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Register a new userCovariant Hysteretic Constitutive Theory for Maxwells equations: Application to Axially Rotating Media
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This paper explores a class of non-linear constitutive relations for materials with memory in the framework of covariant macroscopic Maxwell theory. Based on earlier models for the response of hysteretic ferromagnetic materials to prescribed slowly varying magnetic background fields, generalized models are explored that are applicable to accelerating hysteretic magneto-electric substances coupled self-consistently to Maxwell fields. Using a parameterized model consistent with experimental data for a particular material that exhibits purely ferroelectric hysteresis when at rest in a slowly varying electric field, a constitutive model is constructed that permits a numerical analysis of its response to a driven harmonic electromagnetic field in a rectangular cavity. This response is then contrasted with its predicted response when set in uniform rotary motion in the cavity.
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A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. Whereas the covariant canonical field equations are equivalent to the Euler-Lagrange field equations, the covariant canonical transformation theory offers more general means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proved that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinitesimal canonical transformation furnishes the most general form of Noethers theorem. We furthermore specify the generating function of an infinitesimal space-time step that conforms to the field equations.
We consider the static Maxwell system with an axially symmetric dielectric permittivity and construct complete systems of its solutions which can be used for analytic and numerical solution of corresponding boundary value problems.
The representation of the conformal group (PSU(2,2)) on the space of solutions to Maxwells equations on the conformal compactification of Minkowski space is shown to break up into four irreducible unitarizable smooth Frechet representations of moderate growth. An explicit inner product is defined on each representation. The frequency spectrum of each of these representations is analyzed. These representations have notable properties; in particular they have positive or negative energy, they are of type $A_{frak q}(lambda)$ and are quaternionic. Physical implications of the results are explained.
We study a class of systems whose dynamics are described by generalized Langevin equations with state-dependent coefficients. We find that in the limit, in which all the characteristic time scales vanish at the same rate, the position variable of the system converges to a homogenized process, described by an equation containing additional drift terms induced by the noise. The convergence results are obtained using the main result in cite{hottovy2015smoluchowski}, whose version is proven here under a weaker spectral assumption on the damping matrix. We apply our results to study thermophoresis of a Brownian particle in a non-equilibrium heat bath.
The contraction of the Poincare group with respect to the space trans- lations subgroup gives rise to a group that bears a certain duality relation to the Galilei group, that is, the contraction limit of the Poincare group with respect to the time translations subgroup. In view of this duality, we call the former the dual Galilei group. A rather remarkable feature of the dual Galilei group is that the time translations constitute a central subgroup. Therewith, in unitary irreducible representations (UIR) of the group, the Hamiltonian appears as a Casimir operator proportional to the identity H = EI, with E (and a spin value s) uniquely characterizing the representation. Hence, a physical system characterized by a UIR of the dual Galilei group displays no non-trivial time evolution. Moreover, the combined U(1) gauge group and the dual Galilei group underlie a non- relativistic limit of Maxwells equations known as the electric limit. The analysis presented here shows that only electrostatics is possible for the electric limit, wholly in harmony with the trivial nature of time evolution governed by the dual Galilei group.
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