Aspects of cauchy elastic materials

The Darboux theorem on the classification and canonical forms of first degree exterior differential forms is used to obtain a canonical representation of Cauehy elastic materials. It is shown that the balance of moment of momentum and the principle of material indifference are independent for materials whose Darboux class is greater than 1. By means of the Caratheodory's theorem on inaccessibility it is shown that any homogeneous deformation state can be reached from any other homogeneous deformation state by a path of zero work whenever the Darboux class of the material is 3 or greater.     

Gauge formulation of general relativity using conformal and spin symmetries

The gauge symmetry inherent in Maxwell's electromagnetics has a profound impact on modern physics. Following the successful quantization of electromagnetics and other higher order gauge field theories, the gauge principle has been applied in various forms to quantize gravity. A notable development in this direction is loop quantum gravity based on the spin-gauge treatment. This paper considers a further incorporation of the conformal gauge symmetry in canonical general relativity. This is a new conformal decomposition in that it is applied to simplify recently formulated parameter-free construction of spin-gauge variables for gravity. The resulting framework preserves many main features of the existing canonical framework for loop quantum gravity regarding the spin network representation and Thiemann's regularization. However, the Barbero-Immirzi parameter is converted into the conformal factor as a canonical variable. It behaves like a scalar field but is somehow non-dynamical since the Hamiltonian constraint does not depend on its momentum. The essential steps of the mathematical derivation of this parameter-free framework for the spin-gauge variables of gravity are spelled out. The implications for the loop quantum gravity programme are briefly discussed.     

Equations for the Momentum and Energy of an X-ray Wave Field in a Crystal

A new approach for describing the interaction of X rays with a crystalline medium is proposed. General relations for a change in the electromagnetic-field momentum and energy have been derived for a nonmagnetic medium with variable permittivity. A special local coordinate system for an inhomogeneous medium is introduced, in which the Maxwell tension tensor has a canonical form determined by the energy and momentum densities. Main relations for changes in the energy and momentum densities have been obtained in the coordinate system related to the propagation direction of a plane electromagnetic wave pulse in a homogeneous medium.     

Canonical equations of hydrodynamics of quantum liquids

A method of constructing canonical equations for quantum liquids is proposed. For superfluid ⁴He the forms of the Hamiltonian equations of hydrodynamics and conservation laws are obtained. The kinetic terms in the equations of hydrodynamics are written out. The set of equations obtained is valid up to the point. For anisotropic quantum liquid ³He-A canonical equations both for spin and orbital hydrodynamics are found with the spin and orbital moments taken into account. The forms of the laws of the conservation of momentum, angular momentum, mass, and energy are obtained. The kinetic terms in the equations of hydrodynamics are considered, and both the dissipative and reactive coefficients are classified. A transfer equation for the order parameter near the A transition point is proposed with the dissipative term taken into account.     

Darboux classification of cauchy elastic bodies and some of its consequences

The Darboux theorem on the classification and canonical forms of first degree differential forms is used to obtain a classification and canonical representation of Cauchy elastic solids. If the stress is assumed to be determined uniquely by the Green deformation tensor C, the double Piola-Kirchhoff stress tensor is shown to belong to one of six possible classes, each of which possesses a unique representation in terms of linear combinations of gradients with respect to C of scalar-valued functions with scalar-valued coefficients. There are three distinct classes for isotropic materials, each of which has a unique representation. It is shown that these three classes are distinguishable in terms of the existence or nonexistence of cycles with nonzero total work and by the accessibility or inaccessibility of any deformation state from any neighboring deformation state by a path of zero work.     

无束腰的磁浸没透镜系统的理论研究和数值模拟

采用曲轴坐标系统 ,导出了磁浸没透镜的宽束曲光轴理论 ,研究了初始正则角动量矩不为零的无束腰磁透镜系统的电子光学性质 ;数值模拟出了能产生缩倍聚场分布的磁透镜结构 ,并计算了具有解析表达式的缩倍聚场的高斯成像性质 ,计算结果与传统的直接计算轨迹的方法一致     

Quantisation of the free electromagnetic field implies quantisation of its angular momentum

It is shown that when the gauge-invariant Bohr–Rosenfeld commutators of the free electromagnetic field are applied to the expressions for the linear and angular momentum of the electromagnetic field interpreted as operators then, in the absence of electric and magnetic charge densities, these operators satisfy the canonical commutation relations for momentum and angular momentum. This confirms their validity as operators that can be used in quantum mechanical calculations of angular momentum.     

