Classical and Quantum Euler equation

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2007Includes bibliographical references (leaves: 74-75)Text in English; Abstract: Turkish and Englishix, 94 leavesIn the present thesis we give generalization of analytical mechanics to describe dynamical systems with dissipation. The Lagrangian function in this case is determined by nonstationary pseudo-Riemannian metric for the kinetic energy, and by general quadratic form, nondiagonal in the generalized coordinates and velocities. Skew symmetric nondiagonal terms in our approach play the role of dissipation coefficients. As an application we study in details the classical damped harmonic oscillator. We show that two known formulations of this oscillator, the Bateman dual and the Caldirola Kanai formulations are particular realizations of our general approach. The Hamiltonian formulation and quantization of the model in both representations are given. Moreover Ostrogradsky generalization of Lagrangian and Hamiltonian formalism for description of systems with higher order derivatives and its application to the constant coefficient equations of an arbitrary order are considered. We construct related with the last one the Euler differential equation of an arbitrary order and its Lagrangian and Hamiltonian structure. Quantum Euler systems are introduced and solved for the stationary Schrodinger picture. Nonstationary nonlinear quantum models corresponding to arbitrary Euler Hamiltonian are solved exactly in the Heisenberg picture

Classical and Quantum Euler equation

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2007Includes bibliographical references (leaves: 74-75)Text in English; Abstract: Turkish and Englishix, 94 leavesIn the present thesis we give generalization of analytical mechanics to describe dynamical systems with dissipation. The Lagrangian function in this case is determined by nonstationary pseudo-Riemannian metric for the kinetic energy, and by general quadratic form, nondiagonal in the generalized coordinates and velocities. Skew symmetric nondiagonal terms in our approach play the role of dissipation coefficients. As an application we study in details the classical damped harmonic oscillator. We show that two known formulations of this oscillator, the Bateman dual and the Caldirola Kanai formulations are particular realizations of our general approach. The Hamiltonian formulation and quantization of the model in both representations are given. Moreover Ostrogradsky generalization of Lagrangian and Hamiltonian formalism for description of systems with higher order derivatives and its application to the constant coefficient equations of an arbitrary order are considered. We construct related with the last one the Euler differential equation of an arbitrary order and its Lagrangian and Hamiltonian structure. Quantum Euler systems are introduced and solved for the stationary Schrodinger picture. Nonstationary nonlinear quantum models corresponding to arbitrary Euler Hamiltonian are solved exactly in the Heisenberg picture

Model Analysis of Ridge and Rib Types of Silicon Waveguides With Void Compositions

Detailed numerical study on mode characteristics and light confinement properties of ridge and rib waveguides with vertical, horizontal, and cross types of void configurations is presented. The study starts from the mode analysis for rectangular ridge and rib waveguides. Then, the analysis is extended to semicircular ridge and rib waveguides with and without slots. The percentages of the electric and magnetic field components and light confinement capabilities are compared. We observe that polarization type of the mode and strong field confinement in the low refractive index medium can be achieved by the incorporation of void structures into the regular optical waveguide.The work of H. Kurt was supported by the Turkish Academy of Sciences. The work of N. Eti was supported by the Scientific and Technical Research Council (TUBITAK) under Grant BIDEB-2218

Adaptive Graded Index Photonic Crystal Lens Design via Nematic Liquid Crystals

An adaptive graded index photonic crystal (GRIN PC) lens system is designed by using liquid crystal (LC) infiltration. LCs have the property to change their refractive indices when an external voltage is applied. This feature allows for the modulation on the effective index profile of the low-index contrast GRIN PC lens. Alongside this property, without changing the length of the designed 2D GRIN PC, the corresponding focal distance can be adapted from infinity to a certain positive or negative focal point by controlling the applied voltage. The effects of random perturbations and intentional line defects on the focal tuning capability of proposed 2D GRIN PC are also studied. Moreover, due to the scalability of PC's dispersion relations, the sizes of the proposed devices can be adapted so that it may operate in either infrared or microwave regimes. Consequently, the proposed tunable polymeric GRIN PC can be implemented in various optical applications, such as near-field imaging and scanning systems, vision correction, and auto-focusing