An example of interacting classical vector fields having a static solution for the Wave equations

1974 ◽  
Vol 10 (11) ◽  
pp. 451-454 ◽  
Author(s):  
B. Ferretti ◽  
G. Velo
Keyword(s):  
Vector Fields ◽  
Wave Equations ◽  
Static Solution ◽  
Classical Vector

Lie Symmetry Analysis, Analytical Solutions, and Conservation Laws of the Generalised Whitham–Broer–Kaup–Like Equations

2017 ◽  
Vol 72 (3) ◽  
pp. 269-279 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Shou-Fu Tian ◽  
Chun-Yan Qin ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Conservation Laws ◽  
Vector Fields ◽  
Wave Equations ◽  
Boussinesq Equations ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Lie Symmetry Analysis ◽  
Bidirectional Propagation

AbstractIn this article, a generalised Whitham–Broer–Kaup–Like (WBKL) equations is investigated, which can describe the bidirectional propagation of long waves in shallow water. The equations can be reduced to the dispersive long wave equations, variant Boussinesq equations, Whitham–Broer–Kaup–Like equations, etc. The Lie symmetry analysis method is used to consider the vector fields and optimal system of the equations. The similarity reductions are given on the basic of the optimal system. Furthermore, the power series solutions are derived by using the power series theory. Finally, based on a new theorem of conservation laws, the conservation laws associated with symmetries of this equations are constructed with a detailed derivation.

scholarly journals Global existence for equivalent nonlinear special scale invariant damped wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sandra Lucente
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Vector Fields ◽  
Wave Equations ◽  
Small Data ◽  
Damped Wave Equation ◽  
Scale Invariant ◽  
Forcing Term ◽  
Time Existence

<p style='text-indent:20px;'>In this paper we give the notion of equivalent damped wave equations. As an application we study global in time existence for the solution of special scale invariant damped wave equation with small data. To gain such results, without radial assumption, we deal with Klainerman vector fields. In particular we can treat some potential behind the forcing term.</p>

BOUNDARY CONDITIONS IN THE MECHANICS OF FIELDS

1952 ◽  
Vol 30 (6) ◽  
pp. 684-698
Author(s):  
S. M. Neamtan ◽  
E. Vogt
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Vector Fields ◽  
Wave Equations ◽  
Equations Of Motion ◽  
Second Order ◽  
Dirac Field ◽  
Energy Tensor ◽  
Complex Scalar ◽  
Set Up

A variational principle has been set up for the description of relativistic fields with the aid of Lagrangians involving second order derivatives of the field functions. This constitutes a generalization of the usual formulation in that, besides the boundary conditions usually imposed, it admits also linear homogeneous boundary conditions. The formulation has been developed for the complex scalar and complex vector fields. The variational principle then yields not only the wave equations but also the allowed boundary conditions. A Hamiltonian and equations of motion in canonical form can be set up. A symmetric stress–energy tensor and a charge–current vector are defined, yielding the usual conservation equations. For the vector field, π4 is not identically zero; also the Lorentz condition arises out of the variational principle and does not have to be separately imposed. For the Dirac field an extension to Lagrangians with second order derivatives is not possible, but for this field also the variational principle yields the allowed boundary conditions.

scholarly journals GRAPH-THEORY INDUCED GRAVITY AND STRONGLY-DEGENERATE FERMIONS IN A SELF-CONSISTENT EINSTEIN UNIVERSE

2012 ◽  
Vol 27 (23) ◽  
pp. 1250131
Author(s):  
NAHOMI KAN ◽  
KOICHIRO KOBAYASHI ◽  
KIYOSHI SHIRAISHI
Keyword(s):  
Vector Fields ◽  
Static Solution ◽  
Scalar Fields ◽  
Induced Gravity ◽  
Einstein Universe ◽  
Mass Matrices ◽  
Self Consistent ◽  
Finite Theory ◽  
Strongly Degenerate ◽  
Matter Fields

We study UV-finite theory of induced gravity. We use scalar fields, Dirac fields and vector fields as matter fields whose one-loop effects induce the gravitational action. To obtain the mass spectrum that satisfies the UV-finiteness condition, we use a graph-based construction of mass matrices. The existence of a self-consistent static solution for an Einstein universe is shown in the presence of degenerate fermions.

