The co-orbital restricted three-body problem and its application

Based on large quantities of co-orbital phenomena in the motion of natural bodies and spacecraft, a model of the co-orbital restricted three-body problem is put forward. The fundamental results for the planar co-orbital circular restricted three-body problem are given, which include the selection of variables and equations of motion, a set of approximation formulas, and an approximate semi-analytical solution. They are applied to the motion of the barycenter of the planned gravitational observatory LISA constellation, which agrees very well with the solution of precise numerical integration.     

Numerical experiments of the planar equal-mass three-body problem: the effects of rotation

The planar three-body problem with angular momentum is numerically and systematically studied as a generalization of the free-fall problem (i.e., the three-body problem with zero initial velocities). The initial conditions in the configuration space exhaust all possible forms of a triangle, whereas the initial conditions in the momentum space are chosen so that position vectors and momentum vectors are orthogonal. Numerical results are organized according to the value of virial ratio k defined as the ratio of the total kinetic energy to the total potential energy. Final motions are mapped in the initial value space. Several interesting features are found. Among others, binary collision curves seem to spiral into the Lagrange point, and for large k, binary collision curves connect the Lagrange point and the Euler point. The existence of a lunar periodic orbit and a periodic orbit of petal-type is suggested. The number of escape orbits as a function of the escape time is analyzed for different k. The behavior of this number for different time and k shows most remarkably the effects of rotation of triple systems. The number of escape orbits increases exponentially for k相似文献   

Stabilization of Lagrange points in circular restricted three-body problem: A port-Hamiltonian approach

Current station keeping strategies target periodic orbits around the unstable Lagrange points. These control strategies are based on the Circular Restricted Three-Body Problem (CRTBP) linearized around an equilibrium point and cannot ensure global stability. In this paper, we use the port-Hamiltonian approach to reformulate the CRTBP with input, which preserves the original nonlinear dynamics. Designing a control strategy based on energy shaping and dissipation injection, we obtain the closed-loop Hamiltonian as the candidate of Lyapunov function, which guarantees asymptotic stability. The control strategy designed here is successfully applied to the stabilization of Lagrange points in CRTBP. Furthermore, the designed control approach shows global stability within the application region of CRTBP model, and it is applicable to set arbitrary equilibrium points. The current framework is also stable against error in the thrust and still works when the third body moves beyond the region of applicability of the linearized dynamics, where the linear controller may fail. Finally, this method has potential to be extended to the three-dimensional CRTBP, where both the perturbation and the thrust out of the plane are considered.     

力梯度辛方法在圆型限制性三体问题中的应用

旋转坐标系下的圆型限制性三体问 题因含非惯性系所附加的影响部分使得动能不是动量的严格二次型, 可能导致力梯度辛积分算法的应用遇到困难. 从Lie算子运算出发, 严格论证了力梯度算子在这种情形下的物理意义 仍然像质心惯性坐标系下的圆型限制性三体问题那样是引力的梯度, 而不是引力与非惯性力所得合力的梯度, 表明了力梯度辛方法适合求解旋转坐标系下的圆型限制性三体问题. 通过应用四阶力梯度辛方法、最优化四阶力梯度辛方法和Forest-Ruth 辛方法分别求解该问题, 进行了数值对比研究, 结果显示最优化型力梯度算法能够取得最好精度. 还应用最优化型算法计算两邻近轨道的Lyapunov指数和快速Lyapunov指标, 确保高精度辛方法能够贯穿于这些混沌指标计算的全过程, 以便准确刻画此系统的动力学定性性质. 关键词： 辛积分器 圆型限制性三体问题 混沌 Lyapunov指数     

A new class of symmetric periodic solutions of the spatial elliptic restricted three-body problem