Derivation of the paraxial form of the angular momentum of the electromagnetic field from the general form

It is shown how the standard forms for the spin and orbital components of the angular momentum of a paraxial wave of electromagnetic radiation are obtained from general expressions for the angular momentum that have been derived recently. This result will enable general expressions for angular momentum to be applied with confidence to the many configurations of electromagnetic fields that are more complicated than plane or paraxial waves.     

On the integration of the nonconservative Hamilton''s dynamical equations

This paper exhibits a method of integrating Hamilton's canonical equations of motion by supposing that one component of the momentum vector can be represented as a field depending on time, generalized coordinates and the rest of the components of the momentum vector. The motion of a conservative or nonconservative dynamical system can be determined by purely algebraic operations if a complete solution of a quasi-linear partial differential equation is known. The method is knually suitable for initial and boundary value problems.     

Radiation pressure and the photon momentum in dielectrics

We review the magnitudes of the photon momenta in free space, derived by Maxwell and Einstein, and those in homogeneous, dispersionless and lossless dielectrics, associated with Abraham and Minkowski. These momenta determine the forces exerted by light beams on material objects, as measured in radiation pressure experiments. It is shown that conservation conditions for components of the electromagnetic energy–momentum tensors derived by Abraham and Minkowski, also by Einstein and Laub, are equivalent relations simply derived from Maxwell's equations. The challenge for theory is the reliable interpretation of experimental momentum transfers from light to matter, which requires extensions of the basic theory to include material dispersion and surface effects. The main experiments are reviewed, together with details of a Lorentz-force theory that accounts for them. It is shown that the Abraham kinetic momentum is associated with the overall motion of a dielectric sample, while the Minkowski canonical momentum applies to the motion of bodies embedded in the dielectric.     

李代数在磁偏转系统三级像差计算中的应用

提出了电子光学偏转像差的李代数理论 ,将这种新型数学工具的应用领域从带电粒子束的聚焦传输系统推广到扫描偏转系统。推导出了磁偏转系统的高斯光学性质及全部三级正则像差 (包括在任意观察平面的正则位置像差和正则动量像差 ) ,所导出的结果与传统的逐次近似像差理论相同 ,为将李代数方法用于扫描探针系统的高阶像差的分析计算奠定了理论基础     

Numerical analysis of cooldown and warmup for the Large Hadron Collider

As the basis of studies, the paper presents the inventory of components and materials for LHC magnets, especially for main dipoles and quadrupoles firstly. Then a mathematical model and analytical method for LHC transient modes, such as cooldown and warmup of a magnet, a standard cell and the eight LHC sectors, has been developed on the basis of the up-to-date layout of the LHC machine, and validated by experimental data. The model and method do not only consider the momentum and continuity equations as well as the energy equations for helium and materials, but also take into account the pressure evolution in the different headers of the cryogenic distribution line and the effect of the pressure drop across the cryogenic valves. Based on the simulation results, the heat transfer in the magnets has been studied and the transient modes optimized.     

Velocity scaling impact melt volume

Velocity scaling relationships for the amount of melted target material produced by hypervelocity impacts are examined. Scaling of the melt volume with nearly the momentum of the impactor has been obtained by Gault and Heitowit using a semi-analytical energy balance approach and energy scaling has been obtained from scaled hydrocode output by O'Keefe and Ahrens. Both results are inconsistent with the point source limit of cratering motions in non-porous media which requires somewhat less than energy scaling. The nearly momentum scaling result from the energy balance approach of Gault and Heitowit has been reexamined and it is shown to be incorrect given their assumptions. A corrected derivation is given which results in energy scaling for all equations of state. The energy scaling result obtained by O'Keefe and Ahrens with hydrocodes has been tested by performing new hydrocode calculations at larger impact velocities. The inconsistency between the coupling parameter approach and O'Keefe and Ahrens hydrocode calculations has been resolved. For aluminum onto aluminum impacts at velocities less than 50 km/s, the new hydrocode calculations have demonstrated that the melt volume forms too close to the impactor for the point source solution to govern the melt volume dimensions, hence energy scaling is a possible result. However, at greater than 50 km/s impact velocity, less than energy scaling is obtained because the flow in the melt volume region approaches the point source limit.     

