3,708 research outputs found
Frame Indifferent Formulation of Maxwell's Elastic Fluid and the Rational Continuum Mechanics of the Electromagnetic Field
We show that the linearized equations of the incompressible elastic medium
admit a `Maxwell form' in which the shear component of the stress vector plays
the role of the electric field, and the vorticity plays the role of the
magnetic field. Conversely, the set of dynamic Maxwell equations are strict
mathematical corollaries from the governing equations of the incompressible
elastic medium. This suggests that the nature of `electromagnetic field' may
actually be related to an elastic continuous medium. The analogy is complete if
the medium is assumed to behave as fluid in shear motions, while it may still
behave as elastic solid under compressional motions. Then the governing
equations of the elastic fluid are re-derived in the Eulerian frame by
replacing the partial time derivatives by the properly invariant (frame
indifferent) time rates. The `Maxwell from' of the frame indifferent
formulation gives the frame indifferent system that is to replace the Maxwell
system. This new system comprises terms already present in the classical
Maxwell equations, alongside terms that are the progenitors of the
Biot--Savart, Oersted--Ampere's, and Lorentz--force laws. Thus a frame
indifferent (truly covariant) formulation of electromagnetism is achieved from
a single postulate that the electromagnetic field is a kind of elastic (partly
liquid partly solid) continuum.Comment: accepte
Near-integrability of low dimensional periodic Klein-Gordon lattices
The low dimensional periodic Klein-Gordon lattices are studied for
integrability. We prove that the periodic lattice with two particles and
certain nonlinear potential is non integrable. However, in the cases of up to
six particles, we prove that their Birkhoff-Gustavson normal forms are
integrable, which allows us to apply KAM theory
Wave Solutions
In classical continuum physics, a wave is a mechanical disturbance. Whether
the disturbance is stationary or traveling and whether it is caused by the
motion of atoms and molecules or the vibration of a lattice structure, a wave
can be understood as a specific type of solution of an appropriate mathematical
equation modeling the underlying physics. Typical models consist of partial
differential equations that exhibit certain general properties, e.g.,
hyperbolicity. This, in turn, leads to the possibility of wave solutions.
Various analytical techniques (integral transforms, complex variables,
reduction to ordinary differential equations, etc.) are available to find wave
solutions of linear partial differential equations. Furthermore, linear
hyperbolic equations with higher-order derivatives provide the mathematical
underpinning of the phenomenon of dispersion, i.e., the dependence of a wave's
phase speed on its wavenumber. For systems of nonlinear first-order hyperbolic
equations, there also exists a general theory for finding wave solutions. In
addition, nonlinear parabolic partial differential equations are sometimes said
to posses wave solutions, though they lack hyperbolicity, because it may be
possible to find solutions that translate in space with time. Unfortunately, an
all-encompassing methodology for solution of partial differential equations
with any possible combination of nonlinearities does not exist. Thus, nonlinear
wave solutions must be sought on a case-by-case basis depending on the
governing equation.Comment: 22 pages, 3 figures; to appear in the Mathematical Preliminaries and
Methods section of the Encyclopedia of Thermal Stresses, ed. R.B. Hetnarski,
Springer (2014), to appea
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