Base-controlled mechanical systems and geometric phases

In this paper, we carry a detailed study of mechanical systems with configuration space Q?Q/G$Q?Q/G$ for which the base Q/G$Q/G$ variables are being controlled. The overall system’s motion is considered to be induced from the base one due to the presence of general non-holonomic constraints. It is shown that the solution can be factorized into dynamical and geometrical parts. Moreover, under favorable kinematical circumstances, the dynamical part admits a further factorization since it can be reconstructed from an intermediate (body) momentum solution, yielding a reconstruction phase formula. Finally, we apply this results to the study of concrete mechanical systems.     

Characteristic classes of bad orbifold vector bundles

We show that every bad orbifold vector bundle can be realized as the restriction of a good orbifold vector bundle to a suborbifold of the base space. We give an explicit construction of this result in which the Chen–Ruan orbifold cohomologies of the two base spaces are isomorphic (as additive groups). This construction is used to indicate an extension of the Chern–Weil construction of characteristic classes to bad orbifold vector bundles. In particular, we apply this construction to the orbifold Euler class and demonstrate that it acts as an obstruction to the existence of nonvanishing sections.     

Homogeneous pseudo-Riemannian structures of linear type

This article is partly a survey and partly a research paper on homogeneous pseudo-Riemannian structures of linear type. In the Riemannian case, these structures furnish characterisations of real, complex and quaternionic hyperbolic spaces. In the Lorentzian case, a related class gives characterisations of singular homogeneous plane waves.     

The fundamental form of almost-quaternionic Hermitian manifolds

We prove that if the fundamental 4-form Ω$\Omega$ of an almost-quaternionic Hermitian manifold (M,Q,g)$(M,Q,g)$ of dimension 4n≥8$4n\ge 8$ satisfies the conformal-Killing equation, then (M,Q,g)$(M,Q,g)$ is quaternionic-Kähler.     

A generalized Schrödinger formalism as a Hilbert space representation of a generalized Liouville equation

A generalized Schrödinger formalism has been presented which is obtained as a Hilbert space representation of a Liouville equation generalized to include the action as a dynamical variable, in addition to the positions and the momenta. This formalism applied to a classical mechanical system had been shown to yield a similar set of Schrödinger like equations for the classical dynamical system of charged particles in a magnetic field. The novel quantum-like predictions for this classical mechanical system have been experimentally demonstrated and the results are presented.     

A causal quantum theory in phase space

The causal theory for the coherent state representation of quantum mechanics is derived. The general conditions for the classical limit are given and it is shown that phase space classical mechanics can be obtained as a limit even for stationary states, in contrast to the de Broglie-Bohm quantum theory of motion.     

Hyperbolic reduction

The notions of weak Darboux integrability and hyperbolic reduction are introduced, and their potential is gauged as a means of extending the range of application of geometric methods for solving hyperbolic partial differential equations. For directness, our work is expressed in local coordinates and formulated for semilinear hyperbolic systems in two independent variables. The theory is applied to the study of 1+1-wave maps into surfaces of revolution. It is shown that the differential system for any such wave map may be viewed as an integrable extension of a certain scalar, semilinear, hyperbolic partial differential equation which is explicitly constructed. Using this we discover a new integrable wave map system for which hyperbolic reduction leads to a large family of explicit wave maps.     

曲面上的动量和动能算符

对约束在曲面上粒子运动的描述可以在内部坐标即曲面局部坐标下进行,也可以在外部坐标即在笛卡尔坐标下进行.在量子力学中,动量和动能算符的表示在这两种描述中各有不同,前者的动量算符仅包含内禀几何量,后者的动量算符包含了曲面的平均曲率.考虑到算符次序问题,动能算符对动量算符的依赖关系也不同,前者的依赖关系仅发现存在一种,后者的依赖关系已经发现有两种. 关键词： 量子力学 微分几何     

曲面上的动量和动能算符

对约束在曲面上粒子运动的描述可以在内部坐标即曲面局部坐标下进行，也可以在外部坐标即在笛卡尔坐标下进行.在量子力学中，动量和动能算符的表示在这两种描述中各有不同，前者的动量算符仅包含内禀几何量，后者的动量算符包含了曲面的平均曲率.考虑到算符次序问题，动能算符对动量算符的依赖关系也不同，前者的依赖关系仅发现存在一种，后者的依赖关系已经发现有两种.     

