Hamiltonian Analysis of 3-Dimensional Connection Dynamics in Bondi-like Coordinates

The Hamiltonian analysis for a 3-dimensional connection dynamics of so(1, 2), spanned by {L_(-+), L_(-2), L_(+2)}instead of {L_(01), L_(02), L_(12)}, is first conducted in a Bondi-like coordinate system. The symmetry of the system is clearly presented. A null coframe with 3 independent variables and 9 connection coefficients are treated as basic configuration variables. All constraints and their consistency conditions, the solutions of Lagrange multipliers as well as the equations of motion are presented. There is no physical degree of freedom in the system. The Ban?ados–Teitelboim–Zanelli(BTZ)spacetime is discussed as an example to check the analysis. Unlike the ADM formalism, where only non-degenerate geometries on slices are dealt with and the Ashtekar formalism, where non-degenerate geometries on slices are mainly concerned though the degenerate geometries may be studied as well, in the present formalism the geometries on the slices are always degenerate though the geometries for the spacetime are not degenerate.     

Macroscopic Time Evolution and MaxEnt Inference for Closed Systems with Hamiltonian Dynamics

MaxEnt inference algorithm and information theory are relevant for the time evolution of macroscopic systems considered as problem of incomplete information. Two different MaxEnt approaches are introduced in this work, both applied to prediction of time evolution for closed Hamiltonian systems. The first one is based on Liouville equation for the conditional probability distribution, introduced as a strict microscopic constraint on time evolution in phase space. The conditional probability distribution is defined for the set of microstates associated with the set of phase space paths determined by solutions of Hamilton’s equations. The MaxEnt inference algorithm with Shannon’s concept of the conditional information entropy is then applied to prediction, consistently with this strict microscopic constraint on time evolution in phase space. The second approach is based on the same concepts, with a difference that Liouville equation for the conditional probability distribution is introduced as a macroscopic constraint given by a phase space average. We consider the incomplete nature of our information about microscopic dynamics in a rational way that is consistent with Jaynes’ formulation of predictive statistical mechanics, and the concept of macroscopic reproducibility for time dependent processes. Maximization of the conditional information entropy subject to this macroscopic constraint leads to a loss of correlation between the initial phase space paths and final microstates. Information entropy is the theoretic upper bound on the conditional information entropy, with the upper bound attained only in case of the complete loss of correlation. In this alternative approach to prediction of macroscopic time evolution, maximization of the conditional information entropy is equivalent to the loss of statistical correlation, and leads to corresponding loss of information. In accordance with the original idea of Jaynes, irreversibility appears as a consequence of gradual loss of information about possible microstates of the system.     

An Ergodic Sampling Scheme for Constrained Hamiltonian Systems with Applications to Molecular Dynamics

This article addresses the problem of computing the Gibbs distribution of a Hamiltonian system that is subject to holonomic constraints. In doing so, we extend recent ideas of Cancès et al. (M2AN 41(2), 351–389, 2007) who could prove a Law of Large Numbers for unconstrained molecular systems with a separable Hamiltonian employing a discrete version of Hamilton’s principle. Studying ergodicity for constrained Hamiltonian systems, we specifically focus on the numerical discretization error: even if the continuous system is perfectly ergodic this property is typically not preserved by the numerical discretization. The discretization error is taken care of by means of a hybrid Monte-Carlo algorithm that allows for sampling bias-free expectation values with respect to the Gibbs measure independently of the (stable) step-size. We give a demonstration of the sampling algorithm by calculating the free energy profile of a small peptide.     

Classical Motion of Complex 2-D Non-Hermitian Hamiltonian Systems

Classical motion of complex 2-D non-Hermitian Hamiltonian systems is investigated to identify periodic, unbounded and chaotic trajectories. The caustic curves, the Lyapunov exponents, and spectral analysis have been used to identify periodic and chaotic trajectories. Using classical trajectories, we were able to predict quantum transition frequaencies of pseudo-Hermitian non-PT symmetric systems accurately. This indicates that there exists a correspondence between classical mechanics and quantum mechanics for certain non-Hermitian Hamiltonians as in the case of real Hermitians.     

Extensions of 1-Dimensional Polytropic Gas Dynamics

Abstract

1-dimensional polytropic gas dynamics is integrable for trivial reasons, having 2 < 3 components. It is realized as a subsystem of two different integrable systems: an infinite-component hydrodynamic chain of Lax type, and a 3-component system not of Lax type.     

Point-Form Hamiltonian Dynamics and Applications

This short review summarizes recent developments and results in connection with point-form dynamics of relativistic quantum systems. We discuss a Poincaré invariant multichannel formalism which describes particle production and annihilation via vertex interactions that are derived from field theoretical interaction densities. We sketch how this rather general formalism can be used to derive electromagnetic form factors of confined quark?Cantiquark systems. As a further application it is explained how the chiral constituent quark model leads to hadronic states that can be considered as bare hadrons dressed by meson loops. Within this approach hadron resonances acquire a finite (non-perturbative) decay width. We will also discuss the point-form dynamics of quantum fields. After recalling basic facts of the free-field case we will address some quantum field theoretical problems for which canonical quantization on a space?Ctime hyperboloid could be advantageous.     

