辛几何算法在计算非磁化等离子体中波传播轨迹时的应用

为了在数值计算中保持哈密顿系统的辛几何结构不变,利用辛几何算法得到了在线性哈密顿系统中射线追踪方程的一般辛差分格式。通过具体算例,利用辛几何算法计算了波在非磁化等离子体中的传播轨迹,并且与传统Runge-Kutta-Fehlberg算法所得结果进行了比较。利用辛几何算法所得传播轨迹与解析解一致,其色散函数值的误差随时间线性增长,能在长时间内保持色散函数值在一个很小的误差范围内。利用传统的Runge-Kutta-Fehlberg算法所得传播轨迹与解析解不一致,其误差随时间做大幅振荡增加。计算结果表明辛几何算法在保持传播轨迹和色散函数值方面具有独特的优势。     

辛几何算法在计算非磁化等离子体中波传播轨迹时的应用

为了在数值计算中保持哈密顿系统的辛几何结构不变,利用辛几何算法得到了在线性哈密顿系统中射线追踪方程的一般辛差分格式。通过具体算例,利用辛几何算法计算了波在非磁化等离子体中的传播轨迹,并且与传统Runge-Kutta-Fehlberg算法所得结果进行了比较。利用辛几何算法所得传播轨迹与解析解一致,其色散函数值的误差随时间线性增长,能在长时间内保持色散函数值在一个很小的误差范围内。利用传统的Runge-Kutta-Fehlberg算法所得传播轨迹与解析解不一致,其误差随时间做大幅振荡增加。计算结果表明辛几何算法在保持传播轨迹和色散函数值方面具有独特的优势。     

射线追踪、辛几何算法与波场的数值模拟

论述了射线追踪、辛几何算法与波场的数值模拟之间的关系,说明了辛几何算法长时间守恒性质及运用辛几何算法进行射线追踪的必要性.对线性层状模型,利用辛几何算法和Maslov渐近理论对波场进行了数值模拟,并与解析解进行了比较.     

Chaplygin系统的Noether对称性与形式不变性

利用d’AlembertLagrange原理的Chaplygin形式在无限小变换下的变形形式得到Chaplygin系统的广义Noether等式和守恒量的形式.研究Chaplygin系统的形式不变性以及它与Noether对称性的关系. 关键词： Chaplygin系统 Noether对称性 形式不变性 守恒量     

广义Chaplygin系统的约化

将广义Chaplygin系统写成含Lagrange函数的形式，它在一定条件下可约化为一个无约束系统.给出这些条件并举例说明结果的应用. 关键词： 分析力学 约化 非完整系统 广义Chaplygin系统     

非完整转动相对论系统的Lindel?f方程

由转动相对论系统的Hamilton原理分别建立在广义坐标和准坐标下的Lindel?f方程及其改进形式,并从改进的Lindel?f方程导出新Chaplygin方程.最后说明由转动系统的相对论分析力学向普通分析力学过渡的方法. 关键词： 非完整约束 转动系统 相对论 Lindel?f方程 Chaplygin方程     

Schrödinger方程的辛结构与量子力学的辛算法

冯康开创的哈密顿力学的辛算法取得了惊人的成功.这是因为哈密顿力学的数学框架是辛几何,一个合理的离散方法自然应使离散哈密顿力学保持辛结构.本文指出,经过适当的变换,Schrödinger方程也具有辛结构,从而把哈密顿力学的辛算法,推广用到量子力学.作为例子计算了中子在旋转磁场中的演化.计算结果表明,辛算法明显优于通常算法,特别是对演化时间长的情况.     

探索高等科教书店物理类书目推荐(VII)

作者书名定价作者书名定价E.Bick物理学中的拓扑和几何(影印)65.0 G.B.Arfken物理学家用的数学方法第6版(影印)189.0R.Ticciati数学家用的量子场论(影印)105.0冯康、秦孟兆哈密尔顿系统的辛几何算法68.0J.S.Liu科学计算中的蒙特卡罗策略(影印)48.0楼森岳非线性数学物理方法59.0     

非完整转动相对论系统的Lindelof方程

由转动相对论系统的Hamilton原理分别建立在广义坐标和准坐标下的Lindelof方程及其改进形式,并从改进的Lindelof方程导出新Chaplygin方程. 最后说明由转动系统的相对论分析力学向普通分析力学过渡的方法.     

Poincar-Chetaev变量下变质量非完整动力学系统的运动方程

研究Poincaré-Chetaev变量下,变质量非线性非完整力学系统的运动方程.首先,由变质量力学系统的D’Alembert-Lagrange原理导出Chaplygin型方程、Nielsen型方程及Appell型方程.其次,研究Chaplygin方程与Appell方程的等价性问题.最后,举例说明新结果的应用.     

