约束系统的变换性质

受约束的系统,它的运动可用非独立的态函数来描述。在约束系统的时空坐标和态函数作无穷小变换下,考虑到作用量和约束方程的变化,导致了普遍的变换结果,沿着系统运动的轨线,得到约束系统变换性质的一般方程,从而给出约束系统在变换下产生守恒律的条件,对连续系统的某些具体变换写出了其变换性质方程,在特殊情况下,变换性质方程可化为经典Noether定理的结果,用于经典力学作了较详细的讨论,并把Poincaré-Cartan不变量推广到了受约束系统。 关键词：     

Perturbative Solutions of the Extended Constraint Equations in General Relativity

The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface in an asymptotically simple space-time satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-linear system of partial differential equations for the induced metric, the second fundamental form and two other tensorial quantities defined on , and are equivalent to the usual constraint equations that satisfies as a space-like hypersurface in a space-time satisfying Einstein’s vacuum equation. This article develops a method for finding perturbative, asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on Euclidean space. This method is fundamentally different from the ‘classical’ method of Lichnerowicz and York that is used to solve the usual constraint equations.     

A Hierarchy of Integrable Nonlinear Lattice Equations and New Integrable Symplectic Map

A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related tothis spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are givenby nonlinearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resultingintegrable lattice equations.     

Novel homoclinic and heteroclinic solutions for the 2D complex cubic Ginzburg-Landau equation

Homoclinic and heteroclinic solutions are two important concepts that are used to investigate the complex properties of nonlinear evolutionary equations. In this Letter, we perform hyperbolic and linear stability analysis, and prove the existence of homoclinic and heteroclinic solutions for two-dimensional cubic Ginzburg-Landau equation with periodic boundary condition and even constraint. Then, using the Hirota's bilinear transformation, we find the closed-form homoclinic and heteroclinic solutions. Moreover, we find that the homoclinic tubes and two families of heteroclinic solutions are asymptotic to a periodic cycle in one dimension.     

On a transformed stochastic equation

A linear stochastic equation is considered. As a result of the transformation used in the theory of integral equations for improving the convergence of successive approximations, transformed stochastic equations are obtained. The latter are exact and are equivalent to the original equation. By solving the transformed stochastic equations by the method of small perturbations the conditions are derived for the applicability of the approximate Keller equations for a value of the field averaged over the ensemble, which satisfies the original stochastic equation. As an application, the applicability boundaries of the Dyson equations are estimated in the Foldy and Burre approximation. In the first case it is assumed that the medium consists of Rayleigh scatters, while in the second case it is assumed that the fluctuations of the permeability of the medium are small-scale ones. If the medium is bounded and has the form of a sphere, the applicability condition of the Dyson equations impose an upper constraint on the radius of the sphere which nevertheless may take values that exceed the extinction length.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 1, pp. 66–72, January, 1972.     

Mei symmetries and Mei conserved quantities for higher-order nonholonomic constraint systems

This paper presents the Mei symmetries and new types of non-Noether conserved quantities for a higher-order nonholonomic constraint mechanical system.On the basis of the form invariance of differential equations of motion for dynamical functions under general infinitesimal transformation,the determining equations,the constraint restriction equations and the additional restriction equations of Mei symmetries of the system are constructed.The criterions of Mei symmetries,weak Mei symmetries and strong Mei symmetries of the system are given.New types of conserved quantities,i.e.the Mei symmetrical conserved quantities,the weak Mei symmetrical conserved quantities and the strong Mei symmetrical conserved quantities of a higher-order nonholonomic system,are obtained.Then,a deduction of the first-order nonholonomic system is discussed.Finally,two examples are given to illustrate the application of the method and then the results.     

A Hierarchy of Integrable Lattice Soliton Equations and New Integrable Symplectic Map

Starting from a discrete spectral problem, a hierarchy of integrable lattice soliton equations is derived. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses discrete bi-Hamiltonian structure. A new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a Bäcklund transformation for the resulting integrable lattice equations. At last, conservation laws of the hierarchy are presented.     

New extension of dispersionless Harry Dym hierarchy

The symmetry constraint for dispersionless Harry Dym (dHD) hierarchy is derived for the first time by taking dispersionless limit of that for 2+1 dimensional Harry Dym hierarchy. Then, the dHD is extended by means of the symmetry constraint which we derived. From the zero-curvature equation of the new extended dHD hierarchy, two types of dHD equations with self-consistent sources (dHDESCS) together with their associated conservation equations are obtained. Moreover, the hodograph solutions to the first type of dHDESCS are given. Finally, Bäcklund transformation between the extended dispersionless mKP hierarchy and extended dHD Hierarchy are also constructed.     

