Geometric algebra in plasma electrodynamics

2013 ◽  
Vol 79 (5) ◽  
pp. 735-738
Author(s):  
D. P. RESENDES
Keyword(s):  
Geometric Algebra ◽  
Maxwell Equations ◽  
Differential Operators ◽  
Differential Forms ◽  
Single Equation ◽  
Mathematical Framework ◽  
Plasma System ◽  
First Order ◽  
Vector Derivative ◽  
Coordinate Geometry

AbstractGeometric algebra (GA) is a recent broad mathematical framework incorporating synthetic and coordinate geometry, complex variables, quarternions, vector analysis, matrix algebra, spinors, tensors, and differential forms. It has been claimed to be a unified language for physics. GA is presented in the context of the Maxwell-Plasma system. In this formalism the divergence and curl differential operators are united in a single vector derivative, which is invertible, in the form of a first-order Green function. The four Maxwell equations can be combined into a single equation (for homogeneous and constant media) or into two equations involving the invertible vector derivative for more complex media. GA is applied to simple examples to illustrate the compactness of the notation and coordinate-free computations.

Multicomponent Beam-Plasma Waves

1967 ◽  
Vol 22 (12) ◽  
pp. 1935-1939
Author(s):  
Frank G. Verheest
Keyword(s):  
Magnetic Induction ◽  
Maxwell Equations ◽  
Equations Of Motion ◽  
Charged Particles ◽  
Plasma Waves ◽  
Plasma System ◽  
First Order ◽  
Multicomponent Plasma ◽  
Beam Plasma ◽  
General Dispersion

The linearization procedure is applied to the equations governing a beam-plasma system, in which the stream velocities and the wavevector are parallel to the external magnetic induction. No special constraints are imposed on the parameters characterizing the constituent fluids in the equilibrium state of this macroscopic picture. From the MAXWELL equations an expression for the electromagnetic field of the wave is obtained and substituted in the equations of motion. The components of the first-order pressure tensors are computed in the low-temperature approximation, but without recurring to the strong magnetic induction CGL hypothesis. Since the equations of motion are now expressed only in the components of the perturbations of the drift velocities, the dispersion relations follow immediately. These relations are applicable to all beam-plasma systems comprised between the now conventional multicomponent plasma and the system of beams of charged particles. Some known cold beam-plasma cases are included in the general dispersion equations.

scholarly journals An Exterior Algebraic Derivation of the Euler–Lagrange Equations from the Principle of Stationary Action

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2178
Author(s):  
Ivano Colombaro ◽  
Josep Font-Segura ◽  
Alfonso Martinez
Keyword(s):  
Maxwell Equations ◽  
Lagrangian Density ◽  
Exterior Algebra ◽  
Field Theories ◽  
Lagrange Equations ◽  
First Order ◽  
Vector Calculus ◽  
Stationary Action ◽  
Vector Derivative ◽  
Derivatives Of

In this paper, we review two related aspects of field theory: the modeling of the fields by means of exterior algebra and calculus, and the derivation of the field dynamics, i.e., the Euler–Lagrange equations, by means of the stationary action principle. In contrast to the usual tensorial derivation of these equations for field theories, that gives separate equations for the field components, two related coordinate-free forms of the Euler–Lagrange equations are derived. These alternative forms of the equations, reminiscent of the formulae of vector calculus, are expressed in terms of vector derivatives of the Lagrangian density. The first form is valid for a generic Lagrangian density that only depends on the first-order derivatives of the field. The second form, expressed in exterior algebra notation, is specific to the case when the Lagrangian density is a function of the exterior and interior derivatives of the multivector field. As an application, a Lagrangian density for generalized electromagnetic multivector fields of arbitrary grade is postulated and shown to have, by taking the vector derivative of the Lagrangian density, the generalized Maxwell equations as Euler–Lagrange equations.

Microstructure Sensitive Design with First Order Homogenization Theories and Finite Element Codes

2005 ◽  
Vol 495-497 ◽  
pp. 23-30 ◽  
Author(s):  
Surya R. Kalidindi ◽  
J. Houskamp ◽  
G. Proust ◽  
H. Duvvuru
Keyword(s):  
Finite Element ◽  
Inverse Problems ◽  
Mechanical Design ◽  
Materials Design ◽  
Mathematical Framework ◽  
First Order

A mathematical framework called Microstructure Sensitive Design (MSD) has been developed recently to solve inverse problems of materials design, where the goal is to identify the class of microstructures that are predicted to satisfy a set of designer specified objectives and constraints [1]. This paper demonstrates the application of the MSD framework to a specific case study involving mechanical design. Processing solutions to obtain one of the elements of the desired class of textures are also explored within the same framework.

scholarly journals On Certain Proximities and Preorderings on the Transposition Hypergroups of Linear First-Order Partial Differential Operators

2014 ◽  
Vol 22 (1) ◽  
pp. 85-103 ◽  
Author(s):  
Jan Chvalina ◽  
Šárka Hošková-Mayerová
Keyword(s):  
Differential Operators ◽  
Partial Differential ◽  
First Order ◽  
Ordered Groups ◽  
Partial Differential Operators ◽  
Power Set

AbstractThe contribution aims to create hypergroups of linear first-order partial differential operators with proximities, one of which creates a tolerance semigroup on the power set of the mentioned differential operators. Constructions of investigated hypergroups are based on the so called “Ends-Lemma” applied on ordered groups of differnetial operators. Moreover, there is also obtained a regularly preordered transpositions hypergroup of considered partial differntial operators.

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