An improved symplectic precise integration method for analysis of the rotating rigid–flexible coupled system

This paper presents an improved symplectic precise integration method (PIM) to increase the accuracy and keep the stability of the computation of the rotating rigid–flexible coupled system. Firstly, the generalized Hamilton's principle is used to establish a coupled model for the rotating system, which is discretized and transferred into Hamiltonian systems subsequently. Secondly, a suitable symplectic geometric algorithm is proposed to keep the computational stability of the rotating rigid–flexible coupled system. Thirdly, the idea of PIM is introduced into the symplectic geometric algorithm to establish a symplectic PIM, which combines the advantages of the accuracy of the PIM and the stability of the symplectic geometric algorithm. In some sense, the results obtained by this method are analytical solutions in computer for a long span of time, so the time-step can be enlarged to speed up the computation. Finally, three numerical examples show the stability of computation, the accuracy of solving stiff equations and the capability of solving nonlinear equations, respectively. All these examples prove the symplectic PIM is a promising method for the rotating rigid–flexible coupled systems.     

Algebraic dynamics algorithm:Numerical comparison with Runge-Kutta algorithm and symplectic geometric algorithm

Based on the exact analytical solution of ordinary differential equations, a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm. A detailed numerical comparison is presented with Runge-Kutta algorithm and symplectic geometric algorithm for 12 test models. The results show that the algebraic dynamics algorithm can better preserve both geometrical and dynamical fidelity of a dynamical system at a controllable precision, and it can solve the problem of algorithm-induced dissipation for the Runge-Kutta algorithm and the problem of algorithm-induced phase shift for the symplectic geometric algorithm.     

Symplectic algebraic dynamics algorithm

Based on the algebraic dynamics solution of ordinary differential equations andintegration of  ,the symplectic algebraic dynamics algorithm sn is designed,which preserves the local symplectic geometric structure of a Hamiltonian systemand possesses the same precision of the na ve algebraic dynamics algorithm n.Computer experiments for the 4th order algorithms are made for five test modelsand the numerical results are compared with the conventional symplectic geometric algorithm,indicating that sn has higher precision,the algorithm-inducedphase shift of the conventional symplectic geometric algorithm can be reduced,and the dynamical fidelity can be improved by one order of magnitude.     

一类特殊非完整力学系统的辛算法计算

对一类特殊的非完整力学系统的动力学性质进行数值研究,采用当前比较优越的保结构算法进行数值计算,并和传统的Runge-Kutta方法进行比较. 通过计算结果的比较而得出辛算法在这类特殊的非完整力学系统的数值计算中的优越性. 关键词： 非完整约束 辛算法 辛差分格式     

Symplectic geometry and geometric quantization on supermanifold withU numbers

The usual theory of supermanifold is extended to the case that contains (anti)commuting variable pairs with oppositeU numbers. The symplectic geometry and geometric quantization on such a special manifold are discussed in detail. As applications, the BRST system with finite dimensional first class constraints and bosonic strings are investigated systematically.     

欧拉-拉格朗日系统的jet辛算法

研究了在欧拉-拉格朗日系统上的jet辛算法.证明了第二作者在1998年给出的一个离散的欧拉-拉格朗日(DEL)方程存在一个离散形式的几何结构,它沿着解是不变的,这个结构可以通过对离散的作用量函数求导得到.由此,可以给出此格式的jet辛性质.利用这个结构证明了与此DEL方程相关的离散Nother定理.最后,给出了一个欧拉-拉格朗日方程上的jet辛差分格式的数值算例,并与其它的差分格式进行了比较.     

辛格式在强激光场一维模型问题计算中的应用

 将辛算法推广到复辛空间，指出了辛算法保定态Schr-dinger方程的Wronskian守恒。将辛算法应用于强场一维模型的计算中，并与Runge-Kutta法作了比较。结果显示，辛算法保持定态Schr-dinger方程的Wronskian守恒，适合于在充分 远空间上计算线性无关解，是计算强激光场一维模型的合理的数值方法。     

Cosmology with a variable Chaplygin gas

We consider a new generalized Chaplygin gas model that includes the original Chaplygin gas model as a special case. In such a model the generalized Chaplygin gas evolves as from dust to quiescence or phantom. We show that the background evolution for the model is equivalent to that for a coupled dark energy model with dark matter. The constraints from the current type Ia supernova data favour a phantom-like Chaplygin gas model.     

Nonholonomic Hamilton–Jacobi theory via Chaplygin Hamiltonization

We develop Hamilton–Jacobi theory for Chaplygin systems, a certain class of nonholonomic mechanical systems with symmetries, using a technique called Hamiltonization, which transforms nonholonomic systems into Hamiltonian systems. We give a geometric account of the Hamiltonization, identify necessary and sufficient conditions for Hamiltonization, and apply the conventional Hamilton–Jacobi theory to the Hamiltonized systems. We show, under a certain sufficient condition for Hamiltonization, that the solutions to the Hamilton–Jacobi equation associated with the Hamiltonized system also solve the nonholonomic Hamilton–Jacobi equation associated with the original Chaplygin system. The results are illustrated through several examples.