Periodic Solutions of the Elliptic Isosceles Restricted Three-body Problem with Collision

The elliptic isosceles restricted three-body problem with collision, is a restricted three-body problem where the primaries move having consecutive elliptic collisions and the infinitesimal mass is moving in the plane perpendicular to the primaries motion that passes through the center of mass of the primary system. Our purpose in this paper is to prove the existence of many families of periodic solutions using Continuation’s method, where the perturbing parameter is related with the energy of the primaries. This work is merely analytic and uses symmetry conditions and appropriate coordinates. Partially supported by Dirección de Investigación UBB, 064608 3/RS.     

基于混合Lie算子辛算法的不变流形计算

平动点附近周期轨道的不变流形因其在低能轨道转移中起着重要作用而受到广泛关注.在设计低能轨道过程中不变流形要实时进行能量匹配,但利用传统数值积分方法进行积分时能量会耗散.显式辛算法具有比隐式辛算法计算效率高的优势,但其要求Hamilton系统必须分成两个可积的部分,而旋转坐标系下的圆型限制性三体问题是不可分的,因而显式辛算法难以用于求解旋转坐标系下的圆型限制性三体问题.本文通过引入混合Lie算子,成功实现了带三阶导数项的力梯度辛算法对圆型限制性三体问题的求解,并将基于混合Lie算子的带三阶导数项的辛算法与Runge-Kutta78算法和Runge-Kutta45算法进行仿真对比,仿真结果表明基于混合Lie算子的含有三阶导数项的辛算法位置精度高、能量误差小且计算效率高.利用基于混合Lie算子的带三阶导数项的辛算法计算不变流形,可以实现低能轨道转移过程中轨道拼接点的能量精准匹配.     

Arnold diffusion in the elliptic restricted three-body problem

In this paper, we show the existence of the Arnold diffusion in the elliptic restricted three-body problem. This gives one of the very few examples of Arnold diffusion in real physical systems. The construction is based on the transversal homoclinic orbits in the circular restricted three-body problem ([6, 7, 14]). We prove that the small perturbations to the horseshoe maps in the neighborhood of the homoclinic orbits creates the Arnold diffusion. The existence of the Arnold diffusion also shows that the elliptic restricted three-body problem is non-integrable.     

Lunar landing trajectory design based on invariant manifold

The low-energy lunar landing trajectory design using the invariant manifolds of restricted three-body problem is studied.Considering angle between the ecliptic plane and lunar orbit plate the four-body problem of sun-earth-moon-spacecraft is divided into two three-body problems,the sun-earth-spacecraft in the ecliptic plane and the earth- moon-spacecraft in the lunar orbit plane.Using the orbit maneuver at the place where the two planes and the invariant manifolds intersect,a general method to design low energy lunar landing trajectory is given.It is found that this method can save the energy about 20% compared to the traditional Hohmann transfer trajectory,The mechanism that the method can save energy is investigated in the point of view of energy and the expression of the amount of energy saved is given.In addition,some rules of selecting parameters with respect to orbit design are provided.The method of energy analysis in the paper can be extended to energy analysis in deep space orbit design.     

The annular twist mapping of the restricted three-body problem

This paper presents a new derivation of the twist mapping in the planar restricted problem. It differs from other treatments in the use of a novel canonical transformation which allows for the utilization of symplectic reduction techniques.     

Stability of triangular libration points in a planar restricted elliptic three body problem in cases of double resonances

Stability of triangular libration points in a planar restricted elliptic three-body problem is investigated for two sets of parameters corresponding to the cases of double resonances. These positions are shown to be formally stable in the case of the fourth-order double resonance. When a weak third-order and a strong fourth-order resonances occur in the system, any motion of an approximate autonomous system is found to be bounded if initial conditions are sufficiently close to the unperturbed motion. Boundedness domain estimate is obtained.     

An improved nonlinear control strategy for deep space formation flying spacecraft

This paper aims to provide further study on the nonlinear modeling and controller design of formation flying spacecraft in deep space missions. First, in the Sun-Earth system, the nonlinear formation dynamics for the circular restricted three-body problem （CRTBP） and elliptic restricted three-body problem （ERTBP） are presented. Then, with the Floquet mode method, an impulsive controller is developed to keep the Chief on the desired Halo orbit. Finally, a nonlinear adaptive control scheme based on Nonzero set- point LQR and neural network is proposed to achieve high precision formation maneuver and keeping. The simulation results indicate that the proposed nonlinear control strategy is reasonable as it considers not only the orbit keeping of the Chief, but also the formation modeling inaccuracy. Moreover, the nonlinear adaptive control scheme is effective to improve the control accuracy of the formation keeping.     

