38 research outputs found
Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations
We use Fr\"olicher-Nijenhuis theory to obtain global Helmholtz conditions,
expressed in terms of a semi-basic 1-form, that characterize when a semispray
is locally Lagrangian. We also discuss the relation between these Helmholtz
conditions and their classic formulation written using a multiplier matrix.
When the semi-basic 1-form is 1-homogeneous (0-homogeneous) we show that two
(one) of the Helmholtz conditions are consequences of the other ones. These two
special cases correspond to two inverse problems in the calculus of variation:
Finsler metrizability for a spray, and projective metrizability for a spray
Geometric properties of electromagnetic waves
This work studies geometrical properties of electromagnetic wave propagation. The work starts by studying geometrical properties of electromagnetic Gaussian beams in inhomogeneous anisotropic media. These are asymptotical solutions to Maxwell's equations that have a very characteristic feature. Namely, at each time instant the entire energy of the solution is concentrated around one point in space. When time moves forward, a Gaussian beam propagates along a curve. In recent work by A. P. Kachalov, Gaussian beams have been studied from a geometrical point of view. Under suitable conditions on the media, Gaussian beams propagate along geodesics. Furthermore, the shape of a Gaussian beam is determined by a complex tensor Riccati equation. The first paper of this dissertation provides a partial classification of media where Gaussian beams geometrize. The second paper shows that the real part of a solution to the aforementioned Riccati equation is essentially the shape operator for the phase front for the Gaussian beam. An important phenomena for electromagnetic Gaussian beams is that their propagation depend on their polarization. The last paper studies this phenomena from a very general point of view in arbitrary media. It also studies a connection between contact geometry and electromagnetism.reviewe
Descending maps between slashed tangent bundles
Suppose and are slashed
tangent bundles of two smooth manifolds and , respectively.
In this paper we characterize those diffeomorphisms that can be written as for a diffeomorphism \phi\colon M\to \wt M. When
one say that \emph{descends}. If is
equipped with two sprays, we use the characterization to derive sufficient
conditions that imply that descends to a totally geodesic map. Specializing
to Riemann geometry we also obtain sufficient conditions for to descent to
an isometry
Characterisation and representation of non-dissipative electromagnetic medium with a double light cone
We study Maxwell's equations on a 4-manifold N with a medium that is
non-dissipative and has a linear and pointwise response. In this setting, the
medium can be represented by a suitable (2,2)-tensor on the 4-manifold N.
Moreover, in each cotangent space on N, the medium defines a Fresnel surface.
Essentially, the Fresnel surface is a tensorial analogue of the dispersion
equation that describes the response of the medium for signals in the geometric
optics limit. For example, in isotropic medium the Fresnel surface is at each
point a Lorentz light cone. In a recent paper, I. Lindell, A. Favaro and L.
Bergamin introduced a condition that constrains the polarisation for plane
waves. In this paper we show (under suitable assumptions) that a slight
strengthening of this condition gives a pointwise characterisation of all
medium tensors for which the Fresnel surface is the union of two distinct
Lorentz null cones. This is for example the behaviour of uniaxial medium like
calcite. Moreover, using the representation formulas from Lindell et al. we
obtain a closed form representation formula that pointwise parameterises all
medium tensors for which the Fresnel surface is the union of two distinct
Lorentz null cones. Both the characterisation and the representation formula
are tensorial and do not depend on local coordinates
A restatement of the normal form theorem for area metrics
An area metric is a (0,4)-tensor with certain symmetries on a 4-manifold that
represent a non-dissipative linear electromagnetic medium. A recent result by
Schuller, Witte and Wohlfarth provides a pointwise normal form theorem for such
area metrics. This result is similar to the Jordan normal form theorem for
(1,1)-tensors, and the result shows that any area metric belongs to one of 23
metaclasses with explicit coordinate expressions for each metaclass. In this
paper we restate and prove this result for skewon-free (2,2)-tensors and show
that in general, each metaclasses has three different coordinate
representations, and each of metaclasses I, II, ..., VI, VII need only one
coordinate representation.Comment: Updated proof of Proposition A.2 (Claim 5). Fixed typo in Theorem 6
(Metaclass XXIII