The dynamical feature of transition of a Hamiltonian system to a dissipative system

The mechanism of generation and annihilation of attractors during transition from a Hamiltonian system to a dissipative system is studied numerically using the dissipative standard map. The transient process related to the formation of attracting basins of periodic attractors is studied by discussing the evolution of the KAM tori of the standard map. The result shows that as damping increases, attractors are mainly generated from elliptic orbits of the Hamiltonian system and annihilated by colliding with unstable periodic orbits originating from the corresponding hyperbolic orbits of the Hamiltonian system. The transient process also exhibits the general feature of bifurcation.     

Symmetry of Hamiltonian and conserved quantity for a system of generalized classical mechanics

This paper focuses on a new symmetry of Hamiltonian and its conserved quantity for a system of generalized classical mechanics.The differential equations of motion of the system are established.The definition and the criterion of the symmetry of Hamiltonian of the system are given.A conserved quantity directly derived from the symmetry of Hamiltonian of the generalized classical mechanical system is given.Since a Hamilton system is a special case of the generalized classical mechanics,the results above are equally applicable to the Hamilton system.The results of the paper are the generalization of a theorem known for the existing nonsingular equivalent Lagrangian.Finally,two examples are given to illustrate the application of the results.     

Correlation properties of anisotropic XY model with a sudden quench

Starting from a general Hamiltonian which may undergo a quantum phase transition (QPT) with the change of a controllable parameter, we obtain a general conclusion that in a sudden quench system, when the final Hamiltonian is fixed, the behavior of the time-averaged expectation of any observable has close relationship with the gapless excitation of the initial Hamiltonian. To clarify our conclusion, we investigate the two-spin correlation of a XY chain in a transverse field under a sudden quench at zero temperature. The critical property of the derivative of quench two-spin correlation and the long-range correlation of the quench system are analyzed.     

Modeling and Analysis of a Three-Terminal-Memristor-Based Conservative Chaotic System

In this paper, a three-terminal memristor is constructed and studied through changing dual-port output instead of one-port. A new conservative memristor-based chaotic system is built by embedding this three-terminal memristor into a newly proposed four-dimensional (4D) Euler equation. The generalized Hamiltonian energy function has been given, and it is composed of conservative and non-conservative parts of the Hamiltonian. The Hamiltonian of the Euler equation remains constant, while the three-terminal memristor’s Hamiltonian is mutative, causing non-conservation in energy. Through proof, only centers or saddles equilibria exist, which meets the definition of the conservative system. A non-Hamiltonian conservative chaotic system is proposed. The Hamiltonian of the conservative part determines whether the system can produce chaos or not. The non-conservative part affects the dynamic of the system based on the conservative part. The chaotic and quasiperiodic orbits are generated when the system has different Hamiltonian levels. Lyapunov exponent (LE), Poincaré map, bifurcation and Hamiltonian diagrams are used to analyze the dynamical behavior of the non-Hamiltonian conservative chaotic system. The frequency and initial values of the system have an extensive variable range. Through the mechanism adjustment, instead of trial-and-error, the maximum LE of the system can even reach an incredible value of 963. An analog circuit is implemented to verify the existence of the non-Hamiltonian conservative chaotic system, which overcomes the challenge that a little bias will lead to the disappearance of conservative chaos.     

New invariant relations for one critical subsystem of a generalized two-field gyrostat

In this paper new invariant relations for one critical subsystem of a completely integrable Hamiltonian system with three degrees of freedom found by V.V. Sokolov and A.V. Tsyganov, known as a generalized two-field gyrostat, are obtained. The dynamic system that is induced on the invariant four-dimensional submanifolds is almost everywhere Hamiltonian with two degrees of freedom. The type of system motions on this invariant manifold is determined.     

四维相空间中非中心势动力学系统的Poisson 结构和Casimir 函数

将非中心势动力学系统的运动微分方程写成Ermakov形式，得到Ermakov不变量. 运用Hamilton理论，把Ermakov不变量当作Hamiltonian 函数，在四维相空间中建立了非中心势动力学系统的Poisson 结构。结果表明：此Poisson 结构是一退化的结构，而系统具有四个不变量，即Hamiltonian 函数，Ermakov不变量及两个Casimir函数。     

Coupled KdV Equations and Their Explicit Solutions Through Two-Dimensional Hamiltonian System with a Quartic Potential

Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differentiM equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable ease of the two-dimensional Hamiltonian system.     

Renormalization invariance of a Hamiltonian system near the saddle point

The rescaling property of a one degree of freedom Hamiltonian system near the saddle point is analytically studied as regards the transformations of time-periodic perturbations. Two kinds of perturbation functions are considered: (a) linear functions and (b) homogeneous polynomial functions of canonical variables with time-periodic coefficients. The simple rescaling law of the phase-space of the Hamiltonian system near the hyperbolic fixed point with respect to a transformation of the amplitude and phase of the time-periodic perturbation is derived.     

Quantum motion on a torus as a submanifold problem in a generalized Dirac’s theory of second-class constraints

A generalization of Dirac’s canonical quantization scheme for a system with second-class constraints is proposed, in which the fundamental commutation relations are constituted by all commutators between positions, momenta and Hamiltonian, so they are simultaneously quantized in a self-consistent manner, rather than by those between merely positions and momenta which leads to ambiguous forms of the Hamiltonian and the momenta. The application of the generalized scheme to the quantum motion on a torus leads to a remarkable result: the quantum theory is inconsistent if built up in an intrinsic geometric manner, whereas it becomes consistent within an extrinsic examination of the torus as a submanifold in three dimensional flat space with the use of the Cartesian coordinate system. The geometric momentum and potential are then reasonably reproduced.     