Quantum-optical Phase and Canonical Conjugation

Abstract

We use a new limiting procedure, developed to study quantum-optical phase, to examine canonically conjugate operators in general. We find that Dirac's assumption that photon number and phase should be canonically conjugate variables, similar to momentum and position, is essentially correct. The difficulties with Dirac's approach are shown to arise through use of a form of the canonical commutator which, although the only possible form in the usual infinite Hilbert space approach, is not sufficiently general to be used as a model for a number-phase commutator. The approach in this paper unifies the theory of conjugate operators, which include photon number and phase, angular momentum and angle, and momentum and position as particular cases. The usual position-momentum commutator is regained from a more generally applicable expression by means of a domain restriction which cannot be used for the phase-number commutator.     

Pseudomomentum and material forces in nonlinear elasticity: variational formulations and application to brittle fracture

Summary This work examines critically the various formulations of the balance of linear momentum innonlinear inhomogeneous elasticity. The corresponding variational formulations are presented. From the point of view of the theory of elastic inhomogeneities, the most interesting formulations are those which, being either completely material or mixed-Eulerian, exhibit explicitly the inhomogeneities in the form ofmaterial forces. They correspond to the balance ofpseudomomentum, a material covector which is seldom used but which we show to play a fundamental role in the Hamiltonian canonical formulation of nonlinear elasticity. The flux associated with pseudomomentum is none other than theEshelby material tensor. Applying this formulation to the case of an elastic body containing a crack of finite extent, the notion of suction force acting at the tip of the crack follows while afracture criterion à la Griffith can be deduced from a variational inequality. Possible extensions to higher-grade elastic materials and inelastic materials are indicated as well as the role played by pseudomomentum in the quantization of elastic vibrations.     

A system–analogue method of identification of geometric shape of the abrasive grain projection

An advanced method of identification and quantitative estimation of geometric shape of the projection of abrasive powder grains has been proposed. The method is based on a system–analogue approach. For analogues the two-dimensional geometric figures (circles, ellipses, triangles, canonical forms of tetragons, regular pentagons, hexagons, and octagons) that admit a nonadditive analytical representation of the area in terms of generating parameters (whose number does not exceed three) have been taken. A differential and integral characteristics of the shape similarity have been introduced and an analytical apparatus has been put forward to determine the values of these characteristics. The results of appraisal (testing) of the method on synthetic diamond and cBN grits are discussed.     

Compound matrix block diagonalization for efficient solution of eigenproblems in structural mechanics

Summary Recently, methods were developed for the decomposition of special matrices, leading to canonical forms I-IV. In this paper, the conditions required for the decomposability of such matrices are studied. It is shown that these canonical forms are obtainable by special block diagonalization of matrices, having certain properties. Here, tri- to penta-block diagonal matrices are studied and methods are developed for their decomposition. These methods are incorporated in the efficient solution of numerous eigenproblems of structural mechanics.     

数控车床丝杠进给系统刚度对定位精度的影响

数控车床丝杠进给系统刚度对系统的失动量和定位精度影响较大。基于进给系统质量和弹簧力学模型，给出了进给系统轴向刚度和扭转刚度对失动量的影响与定位误差的解析表达式，为进给系统结构设计、降低系统的失动量和提高系统的定位精度提供了理论依据；结合实例进行了应用分析计算，取得了明显效果。     

A moving three-level atom interacting with a two-mode field: some atom–field aspects

In this paper, the interaction of a moving three-level atom and a two-mode quantized electromagnetic cavity field is extended to involve the effects of the atomic motion. Detuning parameters, Kerr nonlinearity, Stark shift contributions and arbitrary forms of intensity-dependent atom–field coupling have been taken into account. The constants of motion and the wave function, when the atom is initially prepared in superposition states and the field is initially prepared in squeezed coherent states, have been obtained. We calculate some statistical aspects such as atomic inversion, purity, Mandel Q-parameter, cross-correlation, momentum increment, momentum diffusion and Husimi Q-function.