mean motion commensurabilities in the frame of the planar restricted three-body problem. No assumption is made about the size of the eccentricity of the asteroid. At small to moderate eccentricity, we recover existing results (shape of the phase space and location of secondary resonances). We also provide global pictures of the dynamics in the region of secondary resonances. At high eccentricity, the phase space portraits of the integrable part of the Hamiltonian show new families of stable orbits for the 3:2 and 2:1 cases and the secular resonances ₅ and ₆ are located.     

The validity of the continuum limit in the gravitational N-body problem

This paper attempts to give quantitative as well as qualitative answers to the question of the analogy between smooth potentials and N-body systems. A number of simulations were performed in both integrable and nonintegrable smooth environments and their frozen N-body analogues, and comparisons were made using a number of different tools. The comparisons took place on both statistical and pointwise levels. The results of this study suggest that microscopic chaos associated with discreteness effects is always present in N-body configurations. This chaos is different from the macroscopic chaos which is associated with the bulk potential and persists even for very large N. Although the Lyapunov exponents of orbits evolving in N-body environments do not decrease as N increases, comparisons associated with the statistical properties, as well as with the power spectra of the orbits, affirm the existence of the continuum limit.     

A new analytic approach to the Sitnikov problem

A new analytic approach to the solution of the Sitnikov Problem is introduced. It is valid for bounded small amplitude solutions (z _max = 0.20) (in dimensionless variables) and eccentricities of the primary bodies in the interval (–0.4 < e < 0.4). First solutions are searched for the limiting case of very small amplitudes for which it is possible to linearize the problem. The solution for this linear equation with a time dependent periodic coefficient is written up to the third order in the primaries eccentricity. After that the lowest order nonlinear amplitude contribution (being of order z ³) is dealt with as perturbation to the linear solution. We first introduce a transformation which reduces the linear part to a harmonic oscillator type equation. Then two near integrals for the nonlinear problem are derived in action angle notation and an analytic expression for the solution z(t) is derived from them. The so found analytic solution is compared to results obtained from numeric integration of the exact equation of motion and is found to be in very good agreement. CERN SL/AP     

Periodic orbits and bifurcations in the Sitnikov four-body problem

We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż₀ varies within a finite interval (while z ₀ tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity.     

Hip-hop solutions of the 2N-body problem

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.     

The homographic solutions in the N-body problem with a general attraction

The existence of homographic solutions of the N-body problem with a geneva attraction is verified, and the way which leads to obtaining certain types of homographic solutions is indicated. Basic properties of the solutions, such as the relations between the dynamical quantities and the initial conditions are presented. Furthermore, we proved that, for k is not equal to 3, if a homographic solution is not planar, it must be homothetic. And in this case, another important conclusion is that the configurations corresponding to any homographic solution are central configurations. Finally, we showed that along each homographic solution, motion of any individual mass point observes the same rules as the ones observed by mass points of a certain two-body system.     

Periodic solution of the nonlinear Sitnikov restricted three-body problem

The objective of this paper is to find periodic solutions of the circular Sitnikov problem by the multiple scales method which is used to remove the secular terms and find the periodic approximated solutions in closed forms. Comparisons among a numerical solution (NS), the first approximated solution (FA) and the second approximated solution (SA) via multiple scales method are investigated graphically under different initial conditions. We observe that the initial conditions play a vital role in the numerical and approximated solutions behaviour. The obtained motion is periodic, but the difference of its amplitude is directly proportional with the initial conditions. We prove that the obtained motion by the numerical or the second approximated solutions is a regular and periodic, when the infinitesimal body starts its motion from a nearer position to the common center of primaries. Otherwise when the start point distance of motion is far from this center, the numerical solution may not be represent a periodic motion for along time, while the second approximated solution may present a chaotic motion, however it is always periodic all time. But the obtained motion by the first approximated solution is periodic and has regularity in its periodicity all time. Finally we remark that the provided solutions by multiple scales methods reflect the true motion of the Sitnikov restricted three–body problem, and the second approximation has more accuracy than the first approximation. Moreover the solutions of multiple scales technique are more realistic than the numerical solution because there is always a warranty that the motion is periodic all time.     

On the global solution of the N-body problem

In connection with the publication (Wang Qiu-Dong, 1991) the Poincaré type methods of obtaining the maximal solution of differential equations are discussed. In particular, it is shown that the Wang Qiu-Dong'sglobal solution of the N-body problem has been obtained in Babadzanjanz (1979). First the more general results on differential equations have been published in Babadzanjanz (1978).     

Stability of motion of the restricted circular and charged three-body problem

Stabiliity is applied to characterize type of motion in which the moving body is confined to certain limited regions and in this sense we may say that the motion of the body in question is stable. This method has been used in the past chiefly in connection with the classical restricted problem of three bodies.In this paper we consider a dynamical system defined by the Lagrangian