约束系统的变换和推广的Killing方程

本文从约束系统的作用量和约束方程的普遍变换性质出发,得到了约束系统的推广Killing方程组,此方程组的解所生成的变换可产生经典Noether定理的守恒量。讨论了连续系统的时空变换和内禀变换,在变换保持约束方程不变时,指出此变换能导致守恒流的充要条件,给出了对不可压缩连续介质的应用,讨论了对广义力学和经典力学的应用,并给出了推广Killing方程组解的某些实例,将Poincare不变量推广到了受约束广义力学系统。 关键词：     

Geometry of the Constraint Sets for Yang–Mills–Dirac Equations with Inhomogeneous Boundary Conditions

The constraint equation for minimally coupled Yang–Mills and Dirac fields in bounded domains is studied under the inhomogeneous boundary conditions which admit unique solutions of the evolution equations. For each value of the boundary data, the constraint set is shown to be a submanifold of the extended phase space. It is a prinicipal fibre bundle over the reduced phase space with structure group consisting of the gauge symmetries which coincide on the boundary with the identity transformation up to the first order of contact. Received: 29 June 1998 / Accepted: 28 December 1998     

An Integrable Matrix Camassa-Holm Equation

We present an integrable sl(2)-matrix Camassa-Holm(CH) equation.The integrability means that the equation possesses zero-curvature representation and infinitely many conservation laws.This equation includes two undetermined functions,which satisfy a system of constraint conditions and may be reduced to a lot of known multicomponent peakon equations.We find a method to construct constraint condition and thus obtain many novel matrix CH equations.For the trivial reduction matrix CH equation we construct its N-peakon solutions.     

Some remarks on radiation gauges in general relativity

We prove that it is possible to specify the shift vector field for a given foliation in such a way as to provide automatically the transverse-traceless part of the second fundamental form K associated with the slices of the foliation. We interpret this particular specification as providing a simple way for characterizing the dynamical degrees of freedom of the gravitational field. The resulting gauge is not the minimal distortion gauge devised by Smarr and York. In numerical relativity it permits the maintenance in time of the difference equations associated with the divergence constraint.     

Extension of the York field decomposition to general gravitationally coupled fields

We extend the York decomposition analysis of the initial value constraints to general gravitationally coupled classical field theories. The decomposition is found to be particularly useful in solving the constraint equations for all theories of current physical interest. These include Einstein gravity or Einstein-Cartan (torsion) gravity coupled to the massive or massless version of the following: general scalar (including Klein-Gordon, Brans-Dicke, and Higgs), Dirac spin 1/2, Maxwell (Proca) and Yang-Mills (any gauge group). We show in detail how the program works for the general Yang-Mills field and for the Einstein-Cartan-Proca field.     

Initial data sets of minimal energy

The total mass of a gravitational system, the ADM mass, is considered as a functional of the initial data sets for sources and gravitation. From the condition that such a functional has a minimum under a restricted class of variations, tensorial equations for the gravitational initial data are obtained. The solutions to these equations, whose existence for a large class of source fields is asserted, represent initial data sets for gravitation satisfying the constraint equations and may be considered as having no gravitational radiation.     

Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge

We define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find a quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.     

General relativistic axisymmetric rotating systems: Coordinates and equations

The Einstein equations for rotating axisymmetric configurations in asymptotically flat spacetimes are put in a form suitable for numerical calculations of dynamics. The discussion is motivated by the astrophysical problem of gravitational collapse with generation of gravitational radiation and possibly black hole formation. In the context of topologically spherical coordinates there are two spatial gauge conditions which greatly simplify the Einstein equations and are compatible with regularity at the origin. We focus on one, the radial gauge, which generalizes Schwarzschild coordinates and is asymptotically a transverse-traceless gauge for gravitational radiation. The shift vector equation and the Hamiltonian constraint are parabolic equations in the radial gauge, rather than the usual elliptic equations. Two hypersurface conditions are explored in detail, the maximal hypersurface condition and another “polar” hypersurface condition which fits naturally with the radial gauge.     

The Double Self-Dual Gravity with Cosmological Term and Constraints Under the Double Constant Conformal Transformation

The double constraint equations in the self-dual gravitational theory containing the cosmological term are derived in 3 + 1 gravity. Furthermore, in order to deeply study the Lorentzian and Euclidean reality conditions for this theory, the relations between constraints are discussed by introducing the double constant conformal transformation and the double complex function method.     

Constraints of the KP hierarchy and multilinear forms

We consider the trilinear form of the Kaup-Broer system which gives rise to solutions in Wronskian form. The Kaup-Broer system is connected with AKNS system through a gauge transformation. The AKNS hierarchy can be understood as a generalized 1-constraint of the KP hierarchy. Imposing that constraint on Sato's equation we obtain the basic trilinear form and moreover a hierarchy of trilinear equations governing the AKNS flows. Similary, hierarchies of multilinear forms are derived in the case of generalized k-constraints.     

Isospectral flow and lax pairs

Certain non-linear differential equations may be considered as compatibility conditions for a linear system of equations, that is, as the vanishing-curvature condition for some connection. It is shown how one can obtain this connection from an isospectral-flow condition using a Foldy-Wouthuysen transformation.     

Direct assessment of lattice Boltzmann hydrodynamics and boundary conditions for recirculating flows

A hydrodynamic boundary condition is developed for lattice Boltzmann hydrodynamics using a square, orthogonal grid. A constraint based on energy considerations is developed to provide closure for the equations which govern the particle distribution at the boundaries. This boundary condition is applied to the two-dimensional, steady flow of an incompressible fluid behind a grid, known as Kovasznay flow. The results are compared to those using alternate boundary conditions using the known exact solution. The hydrodynamic boundary condition produces quadratic spatial convergence, while alternate techniques fail to maintain this second-order accuracy.