一类Hamilton系统周期映射的单调性

本文证明了系统 x+α· x+β· xn=0 (α>0 ,β≠ 0 ,n 2 )的周期映射是单调的     

一族离散的可积Hamilton方程与可积的辛映射

该文讨论一个新的离散特征值问题，导出了相应的离散的Hamilton系统的保谱族，并且证明了它们是Liouville可积系。通过谱问题的双非线性化，导出一个新的可积的辛映射 。      

Hamilton体系与辛正交系的完备性*

本文定义了一个Banach空间ZH,并证明了一类Hamilton体系的本征函数系(辛正交系)在ZH空间中具有完备性.还证明了如下结论ZH空间能连续嵌入到L₂[0,1]×L₂[0,1]但ZH≠L₂[0,1]×L₂[0,1].     

相空间非传统Hamilton型变分原理与辛算法

通过作者早已提出的新途径, 建立了多自由度系统弹性动力学的相空间非传统Hamilton型变分原理. 这种变分原理不仅能反映这种动力学初值问题的全部特征, 而且具有自然辛结构. 基于该变分原理, 提出一种称之为辛时间子域法的辛算法, 该方法在时间子域上采用Lagrange插值多项式插值, 构造非差分格式. 并且, 证明了这种辛算法是无条件稳定的. 通过两个不同类型算例的计算结果表明, 这种新方法的精度和计算效率都明显高于国际上常用的Wilson-θ 法和Newmark-β 法. 因此, 这种新算法是一种计算性能更好的高效算法.     

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．     

线性矩阵Hamilton系统振动性的区间判据

采用两种不同的方法,得到了线性矩阵Hamilton系统的振动性判据.这些振动性判据仅依赖于系数矩阵在[to,∞)的某些子区间上的性质,从而改进并推广了许多已知的Kamenev型振动准则.     

Hamilton系统约束的新分类和系统的可解性

本文建议Hamilton系统约束的一种新分类法,并利用这种分类法讨论了系统的可解性。     

一类二阶渐近周期Hamilton系统同宿轨道

运用变分方法讨论二阶渐近周期Hamilton系统 -u+L(t)u=(1+g(t))V′(t,u)的Lagrange泛函在流形上的极小问题,进而证明该系统存在非平凡同宿轨道,其中L,V关于t是周期的,g(t)→0(|t|→∞).     

一种构造可积Hamilton系统的新方法

建议了一种新的构造可积Hamilton系统的方法。对于给定的Poisson流形,本文利用Dirac-Poisson结构构造其上的新Poisson括号[1],进而获得了新的可积Hamilton系统。构造的Poisson括号一般是非线的,并且这种方法也不同于通常的方法[2～4]。本文还给出了两个实例。     

一类无穷维Hamilton算子特征函数系的完备性

本文对分离变量后可转化为Sturm-Liouville问题的偏微分方程,引入Hamilton体系,从而导出无穷维Hamilton算子的特征值问题.然后利用辛空间的知识讨论了无穷维Hamilton算子特征函数系的完备性,为对此类方程应用基于Hamilton体系的分离变量法提供了理论基础.作为应用,还给出了波动方程导出的无穷维Hamilton算子特征函数系的完备性.     

Symplectic Structure of Discrete Hamiltonian Systems

This paper is concerned with the symplectic structure of discrete nonlinear Hamiltonian systems. The results are related to an open problem that was first proposed by C. D. Ahlbrandt [J. Math. Anal. Appl.180 (1993), 498-517] discussed elsewhere in the literature. But we give a different statement and different proof. Under a solvable condition, we show that the solution operator of a discrete nonlinear Halmiltonian system is symplectic. Then its phase flow is a discrete one-parameter family of symplectic transformations and preserves the phase volume.     

Hamiltonian Systems with Positive Topological Entropy and Conjugate Points

In this article, we consider the geodesic flows induced by the natural Hamiltonian systems $H(x,p)=\frac{1}{2}g^{ij}(x) p_{i}p_{j} + V(x) $ defined on a smooth Riemannian manifold$(M = \mathbb{S}^{1} \times N, g)$, where $\mathbb {S}^{1}$ is the one dimensional torus, N is a compact manifold, g is the Riemannian metric on M and V is a potential function satisfying $V \leq 0$. We prove that under suitable conditions, if the fundamental group $\pi_{1}(N)$ has sub-exponential growth rate, then the Riemannian manifold M with the Jacobi metric $(h-V)g$, i.e., $(M, (h-V)g)$, is a manifold with conjugate points for all h with $0 < h <\delta$, where $\delta$ is a small number.     

High-Order Symplectic and Symmetric Composition Methods for Multi-Frequency and Multi-Dimensional Oscillatory Hamiltonian Systems

The multi-frequency and multi-dimensional adapted Runge-Kutta-Nyström (ARKN) integrators, and multi-frequency and multi-dimensional extended Runge-Kutta-Nyström(ERKN) integrators have been developed to efficiently solve multi-frequency oscillatory Hamiltonian systems. The aim of this paper is to analyze and derive high-order symplectic and symmetric composition methods based on the ARKN integrators and ERKN integrators. We first consider the symplecticity conditions for the multi-frequency and multi-dimensional ARKN integrators. We then analyze the symplecticity of the adjoint integrators of the multi-frequency and multi-dimensional symplectic ARKN integrators and ERKN integrators, respectively. On the basis of the theoretical analysis and by using the idea of composition methods, we derive and propose four new high-order symplectic and symmetric methods for the multi-frequency oscillatory Hamiltonian systems. The numerical results accompanied in this paper quantitatively show the advantage and efficiency of the proposed high-order symplectic and symmetric methods.     

Symplectic Topology of Integrable Hamiltonian Systems, II: Topological Classification

The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.     

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.     

Special Quasigraded Lie Algebras and Integrable Hamiltonian Systems

We construct a family of special quasigraded Lie algebras of functions of one complex variables with values in finite-dimensional Lie algebra , labeled by the special 2-cocycles F on . The main property of the constructed Lie algebras is that they admit Kostant-Adler-Symes scheme. Using them we obtain new integrable finite-dimensional Hamiltonian systems and new hierarchies of soliton equations.     

The Hamiltonian Systems of the LCZ Hierarchy by Nonlinearization

In this paper, we first search for the Hamiltonian structure of LCZ hierarchy by use of a trace identity. Then we determine a higher-order constraint condition between the potentials and the eigenfunctions of the LCZ spectral prob lem and under this constraint condition, the Lax pairs of LCZ hierarchy are all nonlinearized into the finite-dimensional integrable Hamiltonian systems in Liouville sense.     

Hamiltonian Systems: Stable or Unstable?

This is a review concerning some topics in the field of Hamiltonian dynamics, with emphasis on the problem of Arnold diffusion. Lecture held in the Seminario Matematico e Fisico on January 16, 2006 Received: May 2006     

Infinite-Dimensional Hamiltonian and Lagrangian Systems with Constraints

Mathematical Notes -     

Brake Subharmonic Solutions of Subquadratic Hamiltonian Systems

The author mainly uses the Galerkin approximation method and the iteration inequalities of the L-Maslov type index theory to study the properties of brake subharmonic solutions for the Hamiltonian systems z(t) = J▽H(t, z(t)), where H(t, z) =1/2(B(t)z, z) +H(t, z),B(t) is a semipositive symmetric continuous matrix andH is unbounded and not uniformly coercive. It is proved that when the positive integers j and k satisfy the certain conditions, there exists a_j T-periodic nonconstant brake solution z_j such that z_j and z_(kj) are distinct.