Relativistic Wave Equations

2021 ◽  
pp. 167-190
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Wave Equations ◽  
Plane Waves ◽  
Canonical Formalism ◽  
Real Field ◽  
Klein Gordon ◽  
Linear Differential ◽  
Propagation Properties

We derive the most general relativistically covariant linear differential equations, having at most two derivatives, for scalar, spinor and vector fields. We introduce the corresponding Lagrangian and Hamiltonian formalisms and present the expansion of the solutions in terms of plane waves. In each case, we study the propagation properties of the corresponding Green functions. We start with the simplest example of the Klein–Gordon equation for a real field and generalise it to that of N real, or complex fields. As a next step we derive the Weyl, Majorana and Dirac equations for spinor fields. They are first order differential equations and we show how to adapt to them the canonical formalism. We end with the Proca and Maxwell equations for massive and massless spin-one fields and, in each case, we determine the physical degrees of freedom.

scholarly journals Wave Equations of Cooper Pairs as Quasiparticles in Superconducting Fermion Systems

1997 ◽  
Vol 50 (6) ◽  
pp. 1035
Author(s):  
Nguyen Van Hieu
Keyword(s):  
Variational Principle ◽  
Effective Action ◽  
Vector Fields ◽  
Wave Equations ◽  
Cooper Pairs ◽  
Ginzburg Landau ◽  
Fermion Systems ◽  
Scalar And Vector Fields ◽  
Ginzburg Landau Equations ◽  
Superconducting Pairing

The superconducting pairing of fermions is studied in the framework of the functional intergral approach. The bi-local composite scalar and vector fields are introduced to describe the singlet and triplet pairings. The static (time-independent) fields are the superconducting order parameters. From the variational principle for the effective action of the composite fields we derive the generalised Ginzburg–Landau equations. They are also the extensions of the BCS gap equations.

Relativistic Wave Equations

2017 ◽  
Author(s):  
Laurent Baulieu ◽  
John Iliopoulos ◽  
Roland Sénéor
Keyword(s):  
Dirac Equation ◽  
Vector Fields ◽  
Wave Equations ◽  
Gordon Equation ◽  
Relativistic Wave ◽  
Wave Solutions ◽  
Lagrangian Formulations ◽  
Klein Gordon ◽  
Conserved Currents ◽  
Klein Gordon Equation

Relativistically covariant wave equations for scalar, spinor, and vector fields. Plane wave solutions and Green’s functions. The Klein–Gordon equation. The Dirac equation and the Clifford algebra of γ‎ matrices. Symmetries and conserved currents. Hamiltonian and Lagrangian formulations. Wave equations for spin-1 fields.

Hamiltonian Field Theory

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Field Theory ◽  
Vector Fields ◽  
Wave Equations ◽  
Classical Field Theory ◽  
Late Nineteenth Century ◽  
Metric Tensor ◽  
Minkowski Metric ◽  
Gauge Fixing ◽  
Electromagnetic Tensor ◽  
Lorentz Force Law

This chapter discusses classical electromagnetism. As an example of a classical field theory, electrodynamics is framed using a Lagrangian density. Until pioneers such as Faraday and Maxwell, electric vector fields and magnetic vector fields were regarded as separate phenomena entirely and it was only in the late nineteenth century that scientists saw them as components of a larger concept, the electromagnetic field. Maxwell’s equations are derived and the wave equations are revisited. The chapter discusses gauge fixing, the Hodge star, the Lorentz force law and molecular multipole moments and closes by defining the electromagnetic tensor and the Minkowski metric tensor.

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