求解Hamilton约束的新算法

利用吴方法对多项式类型带约束的Hamilton系统作了研究.给出了判断系统是否正则的一个新算法.对于正则系统,可以得到Hamilton函数和运动方程,而对退化的系统给出了两个求解约束的新算法,得到带约束的Hamilton函数和运动方程.利用符号计算软件,这几个算法都可以在计算机上实现.     

保辛-保能的数值积分

针对Hamilton动力系统时变非线性问题,应用混合能变分原理,提出Hamilton系统的离散积分保辛算法.在此基础上,对Hamilton系统引入参变量,设计非线性问题迭代算法格式,通过对参变量的调整,在积分格点上实现了Hamilton系统数值积分保辛同时保能的目标.     

无限维Hamilton系统稳态解的保结构算法

基于Hamilton变分原理和Bridges意义下的多辛积分理论,提出了保持无穷维Hamilton系统稳态解能流通量和动量通量的保结构分析方法.针对复杂的无穷维Hamilton系统的多辛对称形式,首先讨论了其稳态解所满足的对称形式的守恒律问题;随后,以一个典型的无穷维Hamilton系统——Zufiria方程为例,采用box离散格式,模拟了其稳态解,并验证了算法的保结构性能.研究结果显示:采用保结构算法能够较好地模拟无穷维Hamilton系统的稳态解,并保持了无穷维Hamilton系统稳态解的能流通量和动量通量两个重要力学参量.这一研究结果将为复杂无穷维Hamilton系统稳态解的数值分析提供新的途径.     

约束Hamilton系统的Lie对称性及其在场论中的应用

研究了约束Hamilton系统的Lie对称性,得到了场论系统的守恒量.首先给出约束Hamilton系统的正则运动方程和固有约束方程;其次构建了约束Hamilton 系统的Lie对称性确定方程和结构方程;然后给出了约束Hamilton系统的Lie守恒定理和守恒量;最后研究了复标量场与Chern-Simons项耦合系统的Lie对称性和另外一个例子以说明此方法在场论中的应用.     

用投影方法求耗散广义Hamilton约束系统的李群积分

针对耗散广义Hamilton约束系统,通过引入拉格朗日乘子和采用投影技术,给出了一种保持动力系统内在结构和约束不变性的李群积分法．首先将带约束条件的耗散Hamilton系统化为无约束广义Hamilton系统, 进而讨论了无约束广义Hamilton系统的李群积分法,最后给出了广义Hamilton约束系统李群积分的投影方法．采用投影技术保证了约束的不变性,引入拉格朗日乘子后,在向约束流形投影时不会破坏原动力系统的李群结构．讨论的内容仅限于完整约束系统, 通过数值例题说明了方法的有效性．     

线性Hamilton系统边值问题的保辛数值方法

以Hamilton系统的正则变换和生成函数为基础研究线性时变Hamilton系统边值问题的保辛数值求解算法.根据第二类生成函数系数矩阵与状态传递矩阵的关系,构造了生成函数系数矩阵的区段合并递推算法,并进一步将递推算法推广到线性非齐次边值问题中;然后利用生成函数的性质将边值问题转化为初值问题,最后采用初值问题的保辛算法求解以达到整个Hamilton系统保辛的目的.数值算例表明该方法能够有效地求解线性齐次与非齐次问题,并能很好地保持Hamilton系统的固有特性.     

Hamilton系统下基于相位误差的精细辛算法

Hamilton系统是一类重要的动力系统，辛算法（如生成函数法、SRK法、SPRK法、多步法等）是针对Hamilton系统所设计的具有保持相空间辛结构不变或保Hamilton函数不变的算法.但是，时域上，同阶的辛算法与Runge-Kutta法具有相同的数值精度，即辛算法在计算过程中也存在相位误差，导致时域上解的数值精度不高.经过长时间计算后，计算结果在时域上也会变得“面目全非”.为了提高辛算法在时域上解的精度，将精细算法引入到辛差分格式中，提出了基于相位误差的精细辛算法（HPD-symplectic method），这种算法满足辛格式的要求，因此在离散过程中具有保Hamilton系统辛结构的优良特性.同时，由于精细化时间步长，极大地减小了辛算法的相位误差，大幅度提高了时域上解的数值精度，几乎可以达到计算机的精度，误差为O（10-13）.对于高低混频系统和刚性系统，常规的辛算法很难在较大的步长下同时实现对高低频精确仿真，精细辛算法通过精细计算时间步长，在大步长情况下，没有额外增加计算量，实现了高低混频的精确仿真.数值结果验证了此方法的有效性和可靠性.     