Binary trees,symplectic matrices and the Jacobi coordinates of celestial mechanics

A graph-theoretic algorithm for constructing the Jacobi coordinates in celestial mechanics is given. To every full binary tree with N leaves, there corresponds a 6N×6N symplectic matrix, which defines a Jacobi transformation. This correspondence yields a direct proof of the symplectic property for all the Jacobi coordinates; hitherto, only special examples of these transformations have been shown to be canonical. Communicated by R. P. McGeehee     

Global Surfaces of Section in the Planar Restricted 3-Body Problem

The restricted planar three-body problem has a rich history, yet many unanswered questions still remain. In the present paper we prove the existence of a global surface of section near the smaller body in a new range of energies and mass ratios for which the Hill’s region still has three connected components. The approach relies on recent global methods in symplectic geometry and contrasts sharply with the perturbative methods used until now.     

三维引力辅助解析分析方法研究

针对星际探测中的引力辅助技术,基于二体问题,导出了三维引力辅助前后飞行器速度、能量、角动量和轨道倾角与引力辅助参数之间的解析公式. 以进行木星引力辅助为例,将解析公式计算的结果与基于圆型限制性三体问题的数值仿真结果进行了对比,各项绝对误差小于0.01,除极少数情况外,相对误差均小于0.7%. 分析结果表明导出的解析公式与实际情况对比具有较高的精度,可以满足引力辅助轨道初步设计的需要. 为了讨论和应用上的方便,对导出的公式进行了简化,据此进一步讨论了引力辅助参数对引力辅助效果的影响.     

First-round design of the flight scenario for Chang’e-2’s extended mission:take off from lunar orbit

Chang'e-2, Chinese second lunar probe, was inserted into a 100 km altitude low lunar orbit on October 9th, 2010, its purpose is to continuously photograph the lunar surface and possibly chosen landing sites for future lunar missions. The probe will still carry considerable amount of propellant after completing all prescribed tasks in about six months. After the successful launch of Chang'e-2, we began designing the probe's subsequent flight scenario, considering a total impulse of 1 100 m/s for takeoff from low lunar orbit and a maximum 3×10 6 km distance for Earth-probe telecommunication. Our first-round effort proposed a preliminary flight scenario that involves consecutive arrivals at the halo orbits around the Earth-Moon L1/L2 and Sun-Earth L1/L2 points, near-Earth asteroid flyby, Earth return, and lunar im- pact. The designed solution of Chang'e-2's subsequent flight scenario is a multi-segment flight trajectory that serves as a reference for making the final decision on Chang'e-2's extended mission, which is a flight to the Sun-Earth L2 point, and a possible scheme of lunar impact via Earth flyby after remaining at the Sun-Earth L2 point was also presented. The proposed flight trajectory, which possesses acceptable solution accuracy for mission analysis, is a novel design that effectively exploits the invariant manifolds in the circular restricted three-body problem and the patched-manifold-conic method.     

Variation principle of piezothermoelastic bodies, canonical equation and homogeneous equation

Combining the symplectic variations theory,the homogeneous control equa- tion and isoparametric element homogeneous formulations for piezothermoelastic hybrid laminates problems were deduced.Firstly,based on the generalized Hamilton variation principle,the non-homogeneous Hamilton canonical equation for piezothermoelastic bod- ies was derived.Then the symplectic relationship of variations in the thermal equilibrium formulations and gradient equations was considered,and the non-homogeneous canoni- cal equation was transformed to homogeneous control equation for solving independently the coupling problem of piezothermoelastic bodies by the inceusement of dimensions of the canonical equation.For the convenience of deriving Hamilton isoparametric element formulations with four nodes,one can consider the temperature gradient equation as constitutive relation and reconstruct new variation principle.The homogeneous equa- tion simplifies greatly the solution programs which are often performed to solve non- homogeneous equation and second order differential equation on the thermal equilibrium and gradient relationship.     

Secular perturbations of quasi-elliptic orbits in the restricted three-body problem with variable masses

In this work we consider the satellite version of the restricted three-body problem when masses of the primary bodies P₀, P₁ vary isotropically with different rates, and their total mass changes according to the joint Meshcherskii law. Equations of motion of the body P₂ of infinitesimal mass are obtained in terms of the osculating elements of the aperiodic quasi-conical motion about the body P₀. Doubly averaging these equations and using the Hill approximation, we have obtained the differential equations determining the secular perturbations of the orbital elements and determined the domains of possible values of the system parameters for which their analytical solutions are expressed in terms of elementary or elliptic functions. The bodies P₀, P₁ mass variation laws for which the corresponding differential equations are integrable, have been found.     

