Hamiltonian formalism for nonlinear Schr\"{o}dinger equations

We study the Hamiltonian formalism for second order and fourth order nonlinear Schr\"{o}dinger equations. In the case of second order equation, we consider cubic and logarithmic nonlinearities. Since the Lagrangians generating these nonlinear equations are degenerate, we follow the Dirac-Bergmann formalism to construct their corresponding Hamiltonians. In order to obtain consistent equations of motion, the Dirac-Bergmann formalism imposes some set of constraints which contribute to the total Hamiltonian along with their Lagrange multipliers. The order of the Lagrangian degeneracy determines the number of the primary constraints. Multipliers are determined by the time consistency of constraints. If a constraint is not a constant of motion, a secondary constraint is introduced to force the consistency. We show that for both second order nonlinear Schr\"{o}dinger equations we only have primary constraints, and the form of nonlinearity does not change the constraint dynamics of the system. However, introducing a higher order dispersion changes the constraint dynamics and secondary constraints are needed to construct a consistent Hamilton equations of motion.Comment: 8 page

Hamiltonian formalism for nonlinear Schrödinger equations

We study the Hamiltonian formalism for second and fourth order nonlinear Schrödinger equations. In the case of the second order equation, we consider cubic and logarithmic nonlinearities. Since the Lagrangians generating these nonlinear equations are degenerate, we follow the Dirac–Bergmann formalism to construct their corresponding Hamiltonians. In order to obtain consistent equations of motion, the Dirac–Bergmann formalism imposes some set of constraints that contribute to the total Hamiltonian along with their Lagrange multipliers. The order of the Lagrangian degeneracy determines the number of primary constraints. If a constraint is not a constant of motion, a secondary constraint is introduced to force the consistency condition. We show that for second order and fourth order nonlinear Schrödinger equations we only have primary constraints, and the form of nonlinearity or the order of derivatives does not change the constraint dynamics of the system. However, we observe that introducing new fields to treat higher derivatives in the Lagrangians of these equations changes the constraint dynamics, and secondary constraints are needed to construct a consistent set of Hamilton equations

Precise monitoring of temporal topographic change detection via unmanned air vehicle

Nowadays, fast developing space-borne and airborne remote sensing technologies became indispensable for land related engineering disciplines such as mapping, geology, environment, mining and forestry. The new technologies, provide more qualified and rapid achievable outcomes, are adopted permanently. The description of the topographic surface became easier by means of very high resolution (VHR), rapid achievable and accurate point clouds acquired by digital photogrammetry and airborne laser scanning (ALS). Optical unmanned air vehicle (UAV), one of the most actual photogrammetric techniques, is much in demand for varied purposes. UAVs provide high resolution data using the advantage of lower flight altitudes. In this study, a construction activity and its environmental influences in Bulent Ecevit University Central Campus were monitored by an optical hand-made UAV. In the application, the temporal change was detected by generating contour-lines, digital terrain models (DTMs) and differential DTMs (DiffDTM) of the topography. By DiffDTMs, temporal changes on the topography were visualized in color height scale where the contour-lines presents the change of morphological structure