Hamiltonian Systems of Hydrodynamic Type in 2 + 1 Dimensions

We investigate multi-dimensional Hamiltonian systems associated with constant Poisson brackets of hydrodynamic type. A complete list of two- and three-component integrable Hamiltonians is obtained. All our examples possess dispersionless Lax pairs and an infinity of hydrodynamic reductions.     

Symplectic Symmetries of Hamiltonian Systems

Abstract

We prove a generalization to the case of s × s matrix linear differential operators of the classical theorem of E. Cotton giving necessary and sufficient conditions for equivalence of eigenvalue problems for scalar linear differential operators. The conditions for equivalence to a matrix Schrödinger operator are derived and formulated geometrically in terms of vanishing conditions on the curvature of a gl(s, R)-valued connection. These conditions are illustrated on a class of matrix differential operators of physical interest, arising by symmetry reduction from Dirac’s equation for a spinor field minimally coupled with a cylindrically symmetric magnetic field.     

Non-Lie Symmetry Groups of (2+1)-Dimensional Nonlinear Systems

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known (2 1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik-Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only special cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.     

Non-Lie Symmetry Groups of （2＋1）-Dimensional Nonlinear Systems

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known （2＋1）-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only speciai cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.     

On the Integrability of (2+1)-Dimensional Quasilinear Systems

A (2+1)-dimensional quasilinear system is said to be integrable if it can be decoupled in infinitely many ways into a pair of compatible n-component one-dimensional systems in Riemann invariants. Exact solutions described by these reductions, known as nonlinear interactions of planar simple waves, can be viewed as natural dispersionless analogues of n-gap solutions. It is demonstrated that the requirement of the existence of sufficiently many n-component reductions provides the effective classification criterion. As an example of this approach we classify integrable (2+1)-dimensional systems of conservation laws possessing a convex quadratic entropy.     

Relaxation Times for Hamiltonian Systems

Usually, the relaxation times of a gas are estimated in the frame of the Boltzmann equation. In this paper, instead, we deal with the relaxation problem in the frame of the dynamical theory of Hamiltonian systems, in which the definition itself of a relaxation time is an open question. We introduce a lower bound for the relaxation time, and give a general theorem for estimating it. Then we give an application to a concrete model of an interacting gas, in which the lower bound turns out to be of the order of magnitude of the relaxation times observed in dilute gases.     

Integrable Coupling of （2＋1）-Dimensional Multi-component DLW Integrable Hierarchy and Its Hamiltonian Structure

A new multi-component Lie algebra is constructed, and a type of new loop algebra is presented. A （2＋1）-dimensional multi-component DLW integrable hierarchy is obtained by using a （2＋1）-dimensional zero curvature equation. Furthermore, the loop algebra is expanded into a larger one and a type of integrable coupling system and its corresponding Hamiltonian structure are worked out.     

Nonpropagating Solitary Waves in (2+1)-Dimensional Nonlinear Systems

By means of extended homogeneous balance method and variable separation approach, quite a general variable separation solution of the (2+1)-dimensional Broer-Kaup-Kupershmidt equation is derived. From the variable separation solution and by selecting appropriate functions, a new class of (2+1)-dimensional nonpropagating solitary waves are found. The novel features exhibited by these new structures are first revealed.     

C2H4价壳层电离连续区光解离研究

在量子数亏损框架下,利用多重散射自洽方法计算了C2 H4 价壳层电离连续区光吸收谱.通过对势形共振的峰位分析,探讨了势形共振对光解离动力学的影响. The photoabsorption spectra in the valence-shell continuum region of C 2H 4 are calculated by the multi-scattering self-consistent-field method in the framework of the quantum defect theory. The energy positions of potential shape resonance are determined. The influence of the potential shape resonance on photodissociation dynamics is discussed.     

Asymptocic Freedom of Gluons in Hamiltonian Dynamics

We derive asymptotic freedom of gluons in terms of the renormalized SU(3) Yang–Mills Hamiltonian in the Fock space. Namely, we use the renormalization group procedure for effective particles to calculate the three-gluon interaction term in the front-form Yang–Mills Hamiltonian using a perturbative expansion in powers of g up to third order. The resulting three-gluon vertex is a function of the scale parameter s that has an interpretation of the size of effective gluons. The corresponding Hamiltonian running coupling constant exhibits asymptotic freedom, and the corresponding Hamiltonian \({\beta}\) -function coincides with the one obtained in an earlier calculation using a different generator.     

Friction in a Model of Hamiltonian Dynamics

We study the motion of a heavy tracer particle weakly coupled to a dense ideal Bose gas exhibiting Bose-Einstein condensation. In the so-called mean-field limit, the dynamics of this system approaches one determined by nonlinear Hamiltonian evolution equations describing a process of emission of Cerenkov radiation of sound waves into the Bose-Einstein condensate along the particle’s trajectory. The emission of Cerenkov radiation results in a friction force with memory acting on the tracer particle and causing it to decelerate until it comes to rest.

“A moving body will come to rest as soon as the force pushing it no longer acts on it in the manner necessary for its propulsion.”—— Aristotle