Search results for "Equations"
showing 10 items of 955 documents
Maxwell Theory as a Classical FieldTheory
2012
Hamilton’s variational principle and the Lagrangian mechanics that rests on it are exceedingly successful in their application to mechanical systems with a finite number of degrees of freedom. Hamilton’s principle characterizes the physically realizable orbits, among the set of all possible orbits, as being the critical elements of the action integral. The Lagrangian function, although not an observable on its own, is not only useful in deriving the equations of motion but is also an important tool for identifying symmetries of the theory and constructing the corresponding conserved quantities, via Noether’s theorem.
The Numerical Simulation of Relativistic Fluid Flow with Strong Shocks
2001
In this review we present and analyze the performance of a Go-dunov type method applied to relativistic fluid flow. Our model equations are the corresponding Euler equations for special relativistic hydrodynamics. By choosing an appropriate vector of unknowns, the equations of special relativistic fluid dynamics (RFD) can be written as a hyperbolic system of conservation laws. We give a complete description of the spectral decomposition of the Jacobian matrices associated to the fluxes in each spatial direction, (see (Donat et al., 1998), for details), which is the essential ingredient of the Godunov-type numerical method we propose in this paper. We also review a numerical flux formula tha…
A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows
2013
In this paper we present a kinetic relaxation scheme for the Euler equations of gas dynamics in one space dimension. The method is easily applicable to solve any complex system of conservation laws. The numerical scheme is based on a relaxation approximation for conservation laws viewed as a discrete velocity model of the Boltzmann equation of kinetic theory. The discrete kinetic equation is solved by a splitting method consisting of a convection phase and a collision phase. The convection phase involves only the solution of linear transport equations and the collision phase instantaneously relaxes the distribution function to an equilibrium distribution. We prove that the first order accur…
The time-harmonic Maxwell equations
1996
In this chapter we shall see that the solution of the time-harmonic Maxwell equations with real coefficients can be transformed to time independent partial differential equations with complex coefficients. Then we introduce a finite element approximation proposed in [Křižek, Neittaanmaki, 1989]. A similar technique is analyzed in [Křižek, Neittaanmaki, 1984b], [Monk, 1992a] (for fully time dependent problems see, e.g., [Monk 1992b,c]).
Maxwell’s Equations
2012
The empirical basis of electrodynamics is defined by Faraday’s law of induction, by Gauss’ law, by the law of Biot and Savart and by the Lorentz force and the principle of universal conservation of electric charge. These laws can be tested – confirmed or falsified – in realistic experiments. The integral form of the laws deals with physical objects that are one-dimensional, two-dimensional, or three-dimensional, that is to say, objects such as linear wires, conducting loops, spatial charge distributions, etc. Thus, the integral form depends, to some extent, on the concrete experimental set-up. To unravel the relationships between seemingly different phenomena, one must switch from the integ…
On the theory of domain structure in ferromagnetic phase of diluted magnetic semiconductors
2006
Abstract We present a comprehensive analysis of domain structure formation in ferromagnetic phase of diluted magnetic semiconductors (DMS) of p-type. Our analysis is carried out on the base of effective magnetic free energy of DMS calculated by us earlier [Yu.G. Semenov, V.A. Stephanovich, Phys. Rev. B 67 (2003) 195203]. This free energy, substituting DMS (a disordered magnet) by effective ordered substance, permits to apply the standard phenomenological approach to domain structure calculation. Using coupled system of Maxwell equations with those obtained by minimization of above free energy functional, we show the existence of critical ratio ν cr of concentration of charge carriers and ma…
Systems of Linear Equations
2016
A linear equation in \(\mathbb {R}\) in the variables \(x_1,x_2,\ldots ,x_n\) is an equation of the kind:
Mixing of Two-Quasiparticle Configurations
2007
In this chapter we discuss configuration mixing of two-quasiparticle states. It is caused by the residual interaction remaining beyond the quasiparticle mean field defined in Chap. 13. We derive the equations of motion by the EOM method developed in Sect. 11.1. To accomplish this we need to express the residual Hamiltonian in terms of quasiparticles.
Quasienergy states of trapped ions
2000
The quantum models for a single trapped ion are extended to the description of the collective dynamics for systems of ions confined in quadrupole electromagnetic traps with cylindrical symmetry. A class of quantum Hamiltonians with suitable axial and radial interaction potentials given by homogeneous functions of degree (-2) and invariant under translations and axial rotations are introduced. The considered axial and radial quantum Hamiltonians for the center-of-mass and relative motions are described by collective dynamical systems associated to the symplectic group \(\). Discrete quasienergy spectra are obtained and the corresponding quasienergy states are explicitly realized as \(\) cohe…
Partially Implicit Runge-Kutta Methods for Wave-Like Equations
2014
Runge-Kutta methods are used to integrate in time systems of differential equations. Implicit methods are designed to overcome numerical instabilities appearing during the evolution of a system of equations. We will present partially implicit Runge-Kutta methods for a particular structure of equations, generalization of a wave equation; the partially implicit term refers to this structure, where the implicit term appears only in a subset of the system of equations. These methods do not require any inversion of operators and the computational costs are similar to those of explicit Runge-Kutta methods. Partially implicit Runge-Kutta methods are derived up to third-order of convergence. We ana…