基于Hamilton体系和辛算法的微分对策数值法

微分对策求解往往涉及到困难的两点边值问题（TPBV）,将线性二次型微分对策问题归结于Hamilton体系．对Hamilton系统,辛几何算法具有能复制Hamilton系统的动态结构并保持相平面上的测度的优点．从Hamilton系统角度,探讨了线性二次型微分对策系统的辛性质;作为尝试,对无限期间线性二次型微分对策的计算引入Symplectic-Runge-Kutta算法．给出了一个数值计算实例,从结果可以说明这种方法的可行,也体现了辛算法对系统的能量具有良好的守恒性．     

辛Runge-Kutta方法在卫星交会对接中的非线性动力学应用研究

卫星交会对接问题是实现太空平台等空间系统的关键问题之一.考虑了由于地球引力作用而引起的卫星交会对接中的非线性动力学问题.首先,采用能量方法给出Lagrange函数;然后,通过引入广义坐标和广义动量,以及Legendre变换,得到Hamilton方程;随后,采用辛Runge-Kutta方法求解该Hamilton方程,并与传统的四阶Runge-Kutta方法对比.数值结果表明:辛Runge-Kutta方法能够在积分过程中长时间保持系统的固有特性,为天体动力学问题的研究提供了良好的数值方法.     

Sine-Gordon方程的高阶保能量算法

正1引言具有能量守恒、辛结构等固有特性的Hamilton系统被广泛地用于描述各种物理现象,并在自然界中具有普遍性.构造保持Hamil‘ton系统的固有特性的数值算法,对于正确求解Hamilton系统具有重要的意义.冯康院士及其研究小组提出了保持Hamilton系统辛结构的辛几何算法~([1-3]),辛几何算法凭借其优异的稳定性和长时间计算能力.被广泛应用于孤立子方程,流体力学和量子系统等的计算中~([4-5]).然而,向后误差分析表明~([6-7]),辛算法只能近似保持:Hamilton系统能量守恒特性.     

Solar sail orbits at the Earth–Moon libration points

Solar sail technology offer new capabilities for the analysis and design of space missions. This new concept promises to be useful in overcoming the challenges of moving throughout the solar system. In this paper, novel families of highly non-Keplerian orbits for solar sail spacecraft at linear order are investigated in the Earth–Moon circular restricted three-body problem, where the third body is a solar sail. In particular, periodic orbits near the collinear libration points in the Earth–Moon system will be explored along with their applications. The dynamics are completely different from the Earth–Sun system in that the sun line direction constantly changes in the rotating frame but rotates once per synodic lunar month. Using an approximate, first-order analytical solution to the nonlinear nonautonomous ordinary differential equations, periodic orbits can be constructed that are displaced above the plane of the restricted three-body system. This new family of orbits have the property of ensuring visibility of both the lunar far-side and the equatorial regions of the Earth, and can enable new ways of performing lunar telecommunications.     

光压摄动对空间太阳能电站轨道的影响研究

针对以重力梯度稳定方式设计的3种典型空间太阳能电站轨道动力学问题,提出了考虑地影和有效截面积变化的太阳光压模型.首先,采用能量方法,通过Legendre变换,引入广义动量,建立了Hamilton体系下轨道的正则方程;其次,采用辛Runge-Kutta方法求解相应的正则方程;最后通过数值试验分析,验证了模型的有效性以及数值求解方法的稳定性.同时,说明了地影和有效截面积变化对空间太阳能电站轨道有显著的影响;给出了空间太阳能电站对其半长轴、离心率以及轨道倾角的轨迹曲线,为空间太阳能电站的设计提供一种理论参考.     

Thermal alternation induced vibration analysis of spacecraft with lateral solar arrays in orbit

The rigid-flexible-thermal coupling dynamic analysis for a spacecraft in orbit is studied in this paper. The spacecraft consists of a central rigid platform and two groups of lateral solar arrays. There exists the relative motion between the rigid platform and solar arrays, thus the spacecraft is a multi-rigid-flexible bodies coupling system. As the spacecraft in orbit experience different light areas, alternations of the heat flux on solar arrays can result in changes of dynamic characteristics. Considering thermal stress effects of solar arrays, the dynamical model of the spacecraft is established by using Hamiltonian principle. Further, multi-rigid-flexible coupling modes of the system are obtained. The finite difference method is developed to obtained the responses of the spacecraft and the variation of temperature gradients under the different solar radiation. Results of natural characteristics illustrate that constrained modes can be used to discrete the system directly and efficiently. Modal shapes and parameters analysis reveal the rigid-flexible coupling effects of such spacecraft. The thermal-structural analysis demonstrates the thermal alternation may induce the vibration and even change the original vibration of the spacecraft.     

