55,275 research outputs found
Hamiltonian Structure for Classical Electrodynamics of a Point Particle
We prove that, contrary to the common belief, the classical Maxwell
electrodynamics of a point-like particle may be formulated as an
infinite-dimensional Hamiltonian system. We derive well defined
quasi-Hamiltonian which possesses direct physical interpretation being equal to
the total energy of the composed (field + particle) system. The phase space of
this system is endowed with an interesting symplectic structure. We prove that
this structure is strongly non-degenerated and, therefore, enables one to
define consistent Poisson bracket for particle's and field degrees of freedom.
We stress that this formulation is perfectly gauge-invariant.Comment: 36 pages, LATE
Burgers' Flows as Markovian Diffusion Processes
We analyze the unforced and deterministically forced Burgers equation in the
framework of the (diffusive) interpolating dynamics that solves the so-called
Schr\"{o}dinger boundary data problem for the random matter transport. This
entails an exploration of the consistency conditions that allow to interpret
dispersion of passive contaminants in the Burgers flow as a Markovian diffusion
process. In general, the usage of a continuity equation , where stands for the
Burgers field and is the density of transported matter, is at variance
with the explicit diffusion scenario. Under these circumstances, we give a
complete characterisation of the diffusive transport that is governed by
Burgers velocity fields. The result extends both to the approximate description
of the transport driven by an incompressible fluid and to motions in an
infinitely compressible medium. Also, in conjunction with the Born statistical
postulate in quantum theory, it pertains to the probabilistic (diffusive)
counterpart of the Schr\"{o}dinger picture quantum dynamics.Comment: Latex fil
Fast shape reconstruction of perfectly conducting cracks by using a multi-frequency topological derivative strategy
This paper concerns a fast, one-step iterative technique of imaging extended
perfectly conducting cracks with Dirichlet boundary condition. In order to
reconstruct the shape of cracks from scattered field data measured at the
boundary, we introduce a topological derivative-based electromagnetic imaging
function operated at several nonzero frequencies. The properties of the imaging
function are carefully analyzed for the configurations of both symmetric and
non-symmetric incident field directions. This analysis explains why the
application of incident fields with symmetric direction operated at multiple
frequencies guarantees a successful reconstruction. Various numerical
simulations with noise-corrupted data are conducted to assess the performance,
effectiveness, robustness, and limitations of the proposed technique.Comment: 17 pages, 27 figure
Invisibility and Inverse Problems
This survey of recent developments in cloaking and transformation optics is
an expanded version of the lecture by Gunther Uhlmann at the 2008 Annual
Meeting of the American Mathematical Society.Comment: 68 pages, 12 figures. To appear in the Bulletin of the AM
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