Solution of the Hamilton – Jacobi Equations in an Electromagnetic Field Using Separation of Variables Method – Staeckel Boundary Conditions

2020 ◽  
Vol 13 (1) ◽  
pp. 59-65
Keyword(s):  
Electromagnetic Field ◽  
Boundary Conditions ◽  
Separation Of Variables ◽  
Separation Of Variables Method ◽  
Jacobi Equations

This manuscript aims to resolve the Hamilton-Jacobi equations in an electromagnetic field by two methods. The first uses the separation of variables technique with Staeckel boundary conditions, whereas the second uses the Newtonian formalism to solve the same example. Our results demonstrate that the Hamilton-Jacobi variables can be completely detached by using separation of variables technique with Staeckel boundary conditions that correspond to other results using Newtonian formalism.

Solution of the Hamilton – Jacobi Equation in a Central Potential Using the Separation of Variables Method with Staeckel Boundary Conditions

2021 ◽  
Vol 14 (4) ◽  
pp. 301-308
Keyword(s):  
Characteristic Function ◽  
Boundary Conditions ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Lagrangian Mechanics ◽  
Principal Function ◽  
Central Potential ◽  
Separation Of Variables Method ◽  
Hamilton Jacobi Equation

Abstract: This manuscript aims at solving Hamilton-Jacobi equation in a central potential using the separation of variables technique with Staeckel boundary conditions. Our results show that the Hamilton – Jacobi variables can be completely separated, which agrees with other results employing different methods. Keywords: Lagrangian mechanics, Hamilton-Jacobi, Staeckel boundary conditions, Staeckel matrix, Staeckel vector, Hamilton's characteristic function, Hamilton's principal function.

scholarly journals Computation of Stray Losses in Transformer Bushing Regions Considering Harmonics in the Load Current

2020 ◽  
Vol 10 (10) ◽  
pp. 3527
Author(s):  
Sohail Khan ◽  
Serguei Maximov ◽  
Rafael Escarela-Perez ◽  
Juan Carlos Olivares-Galvan ◽  
Enrique Melgoza-Vazquez ◽  
...  
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Electromagnetic Field ◽  
Boundary Conditions ◽  
Eddy Current ◽  
Special Functions ◽  
Separation Of Variables ◽  
Practical Implementation ◽  
Load Current ◽  
Good Agreement

The presence of harmonics in the load current considerably increases stray losses in electric transformers. In this research paper, a new model for computing the electromagnetic field (EMF) and eddy current (EC) losses in transformer tank covers is derived considering harmonics. Maxwell’s equations are solved with their corresponding boundary conditions. The differential equation thus obtained is solved using the method of separation of variables. The obtained expressions do not require the use of special functions, accommodating them for practical implementation in the industry. The obtained formulas are evaluated for different spectrum contents of the load current and losses. The results are in good agreement with simulations carried out using the Altair Flux finite element (FE) software.

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

Simulations of electric field at the initiation altitudes of sprites and halos generated by the thundercloud charge structure and lightning channels of causative return strokes

2021 ◽  
Author(s):  
Petr Kaspar ◽  
Ivana Kolmasova ◽  
Ondrej Santolik ◽  
Martin Popek ◽  
Pavel Spurny ◽  
...  
Keyword(s):  
Electromagnetic Field ◽  
Electric Field ◽  
Boundary Conditions ◽  
Free Space ◽  
Nonlinear Effects ◽  
Charge Structure ◽  
Transient Luminous Events ◽  
Lightning Channel ◽  
Free Parameters

<p><span>Sprites and halos are transient luminous events occurring above thunderclouds. They can be observed simultaneously or they can also appear individually. Circumstances leading to initiation of these events are still not completely understood. In order to clarify the role of lightning channels of causative lightning return strokes and the corresponding thundercloud charge structure, we have developed a new model of electric field amplitudes at halo/sprite altitudes. It consists of electrostatic and inductive components of the electromagnetic field generated by the lightning channel in free space at a height of 15 km. Above this altitude we solve Maxwell’s equations self-consistently including the nonlinear effects of heating and ionization/attachment of the electrons. At the same time, we investigate the role of a development of the thundercloud charge structure and related induced charges above the thundercloud. We show how these charges lead to the different distributions of the electric field at the initiation heights of the halos and sprites. We adjust free parameters of the model using observations of halos and sprites at the Nydek TLE observatory and using measurements of luminosity curves of the corresponding return strokes measured by an array of fast photometers. The latter measurements are also used to set the boundary conditions of the model.</span></p>

scholarly journals Memory Efficient Method for Electromagnetic Multi-Region Models Using Scattering Matrices

2018 ◽  
Vol 23 (4) ◽  
pp. 71 ◽  
Author(s):  
C. Custers ◽  
J. Jansen ◽  
E. Lomonova
Keyword(s):  
Electromagnetic Field ◽  
Boundary Conditions ◽  
Efficient Method ◽  
Maximum Size ◽  
Scattering Matrix ◽  
Matrix Approach ◽  
Matrix Formulation ◽  
Scattering Matrices ◽  
Memory Efficient

This paper describes the scattering matrix approach to obtain the solution to electromagnetic field quantities in harmonic multi-layer models. Using this approach, the boundary conditions are solved in such way that the maximum size of any matrix used during the computations is independent of the number of regions defined in the problem. As a result, the method is more memory efficient than classical methods used to solve the boundary conditions. Because electromagnetic sources can be located inside the regions of a configuration, the scattering matrix formulation is developed to incorporate these sources into the solving process. The method is applied to a 3D electromagnetic configuration for verification.

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