CALCULATION OF HAMILTONIAN CONSTRAINT AND MANDELSTAM IDENTITIES OVER EXTENDED KNOT FAMILIES {ψ_i}_2~2, {ψ_i}_4~3 and {ψ_i}_6~4

The actions of the Hamiltonian constraint onto the members of the extended knot families {φi}2^2, {φi}3^4 and {φi}4^6, and the check of their invariance under the Mandelstam identities are given in the extended loop representation of loop quantum gravity.

Projected Runge-Kutta methods for constrained Hamiltonian systems

Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.

Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems

This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of nonholonomic Hamiltonian systems are derived from discrete Hamiltonian action. Secondly, the determining equations and structure equation of Lie symmetry of the system are obtained. Thirdly, the Lie theorems and the conservation quantities are given for the discrete nonholonomic Hamiltonian systems. Finally, an example is discussed to illustrate the application of the results.

Ashtekar-Kodama Gravity as a Classical and Quantum Extension of Loop Quantum Gravity

This paper presents a new theory of gravity, called here Ashtekar-Kodama (AK) gravity, which is based on the Ashtekar-Kodama formulation of loop quantum gravity (LQG), yields in the limit the Einstein equations, and in the quantum regime a full renormalizable quantum gauge field theory. The three fundamental constraints (hamiltonian, gaussian and diffeomorphism) were formulated in 3-dimensional spatial form within LQG in Ashtekar formulation using the notion of the Kodama state with positive cosmological constant Λ. We introduce a 4-dimensional covariant version of the 3-dimensional (spatial) hamiltonian, gaussian and diffeomorphism constraints of LQG. We obtain 32 partial differential equations for the 16 variables Emn (E-tensor, inverse densitized tetrad of the metric) and 16 variables Amn (A-tensor, gravitational wave tensor). We impose the boundary condition: for large distance the E-generated metric g(E) becomes the GR-metric g (normally Schwarzschild-spacetime). The theory based on these Ashtekar-Kodama (AK) equations, and called in the following Ashtekar-Kodama (AK-) gravity has the following properties. • For Λ = 0 the AK equations become Einstein equations, A-tensor is trivial (constant), and the E-generated metric g(E) is identical with the GR-metric g. • When the AK-equations are developed into a Λ-power series, the Λ-term yields a gravitational wave equation, which has only at least quadrupole wave solutions and becomes in the limit of large distance r the (normal electromagnetic) wave equation. • AK-gravity, as opposed to GR, has no singularity at the horizon: the singularity in the metric becomes a (very high) peak. • AK-gravity has a limit scale of the gravitational quantum region 39 μm, which emerges as the limit scale in the objective wave collapse theory of Gherardi-Rimini-Weber. In the quantum region, the AK-gravity becomes a quantum gauge theory (AK quantum gravity) with the Lie group extended SU(2) = ε-tensor-group(four generators) as gauge group and a corresponding covariant derivative. • AK quantum gravity is fully renormalizable, we derive its Lagrangian, which is dimensionally renormalizable, the normalized one-graviton wave function, the graviton propagator, and demonstrate the calculation of cross-section from Feynman diagrams.

A New Finite-dimensional Integrable System Associated to(1+1)-dimensional Soliton Equations

In this paper, a new spectral problem is proposed and the corresponding soliton equations hierarchy are also obtained. Under a constraint between the potentials and the eigenfunctions, the eigenvalue problem is nonlinearized so as to be a new finite-dimensional Hamiltonian system. By resotring to the generating function approach, we obtain conserved integrals and the involutivity of the conserved integrals. The finite-dimensional Hamiltonian system is further proved to be completely integrable in the Liouville sense. Finally, we show the decomposition of the soliton equations.

Bargmann Symmetry Constraint and Binary Nonlinearization of Super NLS-MKdV Hierarchy

An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried out for the super NLS-MKdV hierarchy. Under the obtained symmetry constraint, the n-th flow of the super NLS-MKdV hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems, defined over the super-symmetry manifold R4N|2N with the corresponding dynamical variables x and tn. The integrals of motion required for Liouville integrability are explicitly given.

A NEW CLASSIFICATION OF CONSTRAINTS OF HAMILTONIAN SYSTEMS AND THEIR SOLVABILITY

The present paper proposes a new classification of constraints of Hamilronian systems and using this classification investigates the solvability of such systems.

