The Hamiltonian Canonical Form for Euler-Lagrange Equations

Based on the theory of calculus of variation, some sufficient conditions are given for some Euler-Lagrange equations to be equivalently represented by finite or even infinite many Hamiltonian canonical equations. Meanwhile, some further applications for equations such as the KdV equation, MKdV equation, the general linear Euler-Lagrange equation and the cylindric shell equations are given.     

Formulation of Hamiltonian Equations for Fractional Variational Problems

An extension of Riewes fractional Hamiltonian formulation is presented for fractional constrained systems. The conditions of consistency of the set of constraints with equations of motion are investigated. Three examples of fractional constrained systems are analyzed in details.On leave of absence from Institute of Space Sciences, P.O.BOX MG-23, R 76900, Magurele–Bucharest, Romania     

Bosonic Operator Realization of Hamiltonian for a Superconducting Quantum Interference Device

Based on the appropriate bosonic phase operator diagonalized in the entangled state representation we construct the Hamiltonian operator model for a superconducting quantum interference device. The current operator and voltage operator equations are derived.     

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.     

On Feasibility of Variable Separation Method Based on Hamiltonian System for a Class of Plate Bending Equations

The eigenfunction system of infinite-dimensional Hamiltonian operators appearing in the bending problem of rectangular plate with two opposites simply supported is studied. At first, the completeness of the extended eigenfunction system in the sense of Cauchy＇s principal value is proved. Then the incompleteness of the extended eigenfunction system in general sense is proved. So the completeness of the symplectic orthogonal system of the infinite-dimensional Hamiltonian operator of this kind of plate bending equation is proved. At last the general solution of the infinite dimensional Hamiltonian system is equivalent to the solution function system series expansion, so it gives to theoretical basis of the methods of separation of variables based on Hamiltonian system for this kind of equations.     

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.     

Completeness of Eigenfunction Systems for Off-Diagonal Infinite-Dimensional Hamiltonian Operators

For the off-diagonal infinite dimensional Hamiltonian operators, which have at most countable eigenvalues, a necessary and sufficient condition of the eigenfunction systems to be complete in the sense of Cauchy principal value is presented by using the spectral symmetry and new orthogonal relationship of the operators. Moreover, the above result is extended to a more general case. At last, the completeness of eigenfunction systems for the operators arising from the isotropic plane magnetoelectroelastic solids is described to illustrate the effectiveness of the criterion. The whole results offer theoretical guarantee for separation of variables in Hamiltonian system for some mechanics equations.     

哈密顿Ermakov系统的形式不变性

把形式不变性的方法用于研究哈密顿Ermakov系统，从哈密顿Ermakov系统的形式不变性出发，运用比较系数法得到与形式不变性相应的点对称变换生成元的表达式及势能所满足的偏微分方程.结果表明，在点对称变换下，只有自治的哈密顿Ermakov系统才具有形式不变性. 关键词： 哈密顿Ermakov系统 拉格朗日函数 点对称变换 形式不变性     

A Difference Hamiltonian Operator and a Hierarchy of Generalized Toda Lattice Equations

A difference Ha-miltonian operator with three arbitrary constants is presented. When the arbitrary constants -in the Hamiltonian operator are suitably chosen, a pair of Hamiltonian operators are given. The resulting Hamiltonian pair yields a difference hereditary operator. Using Magri scheme of bi-Hamiltonian formulation, a hierarchy of the generalized Toda lattice equations is constructed. Finally, the discrete zero curvature representation is given for the resulting hierarchy.     

Cooperatively surrounding control for multiple Euler–Lagrange systems subjected to uncertain dynamics and input constraints

In this paper, we investigate cooperatively surrounding control(CSC) of multi-agent systems modeled by Euler–Lagrange(EL) equations under a directed graph. With the consideration of the uncertain dynamics in an EL system, a backstepping CSC algorithm combined with neural-networks is proposed first such that the agents can move cooperatively to surround the stationary target. Then, a command filtered backstepping CSC algorithm is further proposed to deal with the constraints on control input and the absence of neighbors' velocity information. Numerical examples of eight satellites surrounding one space target illustrate the effectiveness of the theoretical results.     

A Difference Hamiltonian Operator and a Hierarchy of Generalized Toda Lattice Equations

A difference Hamiltonian operator with three arbitrary constants is presented. When the arbitrary constants in the Hamiltonian operator are suitably chosen, a pair of Hamiltonian operators are given. The resulting Hamiltonian pair yields a difference hereditary operator. Using Magri scheme of bi-Hamiltonian formulations a hierarchy of the generalized Toda lattice equations is constructed. Finally, the discrete zero curvature representation is given for the resulting hierarchy.     