We show that there exists a new class of symmetric periodic solutions of the spatial elliptic restricted three-body problem. In such a solution, the infinitesimal body is confined to the vicinity of a primary and moves on a nearly circular orbit. This orbit is almost perpendicular to the orbital plane of the primaries, where the line of symmetry of the orbit lies. The existence is shown by applying a corollary of Arenstorf’s fixed point theorem to a periodicity equation system of the problem. And this existence doesn’t require any restriction on the mass ratio of the primaries, nor on the eccentricity of their relative elliptic orbit. Potential relevance of this new class of periodic solutions to real celestial body systems and the follow-up studies in this respect are also discussed. Supported by the National Natural Science Foundation of China (Grant No. 10833001) Recommended by ZHOU JiLin     

Alternative treatment of the three-body problem

For solving the three-body problem with local potentials a model HamiltonianH ₀ containing an interaction between one particle and the centre-of-mass of the other two interacting particles is introduced. The total HamiltonianH is obtained byH=H _o +W whereW is a “residual interaction” in close analogy to the nuclear shell model. At a certain stage of the calculationsH ₀ has to be replaced by a new model Hamiltonian \(\tilde H_0 \) containing plane waves. The resolvent (and thereby theT-matrix) of the three-body problem is calculated by operator techniques. It is possible to draw some conclusions concerning three-body properties from these general expressions. Therefore this attempt may be considered as a supplementary treatment, in addition to the Faddeev-equations, of the three-body problem: it exhibits the discrete spectrum, the simple and the twofold continuum ofH arising from the corresponding states ofH ₀, and provides some approximation methods.     

Chaotic scattering in the gravitational three-body problem

We summarize some results of an ongoing study of the chaotic scattering interaction between a bound pair of stars (a binary) and an incoming field star. The stars are modeled as point masses and their equations of motion are numerically integrated for a large number of initial conditions. The global features of the resulting initial-value space maps are presented, and their evolution as a function of system parameters is discussed. We find that the maps contain regular regions separated by rivers of chaotic behavior. The probability of escape within the chaotic regions is discussed, and a straightforward explanation of the scaling present in these regions is reviewed. We investigate a statistical quantity of interest, namely the cross section for temporarily bound interactions, as a function of the third star's incoming velocity and mass. Finally, a new way of considering long-lived trajectories is presented, allowing long data sets to be qualitatively analyzed at a glance.     

Chaos in the one-dimensional gravitational three-body problem

We have investigated the appearance of chaos in the one-dimensional Newtonian gravitational three-body system (three masses on a line with -1/r pairwise potential). In the center of mass coordinates this system has two degrees of freedom and can be conveniently studied using Poincare sections. We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change. We consider only the physically more interesting case of negative total energy. For two mass choices we have calculated 18 000 full orbits (with initial states on a 100x180 lattice on the Poincare section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincare maps for 10x18 starting points. Our results show that the Poincare section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: (1) There is a region of fast scattering, with a minimum of pairwise collisions. This region consists of 'scallops' bordering the E=0 line, within a scallop the orbits vary smoothly. The number of the scallops increases as the mass of the central particle decreases. (2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincare section. For both (1) and (2) the initial and final states consist of a binary + single particle. (3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is a periodic orbit discovered (numerically) by Schubart in 1956. The stability of the Schubart orbit turns out to correlate strongly with the global behavior.     

Sundman stability in the general three-body problem

A new method for studying the Hill-type stability in the general three-body problem using Sundman??s inequality is presented. Sundman??s surfaces in 3D space are constructed, which are counterparts of Hill??s surfaces. The conditional and unconditional Sundman stability criteria are established and used for determining the stability regions.     

On the three-body Coulomb scattering problem

We describe the smoothness properties and the asymptotic form of the Green's function (in configuration space) for three charged particles. We also discuss the integral equations and the boundary value problems for the Coulomb wavefunctions and we show that they form a complete set. Finally, we study the singularities of the Coulomb scattering operator, and we investigate the connection between the Dollard wave operators and the Coulomb wavefunctions.     

The dissipation of restricted rotation

The decay of a classical hindered rotor over the whole domain of motion from almost free overall rotation to harmonic vibration in a well is examined. The decay is described by a modification of a projection operator formalism involving the separation of dynamical and lattice time scales, and depends on a canonical transformation into a hindered rotating frame that rotates and oscillates in accord with the unperturbed equation of motion of two coaxial dipoles.     