一族Liouville可积系及其双约束流的Hamilton系统

构造了Loop代数 A2的一个子代数,由此建立了一个3×3等谱问题,由屠规彰格式得到了一族Liouville意义下的可积Hamilton方程族。通过建立双对称约束,得到了该方程族的两组约束流,并将其化为广义Hamilton系统。

从无穷维可积Hamilton系统到有限维可积Hamilton系统的一个一般途径

本文综述了作者关于将无穷维可积Ｈａｍｉｌｔｏｎ系统（ＩＤＩＨＳ）分解为两个可交换的有限维可积Ｈａｍｉｌｔｏｎ系统（ＦＤＩＨＳ）的一般途径方面能做的工作，该途径提出了联系势和特征函数的一般约束（包括高阶约束），提供了从ＩＤＩＨＳ到ＦＤＩＨＳ的一般方法；在零曲率表示理论框架内，统一处理了一族ＩＤＩＨＳ的分解；证明了在一般约束下，一族ＩＤＩＨＳ中的每一个系统都可以分解为两个可交换的ｘ－和ｔ＿ｎ－ＦＤＩＨＳ；建立了构造所有这些ｔ＿ｎ－ＦＤＩＨＳ的Ｈａｍｉｌｔｏｎ函数的普适公式；直接利用ＩＤＩＨＳ的可积结构导出了这些ＦＤＩＨＳ的守恒积分的生成函数及Ｌｉｏｕｖｉｌｌｅ可积性．最后，本文通过例子说明这一一般途径，并在显式约束和高阶约束下得到一些新的ＦＤＩＨＳ．

一族Liouville可积系及其双约束流的Hamilton正则表示

构造了高阶ｌｏｏｐ代数Ａ２的一个特殊子代数，由此建立了一个３×３等谱问题，利用屠格式 得到了一族Ｌｉｏｕｖｉｌｌｅ意义下的可积 Ｈａｍｉｌｔｏｎ方程．通过建立双对称约束，得到了该方程族的两组 约束流，并将其化为正则表示 Ｈａｍｉｌｔｏｎ系统．

基于哈密顿方程的非完整约束控制系统分析及应用

针对航天器多体系统受非完整约束作用时,其动力学方程是强非线性微分方程组而难以求解析解、系统的稳定性及其控制难以进行分析的问题,提出通过带非完整约束的哈密顿控制系统对航天器多体系统的姿态稳定及控制进行分析的方法。采用该方法可以避免求解非线性方程组的边值问题。建立了一种简洁的航天器多体系统姿态稳定性控制方法,而且该方法使得系统的运动控制更加易于从物理意义上进行解释,即使得系统的稳定控制可以理解为系统和控制器之间的能量平衡。通过算例进行了验证和说明。

一族新的可积系及其双约束流的Hamilton正则表示

基于一个新的等谱问题,按屠格式导出了一族新的可积系,具有双Hamilton结构,通过建立双对称约束,得到了该方程族的两组约束流,并将其化为正则的Hamilton系统.

基于对偶变量变分原理的完整约束系统保辛算法

基于对偶变量变分原理,选择积分区间两端位移为独立变量,构造了求解完整约束哈密顿动力系统的高阶保辛算法。首先,利用拉格朗日多项式对作用量中的位移、动量及拉格朗日乘子进行近似;然后,对作用量中不包含约束的积分项采用Gauss积分近似,对作用量中包含约束的积分项采用Lobatto积分近似,从而得到近似作用量;最后,在此近似作用量的基础上,利用对偶变量变分原理,将求解完整约束哈密顿动力系统问题转化为一组非线性方程组的求解。算法具有保辛性和高阶收敛性,能够在位移的插值点处高精度地满足完整约束。算法的收敛阶数及数值性质通过数值算例验证。

一个新非线性可积晶格族和它们的可积辛映射

该文引入一个离散特征值问题,导出一族离散可积系,建立了它们的Hamilton结构,证明了它们Louville可积性.通过谱问题双非线性化,得到了一个可积辛映射与一族有限维完全可积系,最后给出了离散可积系统解的表示.

带有约束的Hamilton系统的特征值问题

本文主要研究 C^n 上 Hamilton 函数 H 在约束条件 G(p,q)=C 下的 Hamilton 系统的周期解的存在性问题,证明了当{H,G}=0时此问题有无限多个解.

THE HIGHER-ORDER CONSTRAINTS AND INTEGRABLE SYSTEMS ASSOCIATED WITH THE BOUSSINESQ EQUATION

There are only a few results concerned with the constraints and finite-dimensional integrable systems which are associated with the third-order eigenvalue problem tbr soliton equation. In particular, the higher-order constraints and corresponding integrable systems have not been studied yet. In the present note, using the Boussinesq equation as

特征值问题的对称约束

利用对称约束，得到了一个与谱问题相联系的新的完全可积Hamiltonian系统，并进一步讨论与之相关的发展方程族的解。

AKNS系统的对称约束及其哈密顿结构

本文基于一般的位势约束,提出一个新的方法使得AKNS方程族的线性问题约化为有限维的Liouville可积的Hamilton系统,并由此导出了相应的对称约束。

与一个多项式特征值问题相联系的完全可积系统

在位势函数和特征函数的约束下，二阶特征值问题被非线性化为一个Liouville意义下的完全可积系统，该特征值问题的Lax对的时间部分的非线性化给出了其对合系统。

Binary nonlinearization of the super classical-Boussinesq hierarchy

The symmetry constraint and binary nonlinearization of Lax pairs for the super classical-Boussinesq hierarchy is obtained. Under the obtained symmetry constraint, the n-th flow of the super classical-Boussinesq hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems, defined over the super-symmetry manifold with the corresponding dynamical variables x and tn. The integrals of motion required for Liouville integrability are explicitly given.