No description available.

No description available.

No description available.

No description available.

Thesis (PhD)--Montana State University--Bozeman, 2009. / Typescript. Chairperson, Graduate Committee: Tomas Gedeon. Includes bibliographical references (leaves 234-237).

One of the most important branches of applied mechanics is the theory of plates - defined to be plane structural elements whose thickness is very small when compared to the two planar dimensions. There is an abundance of plate theories in the literature modeling classical elastic solids that fit this description. Recently, however, there has been a steady growth of interest in modeling materials with microstructures that exhibit length-scale dependent behavior, generally known as Cosserat elastic materials. Concurrently, there has also been an increased interest in the construction of reduced dimensional models of such materials owing to advantages like reduced computational effort and a simpler, yet elegant, resulting mathematical formulation. The objective of this work is the formulation and implementation of a theory of elastic plates with microstructure. The mathematical underpinning of the approach used is the Variational Asymptotic Method (VAM), a powerful tool used to construct asymptotically correct plate models. Unlike existing Cosserat plate models in the literature, the VAM allows for a plate formulation that is free of a priori assumptions regarding the kinematics. The result is a systematic derivation of the two-dimensional constitutive relations and a set of geometrically-exact, fully intrinsic equations gov- erning the motion of a plate. An important consequence is the extraction of the drilling degree of freedom and the associated stiffness. Finally, a Galerkin approach for the solution of the fully-intrinsic formulation will be developed for a Cosserat sur- face analysis which will also be compatible with more traditional plate solvers based on the classical theory of elasticity. Results and validation are presented from linear static and dynamic analyses, along with a discussion on some challenges and solution techniques for nonlinear problems.One of the most important branches of applied mechanics is the theory of plates - defined to be plane structural elements whose thickness is very small when compared to the two planar dimensions. There is an abundance of plate theories in the literature modeling classical elastic solids that fit this description. Recently, however, there has been a steady growth of interest in modeling materials with microstructures that exhibit length-scale dependent behavior, generally known as Cosserat elastic materials. Concurrently, there has also been an increased interest in the construction of reduced dimensional models of such materials owing to advantages like reduced computational effort and a simpler, yet elegant, resulting mathematical formulation. The objective of this work is the formulation and implementation of a theory of elastic plates with microstructure. The mathematical underpinning of the approach used is the Variational Asymptotic Method (VAM), a powerful tool used to construct asymptotically correct plate models. Unlike existing Cosserat plate models in the literature, the VAM allows for a plate formulation that is free of a priori assumptions regarding the kinematics. The result is a systematic derivation of the two-dimensional constitutive relations and a set of geometrically-exact, fully intrinsic equations gov- erning the motion of a plate. An important consequence is the extraction of the drilling degree of freedom and the associated stiffness. Finally, a Galerkin approach for the solution of the fully-intrinsic formulation will be developed for a Cosserat sur- face analysis which will also be compatible with more traditional plate solvers based on the classical theory of elasticity. Results and validation are presented from linear static and dynamic analyses, along with a discussion on some challenges and solution techniques for nonlinear problems.

We develop new variational techniques, acting on classes of Lagrangians with the same functional dependence but arbitrary functional form, for the derivation of general, strongly conserved quantities, supplementing the usual procedure for deriving weak conservation laws via Noether's theorem. Using these new techniques we generate and generalize virtually all energy-momentum complexes currently known. In the process we discover and understand the reason for the difficulties associated with energy-momentum complexes in general relativity. We study a Palatini variation of a novel Lagrangian due to Nissani. We find that Nissani's principal claim, that his Lagrangian specifies Riemannian geometry in the presence of a generalized matter tensor, is not in fact justifiable, and prove that his Lagrangian is not unique. We speculate on the possibility of deriving a general-relativistic analog of Maxwell's current equation, a matter current equation, yielding an entirely new approach to the idea of energy-momentum in general relativity. We develop the SL(2,C) x U(1) spinor formalism naturally combining the gravitational and electromagnetic potentials in a single object--the spinor connection. Variably charged matter is rigourously introduced, through the use of spin densities, in the unified potential theories we develop. We generate both the Einstein-Maxwell equations and new equations. The latter generalize both the Maxwell equation and the Einstein equation which includes a new "gravitational stress-energy tensor". This new tensor exactly mimicks the electromagnetic stress-energy tensor with Riemann tensor contractions replacing Maxwell tensor contractions. We briefly consider the introduction of matter. A Lagrangian generalizing the two spinor Dirac equations has no gravitational currents and the electromagnetic currents must be on the light cone. A Lagrangian generalizing the Pauli equations has both gravitational and electromagnetic currents. The equations of both Lagrangians demonstrate beautifully how the divergence of the total stress-energy tensor vanishes in this formalism. In the theory of the generalized Einstein-Maxwell and Pauli equations we succeed in deriving an equation describing a generalized matter-charge current density. / Graduate