Preparation of the Cluster States in a Linear Trap Systems

A scheme for the preparation of four-ion entangled cluster states has been proposed. Two two-level ions are confined in a linear trap and are simultaneously driven with a laser beam. In the Lamb-Dicke regime, we can get the effective Hamiltonian in the interaction picture.The effective Hamiltonian may be used to describe a realistic physical system. The scheme is insensitive to both the initial vibrational state and heating.     

非保守力和非完整约束对Hamilton系统Lie对称性的影响

研究非保守力和非完整约束对Hamilton系统的Lie对称性和守恒量的影响.分别研究了Hamilt on系统受到非保守力和非完整约束作用时，系统的Lie对称性保持不变的条件，同时给出了 系统的结构方程和守恒量保持不变的条件.以著名的Emden方程和Appell-Hamel模型为例进行 了分析讨论. 关键词： 分析力学 Hamilton系统 非保守力 非完整约束 对称性 守恒量     

Kinetic ferromagnetism on a kagome lattice

We study strongly correlated electrons on a kagome lattice at 1/6 (and 5/6) filling. They are described by an extended Hubbard Hamiltonian. We are concerned with the limit |t|相似文献   

Determination of the Electromagnetic Lagrangian from a System of Poisson Brackets

The Lagrangian and Hamiltonian formulations of electromagnetism are reviewed and the Maxwell equations are obtained from the Hamiltonian for a system of many electric charges. It is shown that three of the equations which were obtained from the Hamiltonian, namely the Lorentz force law and two Maxwell equations, can be obtained as well from a set of postulated Poisson brackets. It is shown how the results derived from these brackets can be used to reconstruct the original Lagrangian for the theory aided by some reasoning based on physical concepts.     

Nonlinear dynamics analysis of cluster-shaped conservative flows generated from a generalized thermostatted system

The thermostatted system is a conservative system different from Hamiltonian systems,and has attracted much attention because of its rich and different nonlinear dynamics.We report and analyze the multiple equilibria and curve axes of the cluster-shaped conservative flows generated from a generalized thermostatted system.It is found that the cluster-shaped structure is reflected in the geometry of the Hamiltonian,such as isosurfaces and local centers,and the shapes of cluster-shaped chaotic flows and invariant tori rely on the isosurfaces determined by initial conditions,while the numbers of clusters are subject to the local centers solved by the Hessian matrix of the Hamiltonian.Moreover,the study shows that the cluster-shaped chaotic flows and invariant tori are chained together by curve axes,which are the segments of equilibrium curves of the generalized thermostatted system.Furthermore,the interesting results are vividly demonstrated by the numerical simulations.     

Robust tori in a double-waved Hamiltonian model

A Hamiltonian system perturbed by two waves with particular wave numbers can present robust tori, which are barriers created by the vanishing of the perturbed Hamiltonian at some defined positions. When robust tori exist, any trajectory in phase space passing close to them is blocked by emergent invariant curves that prevent the chaotic transport. Our results indicate that the considered particular solution for the two waves Hamiltonian model shows plenty of robust tori blocking radial transport.     

Lower-Dimensional Invariant Tori for Perturbations of a Class of Non-convex Hamiltonian Functions

We consider a class of quasi-integrable Hamiltonian systems obtained by adding to a non-convex Hamiltonian function of an integrable system a perturbation depending only on the angle variables. We focus on a resonant maximal torus of the unperturbed system, foliated into a family of lower-dimensional tori of codimension 1, invariant under a quasi-periodic flow with rotation vector satisfying some mild Diophantine condition. We show that at least one lower-dimensional torus with that rotation vector always exists also for the perturbed system. The proof is based on multiscale analysis and resummation procedures of divergent series. A crucial role is played by suitable symmetries and cancellations, ultimately due to the Hamiltonian structure of the system.     

Energy Spectrum of a Bipartite Complicated-Coupled-Oscillator System

We present a general discussion on the eigenstate problem of a bipartite complicated-coupled-oscillator system. By use of a generalized intermediate entangled state representation, the eigenvalue and eigenfunction of Hamiltonian are analytically obtained. As its application, we obtain the energy spectrum for two special situations of Hamiltonian.     

Bargmann Symmetry Constraint for a Family of Liouville Integrable Differential-Difference Equations

A family of integrable differential-difference equations is derived from a new matrix spectral problem. The Hamiltonian forms of obtained differential-difference equations are constructed. The Liouville integrability for the obtained integrable family is proved. Then, Bargmann symmetry constraint of the obtained integrable family is presented by binary nonliearization method of Lax pairs and adjoint Lax pairs. Under this Bargmann symmetry constraints, an integrable symplectic map and a sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense are worked out, and every integrable differential-difference equations in the obtained family is factored by the integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system.     

Projection of a quantum state on eigenstates of average Hamiltonian

A scheme for projecting an arbitrary quantum state on eigenstates of average Hamiltonian is described. As an experimental example, projection on entangled Bell states, which are eigenstates of specially constructed average Hamiltonian, is demonstrated for a system of two dipolar-coupled nuclear spins. The results of a direct and time-reversed evolution are added to average out the coherences between different eigenstates and accomplish the projection.