A quantum model with one fermionic degree of freedom

A quantum model with one fermionic degree of freedom is discussed in detail. The operator action of the model has local operator gauge symmetry. A group of constrains on operator gauge potentialB ₀ and gauge transformation operatorU from some physical requirement are obtained. The Euler-Lagrange equation of motion of fermionic operator φ is just the usual equation of motion of fermion type. And the Euler-Lagrange equation of motion of operator gauge potentialB ₀ is just a constraint, which is just. the canonical quantization condition of fermion.     

A compressible Hamiltonian electromagnetic gyrofluid model

A Lie-Poisson bracket is presented for a four-field gyrofluid model with magnetic field curvature and compressible ions, thereby showing the model to be Hamiltonian. The corresponding Casimir invariants are presented, and shown to be associated to four Lagrangian invariants advected by distinct velocity fields. This differs from a cold ion limit, in which the Lie-Poisson bracket transforms into the sum of direct and semidirect products, leading to only three Lagrangian invariants.     

On the Linearization of Hamiltonian Systems on Poisson Manifolds

The linearization of a Hamiltonian system on a Poisson manifold at a given (singular) symplectic leaf gives a dynamical system on the normal bundle of the leaf, which is called the first variation system. We show that the first variation system admits a compatible Hamiltonian structure if there exists a transversal to the leaf which is invariant with respect to the flow of the original system. In the case where the transverse Lie algebra of the symplectic leaf is semisimple, this condition is also necessary.     

Numerical integration of constrained Hamiltonian systems using Dirac brackets

We study the numerical properties of the equations of motion of constrained systems derived with Dirac brackets. This formulation is compared with one based on the extended Hamiltonian. As concrete examples, a pendulum in Cartesian coordinates and a chain molecule are treated.

     

一些数学物理问题中的Hamilton方程

讨论了新的一系列在数学物理方程中微分方程的Hamilton正则表示,其中包括变系数2阶对称方程的Hamilton系统,关于常系数的4阶对称方程新的非齐次Hamilton表示,MKdV方程以及KP方程的正则表示.     

具有不依赖于时间的不变量的三维常微分方程组的Hamilton结构

本文证明了具有不依赖于时间的不变量的三维常微分方程组所描述的动力系统相对于一广义Poisson括号可以改写为Hamilton系统,并且这些不变量就是Hamilton量。作为例子,我们讨论了Kermack-Mckendrick传染病模型,所得结果推广了Y.Nutku的结果。     

Zonal-meridional decomposition and the Hamiltonian description of planetary fluid dynamics

The basic properties of planetary flows are studied within the framework of the noncanonical Hamiltonian approach formulated by Morrison. A zonal-symmetric decomposition is applied in order to characterize the contributions of the different dynamical terms. Steady states and the Lorenz energy and angular momentum cycles are also written within the Lie-Poisson bracket formalism.     

无穷维Hamilton 算子广义本征函数系的完备性及其在弹性力学中的应用

本文利用无穷维Hamilton 算子的结构特性, 得到由算子的基本本征函数和若当型本征函数构成的广义本征函数系在Cauchy 主值意义下完备的充分必要条件. 进而将结果应用于弹性力学中的板弯曲问题. 相应结论为Hamilton 体系下的分离变量法(弹性力学求解新体系) 提供了理论保证.     

The geometry of the generalized algebraic Riccati equation and of the singular Hamiltonian system

This paper analyses the properties of the solutions of the generalized continuous algebraic Riccati equation from a geometric perspective. This analysis reveals the presence of a subspace that may provide an appropriate degree of freedom to stabilize the system in the related optimal control problem even in cases where the Riccati equation does not admit a stabilizing solution.     

一个新的离散的演化方程族及其Hamilton结构

A new discrete isospectral problem is introduced, from which a hierarchy of Laxintegrable lattice equation is deduced. By using the trace identity, the correspondingHamiltonian structure is given and its Liouville integrability is proved.     

A quantum model of a real scalar field

A quantum model of a real scalar field with local operator gauge symmetry is discussed. In the localized theory, in order to keep the local operator gauge symmetry, an operator gauge potential BB _μ, is needed. By combining the constraint of operator gauge potentialB _μ, and the microscopic causality theorem, the usual canonical quantization condition of a real scalar field is obtained. Therefore, a quantum model of a real scalar field without the usual procedure of quantizing a related classical model can be directly constructed. Project supported in part by T.D. Lee’s NNSF Grant, National Natural Science Foundation of China, Foundation of Ph. D. Directing Programme of Chinese Universities and the Chinese Academy of Sciences.     

Permutations of tori in integrable Hamiltonian systems and spectral series of pseudodifferential operators

In the present paper, we study integrable Hamiltonian systems with two degrees of freedom, whose regular level sets consist of several Liouville tori, and the bifurcation diagram has an isolated point. We study assumptions under which going around the singular point causes a permutation of the tori. We also consider a quantum analog of this situation and give model examples.     

Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives

The link between the treatments of constrained systems with fractional derivatives by using both Hamiltonian and Lagrangian formulations is studied. It is shown that both treatments for systems with linear velocities are equivalent.     

一族Liouville可积的有限维Hamilton系统

本文生成了一族Liouville可积的Hamilton相流彼此可交换的有限维Hamilton系统,并且给出了一串对合的显式公共运动积分及其一组对合的显式生成元.     

Regularity of the Attractor for the Dissipative Hamiltonian Amplitude Equation Governing Modulated Wave Instabilities

Abstract In the present paper, the existence of global attractor for dissipative Hamiltonian amplitude equationgoverning the modulated wave instabilities in E_0 is considered.By a decomposition of solution operator,it isshown that the global attractor in E_0 is actually equal to a global attractor in E_1.     

Universality of the relaxation structure of equations for the dynamics of continuous media and dissipative Poisson brackets

We generalize the Hamilton equations for dynamical processes with relaxation. We introduce a dissipative Poisson bracket in terms of the dissipation function. We obtain the universal structure of the relaxation terms in the equations for the dynamics of condensed media and verify this result for structureless liquids, elastic solids, and quantum liquids. In the examples of the condensed media under consideration, we obtain expressions for the dissipative Poisson brackets for the complete set of dynamical parameters. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 2, pp. 277–291, February, 2009.     

Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd-dimensional. Under Rüssmann-type non-degenerate condition, by introducing a modified linear KAM iterative scheme, we proved that the majority of the lower-dimensional hyperbolic invariant tori persist under small perturbations for generalized Hamiltonian systems.     

The Proof of the Dirac Conjecture for a Class of Finite-Dimensional Dirac Hamiltonian Systems

Mathematical Notes -     

Hamiltonian Dynamics of Biaxial Nematics with the Molecular Shape Taken into Account

We consider the dynamics of biaxial nematics following the Hamiltonian approach. The hydrodynamic parameters related to the broken symmetry are introduced in terms of the distortion tensor. The densities and flows of additive integrals of motion are represented in terms of the thermodynamic potential. We obtain the ideal hydrodynamic equations and study the spectra of collective excitations of biaxial nematics taking the molecular shape into account.