A Unified Framework for Relativity and Curvilinear-Time Newtonian Mechanics

Classical mechanics is presented so as to render the new formulation valid for an arbitrary temporal variable, as opposed to Newton’s Absolute Time only. Newton’s theory then becomes formally identical (in a precise sense) to relativity, albeit in a three-dimensional manifold. (The ultimate difference between the two dynamics is traced to the existence of the relativistic ‘mass-shell’ condition.) A classical Lagrangian is provided for our formulation of the equations of motion and it is related to one of the known forms of the corresponding relativistic Lagrangian, which is the analogue of the Polyakov Lagrangian of string theory. Dedicated to Emeritus Professor D. Speiser (Université Catholique de Louvain, Belgium) who provided the inspiration for this article.     

Some Further Properties of the Accelerated Kerr-Schild Metrics

We extend the previously found accelerated Kerr-Schild metrics for Einstein-Maxwell-null dust and Einstein-Born-Infeld-null dust equations to the cases including the cosmological constant. This way we obtain the generalization of the charged de Sitter metrics in static space-times. We also give a generalization of the zero acceleration limit of our previous Einstein-Maxwell and Einstein-Born-Infeld solutions.     

A thermostatistical approach to protein structures

In this study, the relation between the additional heat capacities of proteins have been obtained. The expressions for the increments of enthalpy , entropy and Gibbs energy which are encountered in the dissolving of the proteins in water, have been calculated. Partition functions of proteins are related to the macrothermodynamical quantities with the help of free energies. In order to endeavour the structure of the proteins, the theoretical calculations of thermostatistical mechanics are fitted to the data of experimental results.     

Morphological thermodynamics for hard bodies from a controlled expansion

ABSTRACT

The morphometric approach is a powerful ansatz for decomposing the chemical potential for a complex solute into purely geometrical terms. This method has proven accuracy in hard spheres, presenting an alternative to comparatively expensive (classical) density functional theory approaches. Despite this, fundamental questions remain over why it is accurate and how one might include higher-order terms to improve accuracy. We derive the morphometric approach as the exact resummation of terms in the virial series, providing further justification of the approach. The resulting theory is less accurate than previous morphometric theories, but provides fundamental insights into the inclusion of higher-order terms and to extensions to mixtures of convex bodies of arbitrary shape.     

An electromagnetic parameters extraction method for metamaterials based on phase unwrapping technique

Abstract

A new method of extracting effective permittivity and permeability of metamaterials has been developed in this paper. Scattering parameters are used to represent effective wave impedance and refractive index of the metamaterials. Considering analytical continuation of effective refractive index of the metamaterials in the upper half of a complex angular frequency plane, a phase unwrapping technique has been implemented to solve branch ambiguity of the refractive index caused by complex inverse cosine function. An empirical formula to estimate the starting frequency in the phase unwrapping technique has been developed to improve the efficiency of the parameter retrieval procedure. Numerical results including five slabs of rods and split-ring resonators (SRRs) and a structure composed of two SRRs and four capacitively loaded strips are given to demonstrate good accuracy, high efficiency, and noise insensitivity of the proposed retrieval method.     

Thermodynamics of rotating self-gravitating systems

We investigate the statistical equilibrium properties of a system of classical particles interacting via Newtonian gravity, enclosed in a three-dimensional spherical volume. Within a mean-field approximation, we derive an equation for the density profiles maximizing the microcanonical entropy and solve it numerically. At low angular momenta, i.e. for a slowly rotating system, the well-known gravitational collapse “transition” is recovered. At higher angular momenta, instead, rotational symmetry can spontaneously break down giving rise to more complex equilibrium configurations, such as double-clusters (“double stars”). We analyze the thermodynamics of the system and the stability of the different equilibrium configurations against rotational symmetry breaking, and provide the global phase diagram. Received 8 July 2002 Published online 15 October 2002 RID="a" ID="a"e-mail: demartino@hmi.de