Geometric formulations and variational integrators of discrete autonomous Birkhoff systems

The variational integrators of autonomous Birkhoff systems are obtained by the discrete variational principle. The geometric structure of the discrete autonomous Birkhoff system is formulated. The discretization of mathematical pendulum shows that the discrete variational method is as effective as symplectic scheme for the autonomous Birkhoff systems.     

激光场中一维和三维氢分子离子模型经典动力学行为的比较

应用经典轨迹方法,采用辛算法数值求解激光场中的一维和三维氢分子离子(H2 )的Hamilton正则方程,得到氢分子离子在激光场作用下的经典轨迹,并比较分析氢分子离子一维模型与三维模型的存活、解离、电离和库仑爆炸等动力学行为,以及电子的运动情况的相似之处.数值结果表明,采用一维模型能近似定性反映氢分子离子的动力学行为,并且简便可行.     

非完整系统Chetaev动力学和vakonomic动力学的等价条件

就一般非完整约束系统，从约束方程满足的变分恒等式出发，利用增广位形流形上的向量场定义三类非自由变分，即非完整变分：vakonomic变分、Hlder变分、Suslov变分，并讨论它们之间的关系以及它们成为自由变分的充要条件.利用非完整变分以及相应的积分变分原理建立两类动力学方程：vakonomic方程和Routh方程或Chaplygin方程.通过vakonomic方程分别与Routh方程和Chaplygin方程比较，得到它们具有共同解的两类充分必要条件.这些条件并不是约束的可积性条件. 关键词： 非完整约束 非完整变分 Chetaev条件 vakonomic动力学     

双色激光场中1维共线氢分子离子的经典动力学研究

 运用经典理论方法，并采用辛算法数值求解了双色激光场作用下1维共线氢分子离子(H₂⁺)的哈密顿正则方程，得到了氢分子离子在激光场下的经典轨迹。计算了单色场和双色场下氢分子离子(H₂⁺)的存活几率、电离几率、解离几率、库仑爆炸几率随时间的演化，分析了双色场的相位、强度、强度比及倍频的变化对氢分子离子动力学行为的影响,并给出了相应的物理解释。     

Conformal invariance and Hojman conserved quantities of canonical Hamilton systems

This paper discusses the conformal invariance by infinitesimal transformations of canonical Hamilton systems.The necessary and sufficient conditions of conformal invariance being Lie symmetrical simultaneously by the action of infinitesimal transformations are given.The determining equations of the conformal invariance are gained.Then the Hojman conserved quantities of conformal invariance by special infinitesimal transformations are obtained.Finally an illustrative example is given to verify the results.     

Nonlinear resonance phenomenon of one-dimensional Bose–Einstein condensate under periodic modulation

We investigate the effect of an external periodic modulation on the one-dimensional （1D） Bose-Einstein conden- sate with harmonic trapping potential. By numerically solving the Gross-Pitaevskii equation with symplectic algorithm, the nonlinear resonance phenomenon is shown and the corresponding Fourier spectrum is given. The autoresonance phe- nomenon is also presented under almost periodic external modulation, and it shows that the condensate eventually evolves into quasi-periodic oscillation.     

Research on the discrete variational method for a Birkhoffian system

In this paper, we present a new integration algorithm based on the discrete Pfaff-Birkhoff principle for Birkhoffian systems. It is proved that the new algorithm can preserve the general symplectic geometric structures of Birkhoffian systems. A numerical experiment for a damping oscillator system is conducted. The result shows that the new algorithm can better simulate the energy dissipation than the R-K method, which illustrates that we can numerically solve the dynamical equations by the discrete variational method in a Birkhoffian framework for the systems with a general symplectic structure. Furthermore, it is demonstrated that the results of the numerical experiments are determined not by the constructing methods of Birkhoffian functions but by whether the numerical method can preserve the inherent nature of the dynamical system.     

辛几何算法在特殊Chaplygin系统中的应用研究

采用辛几何算法对广义力具有广义势的特殊Chaplygin系统的动力学性质进行数值研究,并和传统的Runge-Kutta算法进行比较,得出辛几何算法在这类非完整力学系统的计算中具有优越性. 关键词： 辛几何算法 Chaplygin系统 Helmholtz条件     

A necessary and sufficient condition for transforming autonomous systems into linear autonomous Birkhoffian systems

The problem of transforming autonomous systems into Birkhoffian systems is studied. A reasonable form of linear autonomous Birkhoff equations is given. By combining them with the undetermined tensor method, a necessary and sufficient condition for an autonomous system to have a representation in terms of linear autonomous Birkhoff equations is obtained. The methods of constructing Birkhoffian dynamical functions are given. Two examples are given to illustrate the application of the results.     

双色场中氦离子高次谐波的保结构算法研究及应用

 通过数值求解含时Schr ö dinger方程，研究了氦离子在不同双色场下的高次谐波。结果显示，尽管倍频光强度仅为基频光的1/10，高次谐波辐射却发生了极大变化。虽然谐波级次推进不多，但效率大大提高了，平台区平均提高103倍，高倍频光对应的高次谐波产生效率提高104~105倍。氦离子在添加倍频光的双色场作用下高次谐波产生效率极大提高的原因是：倍频光极大地加快了电子电离到连续态及返回基态这一过程，使辐射出的高次谐波光子数大大增加，该初步阐释可为高次谐波真正应用到实际中提供一些理论启示。