日地系统L_2点Halo轨道绳系卫星编队动力学

研究平动点附近周期轨道上旋转多体绳系卫星编队系统的非线性耦合动力学问题。编队系统为各卫星质量接近的轮辐状结构,位于日地系统第二平动点附近,整个系统的旋转保持系绳处于张紧状态,建立Hill限制性三体问题的绳系编队系统动力学模型。针对处于留位阶段的典型对称三星编队,在位于较大Halo轨道上无控制力作用的情况下,进行母卫星轨道运动与系绳摆动耦合运动的动力学模拟,分析轨道方向、卫星质量比、系绳长度以及初始旋转速度对编队系统整体稳定性的影响。     

非线性方程的保辛近似算法

保守体系的微分方程可用Hamilton体系的方法描述,其特点是保辛。两个辛矩阵之和不能保辛,两个辛矩阵的乘积仍是辛矩阵。最常用的小参数摄动法用的是加法,因此对辛矩阵不能保辛。从保辛的角度,要用正则变换。本文针对非线性微分方程,运用自变量坐标变换,对原系统进行变换。由此推导出变换后系统的变分原理。引入Hamilton对偶变量,通过数学变换,得到变系数非线性方程。针对该方程,本文提出了保辛摄动算法。通过数值算例,对不同步长下,保辛摄动法、多尺度摄动法、龙格库塔法和精确解的结果做了比较。数值例题表明,对于非线性方程,本文提出的保辛摄动算法有良好的精度。在步长增大的情况下,保辛摄动保持了良好的稳定性。     

High order symplectic conservative perturbation method for time-varying Hamiltonian system

This paper presents a high order symplectic conservative perturbation method for linear time-varying Hamiltonian system.Firstly,the dynamic equation of Hamiltonian system is gradually changed into a high order perturbation equation,which is solved approximately by resolving the Hamiltonian coefficient matrix into a "major component" and a "high order small quantity" and using perturbation transformation technique,then the solution to the original equation of Hamiltonian system is determined through a series of inverse transform.Because the transfer matrix determined by the method in this paper is the product of a series of exponential matrixes,the transfer matrix is a symplectic matrix;furthermore,the exponential matrices can be calculated accurately by the precise time integration method,so the method presented in this paper has fine accuracy,efficiency and stability.The examples show that the proposed method can also give good results even though a large time step is selected,and with the increase of the perturbation order,the perturbation solutions tend to exact solutions rapidly.     

平面刚体系统的参数预调节保辛算法

保辛积分方法在约束哈密顿系统中有着重要的应用,是因为其在长时间仿真中表现出极好的稳定性。然而随着仿真时长增加,保辛格式通常具有较大的相位误差累积。本文提出了一种平面多刚体系统的参数预调节保辛积分方法。通过推导具有待定参数的改进的拉格朗日方程,并将其与已有保辛格式相结合并预先调节相关参数取值,可以大幅降低数值解的相位误差。理论分析与数值结果表明参数预调节保辛积分方法不仅保持了辛结构,而且具有很低的相位误差累积。因此,参数预调节保辛积分方法可应用于长时间仿真分析。     

Asymptotic approximation method for elliptic variational inequality of first kind简

An asymptotic approximation method is proposed to solve a particular elliptic variational inequality of first kind associated with unilateral obstacle problems. In this method, the free boundary is first captured, and then the method of the fundamental solution (MFS) is used to find the solution of the Dirichlet problem for Laplace’s equation in the non-coincidence set. Numerical examples are given to show the efficiency of the method.     

Cross Sections in the Three-Body Problem

In Dynamical Systems, Birkhoff gave a clear formulation of a cross section, suggested a possible generalization to cross sections with boundary, and raised the question of whether or not such cross sections exist in the three-body problem. In this work, we explicitly develop Birkhoff's notion of a generalized cross section, formulate homological necessary conditions for the existence of a cross section or generalized cross section, and show that these conditions are not satisfied in the three-body problem.     

Averaged Momentum Equation for Flow Through a Nonhomogenenous Porous Structure

This paper addresses the derivation of the macroscopic momentum equation for flow through a nonhomogeneous porous matrix, with reference to dendritic structures characterized by evolving heterogeneities. A weighted averaging procedure, applied to the local Stokes' equations, shows that the heterogeneous form of the Darcy's law explicitly involves the porosity gradients. These extra terms have to be considered under particular conditions, depending on the rate of geometry variations. In these cases, the local closure problem becomes extremely complex and the full solution is still out of reach. Using a simplified two-phase system with continuous porosity variations, we numerically analyze the limits where the usual closure problem can be retained to estimate the permeability of the structure.