scholarly journals On Maxwell Electrodynamics in Multi-Dimensional Spaces

Universe ◽  
2021 ◽  
Vol 8 (1) ◽  
pp. 20
Author(s):  
Alexei M. Frolov
Keyword(s):  
Electromagnetic Field ◽  
Maxwell Equations ◽  
Three Dimensional ◽  
Governing Equations ◽  
Tensor Form ◽  
Action Function ◽  
One Dimensional ◽  
Least Action ◽  
Maxwell Electrodynamics ◽  
Principle Of Least Action

The governing equations of Maxwell electrodynamics in multi-dimensional spaces are derived from the variational principle of least action, which is applied to the action function of the electromagnetic field. The Hamiltonian approach for the electromagnetic field in multi-dimensional pseudo-Euclidean (flat) spaces has also been developed and investigated. Based on the two arising first-class constraints, we have generalized to multi-dimensional spaces a number of different gauges known for the three-dimensional electromagnetic field. For multi-dimensional spaces of non-zero curvature the governing equations for the multi-dimensional electromagnetic field are written in a manifestly covariant form. Multi-dimensional Einstein’s equations of metric gravity in the presence of an electromagnetic field have been re-written in the true tensor form. Methods of scalar electrodynamics are applied to analyze Maxwell equations in the two and one-dimensional spaces.

scholarly journals Numerical Calculation of Flow for Cascade With Tip Clearance

1994 ◽  
Author(s):  
Shimpei Mizuki ◽  
Hoshio Tsujita
Keyword(s):  
Three Dimensional ◽  
Tip Clearance ◽  
Turbine Cascade ◽  
Governing Equations ◽  
Tensor Form ◽  
Pressure Surface ◽  
Rear Part ◽  
Blade Tip ◽  
Flow Through ◽  
Blade Tip Geometry

Three-dimensional incompressible turbulent flow within a linear turbine cascade with tip clearance is analyzed numerically. The governing equations involving the standard k-ε model are solved in the physical component tensor form with a boundary-fitted coordinate system. In the analysis, the blade tip geometry is treated accurately in order to predict the flow through the tip clearance in detail when the blades have large thicknesses. Although the number of grids employed in the present study is not enough because of the limitation of computer storage memory, the computed results show good agreements with the experimental results. Moreover, the results clearly exhibit the locus of minimum pressure on the rear part of the pressure surface at the blade tip.

Time-Split Inflow Boundary Treatment

1985 ◽  
Author(s):  
Siu Shing Tong
Keyword(s):  
Three Dimensional ◽  
Internal Flow ◽  
Leading Edge ◽  
Two Dimensional ◽  
Governing Equations ◽  
Internal Flows ◽  
One Dimensional ◽  
Boundary Treatment

This paper describes a new non-reflective inflow treatment for viscous and inviscid internal flow calculations. The method approximates the multi-dimensional governing equations at the inflow boundary in a series of one-dimensional split equations. This treatment allows the artificial inflow boundary to be brought in just in front of the leading edge, while allowing upstream running waves to penetrate without significant reflection. Calculation examples of two dimensional inviscid internal flows are presented. Extension of the method to three-dimensional problems is also discussed.

scholarly journals On the principle of least action in wave mechanics

1928 ◽  
Vol 121 (788) ◽  
pp. 543-557 ◽  
Keyword(s):  
Wave Equations ◽  
Present Theory ◽  
Second Order ◽  
Wave Mechanics ◽  
Electron Wave ◽  
Tensor Form ◽  
Least Action ◽  
Principle Of Least Action ◽  
The Subject ◽  
Second Order Equations

It is well known that de Broglie’s wave mechanics is based on the analogy between the mechanical principle of least action and the optical principle of least time. Schrodinger’s original theory of the hydrogen atom was likewise based on a variational principle, and the method has been used extensively in subsequent developments of the subject. Very recently Darwin has given the form of the principle appropriate to Dirac’s theory and has shown that expressions for the current vector and electric density can be deduced from it. In Dirac’s work the electron wave is specified by a four-rowed matrix. The theory has been brilliantly successful in accounting for the “duplexity” phenomena of the atom, but has the defect that the wave equations are unsymmetrical and have not the tensor form. In a recent paper an attempt has been made to surmount this difficulty. Eight wave functions are employed instead of Dirac’s four. These are grouped together to form two four-vectors and satisfy tensor equations of the second order. It is shown in 2 of the present paper that these eight wave equations can be reduced, by addition and subtraction, to the four second order equations satisfied by Dirac’s functions, taken twice over; and that, in a sense, the present theory includes Dirac’s.

scholarly journals A TWO-GRID TIME-DEPENDENT FORMALISM FOR THE MAXWELL EQUATION

2003 ◽  
Vol 02 (04) ◽  
pp. 537-546 ◽  
Author(s):  
DANIEL NEUHAUSER ◽  
ROI BAER
Keyword(s):  
Maxwell Equations ◽  
Three Dimensional ◽  
Reaction Dynamics ◽  
Iterative Solution ◽  
Time Dependent ◽  
Scattering Problems ◽  
One Dimensional ◽  
Initial Wave ◽  
Chemical Reaction Dynamics ◽  
Iterative Solutions

We develop a formalism for efficient iterative solutions of scattering problems involving the Maxwell equations. The methods, borrowed from recent advancements in chemical reaction dynamics, represent the scattering wavefunctions on two grids; one used for the initial wave and is one-dimensional; the other is a small three-dimensional grid padded with absorbing-potentials on which the scattered function is represented. The formalism is automatically suitable for scattering studies of transmission, reflection and scattering components of a wave. The simulations can be done with time-dependent wavepackets or direct iterative solution for the Green's function, but the results are rigorous time-independent (frequency-dependent) scattering amplitudes. Model time-dependent simulations involving up to a 100×100×100 grid for the inner wavefunction were numerically done on a simple PC.