Hamiltonian approach to studying thin shells of revolution made of composite materials

A method for constructing defining relations of the linear theory of shells of revolution in complex Hamiltonian form has been proposed. Based on the Lagrange variational principle, we have constructed a mathematical model of a multilayer orthotropic shell of revolution. We have obtained explicit expressions for the coefficients and right-hand sides of the Hamiltonian complex system of equations describing the statics of shells of revolution in terms of their rigid characteristics and acting loads. The Hamiltonian resolving system of linear differential equations, formulated in the axially symmetric case, has some specific properties facilitating both analytical studies and numerical procedures of their solution.     

Use of the Bogoliubov method of compensating “dangerous” diagrams to investigate magnetic superconductors

The gauge model in superconductivity theory describes a multiparticle dynamic system in a constant external field of the vector (electromagnetic) potential. The Hamiltonian of this dynamic system models a superconducting antiferromagnet and contains electron-boson interactions of the second (Fröhlich Hamiltonian) and fourth (exchange interaction)_orders in the electron operators. This Hamiltonian accounts for the interaction of the magnetic moments of the conductance electrons. The bosonic spectrum of the system consists of normal modes of the coupled phonon-magnon oscillations. We obtain a system of equations describing a simultaneous compensation of the “dangerous” diagrams (those leading to energy divergence in the perturbation theory) corresponding to the creation of two (bivertex) or four (tetravertex) electron excitations from the vacuum. We find a solution of this system of equations corresponding to the superconducting state.     

Dynamical modeling and attitude analysis for the spacecraft with lateral solar arrays

The dynamics and attitude motion of the three-axis stabilized spacecraft installed with lateral solar arrays is investigated in terms of the rigid-flexible coupled global modes of the system. The spacecraft consists of a rigid platform with small moment of inertia and two groups of flexible solar arrays with relatively large moment of inertia installed on the rigid rotation shafts. The rigid-flexible coupled dynamic model of the spacecraft is established by using the Hamiltonian Principle. The global mode method is employed to work out the natural frequency and global modal shapes of the rigid-flexible coupled dynamic model combined with corresponding boundary conditions. To validate the effectiveness of the analytical results obtained by global mode method, the natural frequencies and mode shapes obtained from finite element model using MSC.Patran software are used as a reference. A numerical example is given to show that the results obtained from both methods are matched very well (the relative errors of the corresponding frequencies are small enough) and the rigid motion of the platform is coupled with the vibration mode of the flexible solar arrays. This implies that the global analytical modes can be used to accurately describe the rigid-flexible coupled motion of the spacecraft. By comparing with the finite element model, the reduced dynamical model derived in terms of the global modes of the system has a lower dimension. Numerical simulations for the system with variations of parameters and dynamic responses analysis for different applied forces are performed to illustrate that, the characteristics of the model are affected by inner and external factors.     

Small time and semiclassical asymptotics for stochastic heat equations driven by Lévy noise

The paper is devoted to the study of stochastic heat equations driven by Lévy noise. Applying the WKB method, we obtain multiplicative small time and semiclassical asymptotics for the Green functions and for solutions of the Cauchy problem for the heat equation under some natural additional assumptions on their coefficients. The first step in this construction consists in solving the corresponding stochastic Hamilton-Jacobi equations which constitute the "classical part" of the semiclassical approximation. In its turn, the corresponding Hamilton-Jacobi equations can be solved via solutions of the corresponding Hamiltonian systems, which gives rise to the method of stochastic characteristics. The relevant theory of stochastic Hamiltonian systems and stochastic Hamilton-Jacobi equations was developed in our previous papers. Here we put the final rung on the ladder: stochastic Hamiltonian systems, stochastic Hamilton-Jacobi equations, stochastic heat equations.     

非线性振动分析的均向量场法

通过构造向量形式的振动微分方程组，利用均向量场（AVF）法得到振动响应的向量差分迭代格式.该离散格式能够保能量，同时具有二阶精度的特征，从而给出非线性振动问题的均向量场法.介绍了均向量场法的基本步骤.在建立AVF格式时，对于微分方程中若干常见的项，直接给出相应的映射项.应用均向量场法研究了非线性单摆问题和Kepler(开普勒)问题，数值结果说明了该方法保能量和具有长时间求解能力的特性.