A solution of the Coulomb three-body problem

In this work, a final state wave function is constructed which represents a solution of the three-body Schr?dinger equation. The formulated wave function is superimposed of one basic analytical function with various parameters. The coefficients of these basic functions involved in final state wave function can be easily calculated from a set of linear equations. The coefficients depend only on incident energy of the system. The process can also be prolonged for application to the problems more than three bodies.     

The three-body problem in the path integral formalism

The propagator related to the Calogero potential is calculated in the phase space by way of Feynman formalism. The energy spectrum is determined along with the corresponding wave functions. In case some constraints are introduced, the problem may be reduced to the one corresponding to a particle constrained to move into a sector of opening angle α. It is shown as well that complicated potentials, may be transformed to allow the calculation of the energy spectrum via the Kleinert method.     

Model three-body problem in the molecular mass limit

The bound states of a three-body molecule composed of two identical heavy nuclei and a light “electron” interacting through short-range s-wave potentials are studied. The spectrum of three-body bound states grows as the mass ratio m between the heavy and light particles increases, and presents a remarkable vibration rotation structure that can be fitted with the usual empirical energy formulas of molecular spectroscopy. The results of the exact three-body calculation for the binding energy and bound-state wavefunction are compared with the predictions of the Born-Oppenheimer method for the same system. We find that for m > 30, the Born-Oppenheimer approximation yields very good results for both the binding energies and wavefunctions. For smaller m (1 <m < 30) the Born-Oppenheimer results are still surprisingly good and this is shown to be related to the range of the two-body interactions.     

Discrete Newtonian gravitation and the three-body problem

Newtonian gravitation is studied from a discrete point of view, in that the dynamical equation is an energy-conserving difference equation. Application is made to planetary-type, nondegenerate three-body problems and several computer examples of perturbed orbits are given.     

Variational calculations on the three-body scattering problem

An extended resonating-group method is used to calculate the elastic scattering amplitudes (up to L = 2 for a system of three identical bosons interacting through local Yukawa potentials. The results are compared to approximate solutions of the Faddeev equations.     

Choreographic solution to the general-relativistic three-body problem

We reexamine the three-body problem in the framework of general relativity. The Newtonian N-body problem admits choreographic solutions, where a solution is called choreographic if every massive particle moves periodically in a single closed orbit. One is a stable figure-eight orbit for a three-body system, which was found first by Moore (1993) and rediscovered with its existence proof by Chenciner and Montgomery (2000). In general relativity, however, the periastron shift prohibits a binary system from orbiting in a single closed curve. Therefore, it is unclear whether general-relativistic effects admit choreography such as the figure eight. We examine general-relativistic corrections to initial conditions so that an orbit for a three-body system can be choreographic and a figure eight. This illustration suggests that the general-relativistic N-body problem also may admit a certain class of choreographic solutions.     

A variation-iteration method for the three-body problem

We construct a practicable formulation of the variation-iteration method for the calculation of properties of a tri-nucleon system interacting via a modern potential. Our general approach is to take some integral form of the Schrödinger equation and reduce it to a set of simpler integral equations which may be solved by iteration. We concentrate particularly on the simplicity of the kernels which appear in this last set. The best method appears to be a two-step one, essentially equivalent to the Green function. In constructing the formalism, we are led to define a set of functions $\text{?}{}_{\mathrm{MN}}{}^{\mathrm{L}}\text{(k}{}_{12}\text{, k}{}_{23}\text{, k}{}_{31}\text{, r}{}_{12}\text{, r}{}_{23}\text{, r}{}_{31}\text{)}$ which play the role of iteration kernels. We give various properties of $\text{?}{}_{\mathrm{MN}}{}^{\mathrm{L}}$ and indicate very briefly how $\text{?}{}^{\mathrm{L}}{}_{\mathrm{MN}}$ may be reduced to a function of only two variables.