The wave equation in conformal space

1936 ◽  
Vol 32 (4) ◽  
pp. 622-631 ◽  
Author(s):  
H. J. Bhabha
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Euclidean Space ◽  
Conformal Invariance ◽  
Maxwell Equations ◽  
Conformal Transformations ◽  
Conformal Space ◽  
Tensor Form ◽  
Direct Transformation ◽  
Long Time

In a recent paper Dirac has shown that by passing from the ordinary Euclidean space to a four-dimensional conformal space, some of the equations of physics can be written in a tensor form, the indices of which take on six values. Those equations which can be written in this form are then invariant under conformal transformations of the Euclidean space. Among the equations of physics which have this more general invariance are the Maxwell equations, as was proved by a direct transformation a long time ago by Cunningham, and Bateman, so that Dirac's paper provides an alternative and more general proof of this result. Certain errors ∥ in Dirac's paper, however, necessitate a reformulation of the proof. Before we do this in § 2, we briefly recapitulate in § 1 some of the general results derived there. In § 3 we investigate further the conformal invariance of the wave equation for an electron in the presence of a general electromagnetic field.

Transport Processes in Magnetosolidmechanics

1966 ◽  
Vol 33 (4) ◽  
pp. 770-776 ◽  
Author(s):  
M. R. El-Saden ◽  
O. A. Arnas
Keyword(s):  
Three Dimensional ◽  
Second Law Of Thermodynamics ◽  
Irreversible Thermodynamics ◽  
Transport Processes ◽  
Governing Equations ◽  
Electrical Currents ◽  
One Dimensional ◽  
Current Densities ◽  
Galvanomagnetic Effects ◽  
State Case

Consideration is given to the problem of a solid conducting heat and electrical currents in the presence of externally applied magnetic fields. Interactions due to the thermoelectric, thermomagnetic, and galvanomagnetic effects are taken into consideration. The general three-dimensional governing equations are established. The generalized Ohm’s law and Fourier’s law of heat conduction as formulated by the methods of nonequilibrium (irreversible) thermodynamics are employed. The governing equations are used for the analysis of a one-dimensional steady-state case, assuming constant thermophysical properties. The variations of temperature, electric field, magnetic field, and electric and thermal current densities are presented analytically and graphically. The second law of thermodynamics imposes certain limitations on the validity of the results. These limitations are discussed in detail.

The action in a uniform field

1941 ◽  
Vol 37 (2) ◽  
pp. 168-176
Author(s):  
L. A. Pars
Keyword(s):  
Uniform Field ◽  
Action Function ◽  
Cartesian Coordinates ◽  
Constant Energy ◽  
Least Action ◽  
Plane Field ◽  
Principle Of Least Action ◽  
Image Position

We consider the motion of a particle in a plane field of force. We take rectangular cartesian coordinates (x, y) in the plane, and denote by V(x, y) the potential of the field, and by h the (constant) energy of the motion. If A and B, whose coordinates are (x0, y0) and (x1, y1), are any two points on an orbit, the orbit is characterized by the property thattaken along the orbit is stationary as compared with the integral taken along a neighbouring curve joining the same points. Here s denotes the length of the are measured from A to B. This is one form, sometimes called Jacobi's form, of the principle of least action. The value of the integral, taken along the orbit, and expressed in terms of the coordinates of the termini and the constant of energy,is the action function for the field V.

Performance Characteristics of Regenerative Flow Compressors for Natural Gas Compression Application

2005 ◽  
Vol 127 (1) ◽  
pp. 7-14 ◽  
Author(s):  
Mukarrum Raheel ◽  
Abraham Engeda
Keyword(s):  
Mathematical Model ◽  
Natural Gas ◽  
Three Dimensional ◽  
Governing Equations ◽  
One Dimensional ◽  
Gas Compression ◽  
And Performance ◽  
Regenerative Flow ◽  
Performance Results ◽  
Channel Region

In this paper we discuss the application of regenerative flow compressors (RFC) for low-pressure natural gas compression required by microturbine systems. A brief overview of fundamentals and the hypothesis of the operation of RFC is presented. A mathematical model to describe the complex three-dimensional corkscrew flow pattern in RFC is discussed. Governing equations for the blade and channel region are developed. A one-dimensional (1-D) performance prediction code for RFC based on governing equations and loss models is developed and performance results are compared with experimental data on a multistage RFC. Excellent agreement between theoretical and experimental results is observed, thus validating the proposed mathematical model.

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