Integrability of S-deformable surfaces: Conservation laws,Hamiltonian structures and more

We present infinitely many nonlocal conservation laws, a pair of compatible local Hamiltonian structures and a recursion operator for the equations describing surfaces in three-dimensional space that admit nontrivial deformations which preserve both principal directions and principal curvatures (or, equivalently, the shape operator).     

量子系统哈密顿量的必要条件

在此前的量子理论中,哈密顿量被要求是厄米算符,这既保证了其本征值谱为实又保证了动力学演化过程几率守恒。近年来,一些非厄米哈密顿量被发现同样满足这两条要求。然而,这两条要求都是哈密顿量可描述量子系统动力学的充分条件而非必要条件。文章中我们从量子化条件和动力学演化方程出发,考察一般形式的哈密顿量,表述为产生算符和湮灭算符之正规积的形式,欲为恰当的哈密顿量所应满足的必要条件。对于形如$\\hat{H}=\\hat{p}^2+\\mathrm{i} \\hat{x}^3$和$\\hat{H}=\\hat{p}^2-\\hat{x}^4$这样的近年来得到广泛关注的非厄米哈密顿量,容易验证它们满足必要性条件。必要性条件可以用来及时排除那些不恰当的非厄米哈密顿量形式。     

Cooperative impulsive formation control for networked uncertain Euler–Lagrange systems with communication delays

This paper investigates the cooperative formation problem via impulsive control for a class of networked Euler–Lagrange systems. To reduce the energy consumption and communication frequency, the impulsive control method and cooperative formation control approach are combined. With the consideration of system uncertainties and communication delays among agents, neural networks-based adaptive technique is used for the controller design. Firstly, under the constraint that each agent interacts with its neighbors only at some sampling moments, an adaptive neural-networks impulsive formation control algorithm is proposed for the networked uncertain Euler–Lagrange systems without communication delays. Using Lyapunov stability theory and Laplacian potential function in the graph theory, we conclude that the formation can be achieved by properly choosing the constant control gains. Further, when considering communication delays,a modified impulsive formation control algorithm is proposed, in which the extended Halanay differential inequality is used to analyze the stability of the impulsive delayed dynamical systems. Finally, numerical examples and performance comparisons with continuous algorithm are provided to illustrate the effectiveness of the proposed methods.     

Form Invariance of Raitzin's Canonical Equations of Relativistic Mechanical System

A form invariance of Raitzin's canonical equations of relativistic mechanical system is studied. First, the Raitzin's canonical equations of the system are established. Next, the definition and criterion of the form invariance in the system under infinitesimal transformations of groups are given. Finally, the relation between the form invariance and the conserved quantity of the system is obtained and an example is given to illustrate the application of the result.     

A Canonical Null Gauge for the Einstein Equations

Associated with any choice of outgoing null (characteristic) coordinate, we construct a null tetrad which is determined uniquely by purely local information. Unlike other well-known characteristic gauge conditions [1–4], this canonical tetrad is determined at any point solely by the metric and its first derivatives at that point. The tetrad leads to a radial coordinate which is also uniquely determined by purely local information. These properties greatly simplify the structure of the hypersurface Einstein equations and may be used to directly compare metrics with compatible null foliations.     

An algorithm and its application for obtaining some kind of infinite-dimensional Hamiltonian canonical formulation

Using factorization viewpoint of differential operator, this paper discusses how to transform a nonlinear evolution equation to infinite-dimensional Hamiltonian linear canonical formulation. It proves a sufficient condition of canonical factorization of operator, and provides a kind of mechanical algebraic method to achieve canonical evolution equation, infinite-dimensional Hamiltonian canonical system, factorization of differential operator, commutator,evolution equation, infinite-dimensional Hamiltonian canonical system, factorization of differential operator, commutatorProject supported by the National Natural Science Foundation of China (Grant No~10562002) and the Natural Science Foundation of Nei Mongol, China (Grant No~200508010103).2007-05-09{\partial}/{\partial x}Corresponding author．E-mail：alatanca@imu．edu.cn/qk/85823A/200711/25754042.html0200, 03405/9/2007 12:00:00 AMUsing factorization viewpoint of differential operator, this paper discusses how to transform a nonlinear evolution equation to infinite-dimensional Hamiltonian linear canonical formulation. It proves a sufficient condition of canonical factorization of operator, and provides a kind of mechanical algebraic method to achieve canonical $`{\partial}/{\partial x}'$-type expression, correspondingly. Then three examples are given, which show the application of the obtained algorithm. Thus a novel idea for inverse problem can be derived feasibly.http://cpb.iphy.ac.cn/CN/10.1088/1009-1963/16/11/002https://cpb.iphy.ac.cn/CN/article/downloadArticleFile.do?attachType=PDF&id=1088382007-11-20'-type expression, correspondingly. Then three examples are given, which show the application of the obtained algorithm. Thus a novel idea for inverse problem can be derived feasibly.