Consider a multidimensional shift space with a countably infinite alphabet, which serves in mathematical physics as a classical lattice gas or lattice spin system. A new definition of a Gibbs measure is introduced for suitable real-valued functions of the configuration space, which play the physical role of specific internal energy. The variational principle is proved for a large class of functions, and then a more restrictive modulus of continuity condition is provided that guarantees a function's Gibbs measures to be a nonempty, weakly compact, convex set of measures that coincides with the set of measures obeying a form of the DLR equations (which has been adapted so as to be stated entirely in terms of specific internal energy instead of the Hamiltonians for an interaction potential). The variational equilibrium measures for a such a function are then characterized as the shift invariant Gibbs measures of finite entropy, and a condition is provided to determine if a function's Gibbs measures have infinite entropy or not. Moreover the spatially averaged limiting Gibbs measures, i.e. constructive equilibria, are shown to exist and their weakly closed convex hull is shown to coincide with the set of true variational equilibrium measures. It follows that the "pure thermodynamic phases", which correspond to the extreme points in the convex set of equilibrium measures, must be constructive equilibria. Finally, for an even smoother class of functions a method is presented to construct a compatible interaction potential and it is checked that the two different structures generate the same sets of Gibbs and equilibrium measures, respectively.

Quantum chaos has emerged in the half of the last century with the notorious problem of scattering of heavy nuclei. Since then, theoreticians have developed powerful techniques to approach disordered quantum systems. In the late 70's, Casati and Chirikov initiated a new field of research by studying the quantum counterpart of classical problems that are known to exhibit chaos. Among the several quantum-classical chaotic systems studied, the kicked rotor stimulated a lot of enthusiasm in the scientific community due to its equivalence to the Anderson tight binding model. This equivalence allows one to map the random Anderson model into a set of fully deterministic equations, making the theoretical analysis of Anderson localization considerably simpler. In the one-dimensional linear regime, it is known that Anderson localization always prevents the diffusion of the momentum. On the other hand, for higher dimensions it was demonstrated that for certain conditions of the disorder parameter, Anderson localized modes can be inhibited, allowing then a phase transition from localized (insulating) to delocalized (metallic) states. In this thesis we will numerically and theoretically investigate the properties of a multidimensional quantum kicked rotor in a nonlinear medium. The presence of nonlinearity is particularly interesting as it raises the possibility of having soliton waves as eigenfunctions of the systems. We keep the generality of our approach by using an adjustable diffusive nonlinearity, which can describe several physical phenomena. By means of Variational Calculus we develop a chaotic map which fully describes the soliton dynamics. The analysis of such a map shows a rich physical scenario that evidences the wave-particle behavior of a soliton. Through the nonlinearity, we trace a correspondence between quantum and classical mechanics, which has no equivalent in linearized systems. Matter waves experiments provide an ideal environment for studying Anderson localization, as the interactions in these systems can be easily controlled by Feshbach resonance techniques. In the end of this thesis, we propose an experimental realization of the kicked rotor in a dipolar Bose Einstein Condensate.

Throughout the last several decades many techniques have been developed in establishing solutions to nonlinear partial differential equations (NPDE). These techniques are characterized by their limited reach in solving large classes of NPDE. This body of work will study the analysis of NPDE using two of the most ubiquitous techniques developed in the last century. In this body of work, the analysis and techniques herein are applied to unsolved physical problems in both the fields of variational approximations and inverse scattering transform. Additionally, a new technique for estimating the error of a variational approximation is established. Note that the material in chapter 2, "Quantitative Measurements of Variational Approximations" has recently been published. Variational problems have long been used to mathematically model physical systems. Their advantage has been the simplicity of the model as well as the ability to deduce information concerning the functional dependence of the system on various parameters embedded in the variational trial functions. However, the only method in use for estimating the error in a variational approximation has been to compare the variational result to the exact solution. In this work, it is demonstrated that one can computationally obtain estimates of the errors in a one-dimensional variational approximation, without any a priori knowledge of the exact solution. Additionally, this analysis can be done by using only linear techniques. The extension of this method to multidimensional problems is clearly possible, although one could expect that additional difficulties would arise. One condition for the existence of a localized soliton is that the propagation constant does not fall into the continuous spectrum of radiation modes. For a higher order dispersive systems, the linear dispersion relation exhibits a multiple branch structure. It could be the case that in a certain parameter region for which one of the components of the solution has oscillations (i.e., is in the continuous spectrum), there exists a discrete value of the propagation constant, k(ES), for which the oscillations have zero amplitude. The associated solution is referred to as an embedded soliton (ES). This work examines the ES solutions in a CHI(2):CHI(3), type II system. The method employed in searching for the ES solutions is a variational method recently developed by Kaup and Malomed [Phys. D 184, 153-61 (2003)] to locate ES solutions in a SHG system. The variational results are validated by numerical integration of the governing system. A model used for the 1-D longitudinal wave propagation in microstructured solids is a KdV-type equation with third and fifth order dispersions as well as first and third order nonlinearities. Recent work by Ilison and Salupere (2004) has identified certain types of soliton solutions in the aforementioned model. The present work expands the known family of soliton solutions in the model to include embedded solitons. The existence of embedded solitons with respect to the dispersion parameters is determined by a variational approximation. The variational results are validated with selected numerical solutions.