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.     

Birkhoff系统的时间积分定理

研究Birkhoff系统的时间积分定理.建立Birkhoff系统的时间积分等式，由此等式导出系统的类功率方程，Pfaff-Birkhoff积分变分原理以及Pfaff-Birkhoff-d′Alembert微分变分原理. 关键词： Birkhoff系统 时间积分等式 类功率方程 变分原理     

Structure-preserving algorithms for Birkhoffian systems

We propose a new approach to construct structure-preserving algorithms for Birkhoffian systems. First, the Pfaff–Birkhoff variational principle is discretized, and based on the discrete variational principle the discrete Birkhoffian equations are obtained. Then, taking the discrete equations as an algorithm, the corresponding discrete flow is proved to be symplectic. That means the algorithm preserves the symplectic structure of Birkhoffian systems. Simulation results of the given example indicate that structure-preserving algorithms obtained by this method have great advantage in conserving conserved quantities.     

相对论Birkhoff系统的平衡稳定性

研究相对论Birkhoff系统的平衡稳定性.给出相对论Birkhoff自治系统、半自治系统和非自治系统的平衡方程、受扰运动方程和一次近似方程;给出判定平衡稳定性的一次近似方法及其判据;讨论相对论Birkhoff系统平衡稳定性和经典Birkhoff系统平衡稳定性的关系.给出实例以说明方法的应用 关键词： 相对论 Birkhoff系统 平衡稳定性 一次近似方法     

Discrete variational principle and first integrals for Lagrange--Maxwell mechanico-electrical systems

This paper presents a discrete variational principle and a method to build first-integrals for finite dimensional Lagrange--Maxwell mechanico-electrical systems with nonconservative forces and a dissipation function. The discrete variational principle and the corresponding Euler--Lagrange equations are derived from a discrete action associated to these systems. The first-integrals are obtained by introducing the infinitesimal transformation with respect to the generalized coordinates and electric quantities of the systems. This work also extends discrete Noether symmetries to mechanico-electrical dynamical systems. A practical example is presented to illustrate the results.     

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.     

广义Birkhoff系统的时间积分定理

研究了广义Birkhoff系统的时间积分定理.给出系统的时间积分等式,并由此等式导出类能量方程、类维里定理、一个积分变分原理和一个微分变分原理. 关键词： 广义Birkhoff系统 时间积分定理 类能量方程 变分原理     

The Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems

This paper investigates the Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems. The variational principle of discrete mechanics, from which discrete motion equations of systems are deduced, is generalized to the case of including the time variational. The requirement for an invariant group transformation is defined to be the Lie symmetry and the criterion when the Noether conserved quantities may be obtained from Lie symmetries is also presented. An example is discussed for applications of the results.     

基于二阶离散变分方法的非线性映射参数估计

提出一种估计非线性映射未知参数的二阶离散变分方法.首先针对非线性离散混沌系统, 利用变分方法导出了伴随方程和目标泛函梯度, 以此为基础利用二阶离散变分方法给出了二阶伴随方程和精确计算Hessian矩阵-向量乘积的显式表达式; 其次设计了估计非线性映射未知参数的新算法, 并以此对Hyperhenón映射和二维抛物映射中的未知参数进行了精确的估计. 数值仿真结果表明了该方法的有效性和优点. 关键词： 非线性映射 参数估计 二阶离散变分方法 伴随方程     

非线性映射参数辨识的离散变分方法

提出一种辨识非线性映射系统中未知参数的离散变分方法, 对以x_k+1 = F(x_k,θ)为状态控制方程的所有映射混沌系统具有通用性. 对典型的Logistic映射和Henón映射中的未知参数进行了估计, 仿真结果表明了该方法的有效性.     

离散Lagrange系统的Lie对称性

研究离散Lagrange系统的Lie对称性. 根据离散变分原理建立离散系统的运动方程. 给出离散运动方程Lie对称性的定义和确定方程. 举例说明结果的应用. 关键词： 离散Lagrange系统 离散变分原理 Lie对称性 确定方程     

Variational principle and dynamical equations of discrete nonconservative holonomic systems

By analogue with the methods and processes in continuous mechanics, a Lagrangian formulation and a Hamiltonian formulation of discrete mechanics are obtained. The dynamical equations including Euler--Lagrange equations and Hamilton's canonical equations of the discrete nonconservative holonomic systems are derived on a discrete variational principle. Some illustrative examples are also given.     

Symmetries and variational calculation of discrete Hamiltonian systems

We present a numerical simulation method of Noether and Lie symmetries for discrete Hamiltonian systems. The Noether and Lie symmetries for the systems are proposed by investigating the invariance properties of discrete Lagrangian in phase space. The numerical calculations of a two-degree-of-freedom nonlinear harmonic oscillator show that the difference discrete variational method preserves the exactness and the invariant quantity.     

广义Birkhoff自治系统平衡状态的流形稳定性

研究广义Birkhoff自治系统平衡状态流形稳定性.建立广义Birkhoff自治系统的受绕运动方程和平衡方程.由Liapunov稳定性理论给出广义Birkhoff自治系统的平衡状态流形稳定性的有关判据.举例说明结果的应用.     

Noether-type theorem for discrete nonconservative dynamical systems with nonregular lattices

We investigate Noether symmetries and conservation laws of the discrete nonconserved systems with nonregular lattices. The operators of discrete transformation and discrete differentiation to the right and left are introduced for the systems. Based on the invariance of discrete Hamilton action on nonregular lattices of the systems with the nonconserved forces under the infinitesimal transformations with respect to the time and generalized coordinates, we give the discrete analog of generalized variational f...     

Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-four matrices is presented, and a discrete four-by-four matrix spectral problem is introduced. A hierarchy of discrete integrable coupling systems is derived. The obtained integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discrete Hamiltonian systems.     

Multi-symplectic Birkhoffian structure for PDEs with dissipation terms

A generalization of the multi-symplectic form for Hamiltonian systems to self-adjoint systems with dissipation terms is studied. These systems can be expressed as multi-symplectic Birkhoffian equations, which leads to a natural definition of Birkhoffian multi-symplectic structure. The concept of Birkhoffian multi-symplectic integrators for Birkhoffian PDEs is investigated. The Birkhoffian multi-symplectic structure is constructed by the continuous variational principle, and the Birkhoffian multi-symplectic integrator by the discrete variational principle. As an example, two Birkhoffian multi-symplectic integrators for the equation describing a linear damped string are given.     

Difference Discrete Variational Principle in Discrete Mechanics and Symplectic Algorithm

We propose the difference discrete variational principle in discrete mechanics and symplectic algorithmwith variable step-length of time in finite duration based upon a noncommutative differential calculus established inthis paper. This approach keeps both symplecticity and energy conservation discretely. We show that there exists thediscrete version of the Euler-Lagrange cohomology in these discrete systems. We also discuss the solution existencein finite time-length and its site density in continuous limit, and apply our approach to the pendulum with periodicperturbation. The numerical results are satisfactory.     

Difference Discrete Variational Principle in Discrete Mechanics and Symplectic Algorithm

We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This approach keeps both symplecticity and energy conservation discretely. We show that there exists the discrete version of the Euler-Lagrange cohomology in these discrete systems. We also discuss the solution existence in finite time-length and its site density in continuous limit, and apply our approach to the pendulum with periodic perturbation. The numerical results are satisfactory.