arXiv subjects

D. H. Delphenich

Publications and source records attributed to D. H. Delphenich.

At least 19 recordsLinked to original sources

arXiv2023

A pre-metric formulation of the Fresnel wave and ray surfaces

The conventional formulation of the Fresnel wave and ray surfaces typically involves the implicit use of Euclidian metrics in order to justify treating vectors and covectors as indistinguishable. This tends to disguise the fact that one of the surfaces lives in the cotangent spaces to the spatial (or space-time) manifold, while the other lives in the tangent spaces. Moreover, no mention is made of how to get from tangent spaces to cotangent spaces in the absence of a metric. The following analysis shows how to resolve those issues within the framework of pre-metric electromagnetism. The way that electromagnetic waves imply spatial frames, coframes, and metrics will be explained.

gr-qc↗

arXiv2022

A generalized Hamiltonian formulation of the principle of virtual work

The authors previous derivation of a variational principle from the total work functional, as a generalization of the first variation of an action functional, is extended by deriving a corresponding generalization of the Hamiltonian formulation of that action functional. Some consequences of it are that one can decompose the Lie brackets of arbitrary vector fields on symplectic mechanics into a sum of terms that involve the Poisson brackets of the functions that appear in the normal form of the Pfaffian that is symplectic-dual to the vector field and that one can also generalize the Hamilton-Jacobi equation to a system of nonlinear first-order partial differential equations for the contact field that one uses in order to obtain them.

math-ph↗

arXiv2020

On the equations of diffracted geodesics

An explicit form for the geodesic equations that would describe diffracted light rays is obtained and the Levi-Civita connection that enters into it is shown to be a sum of contributions from the metric of the ambient space, the index of refraction of the optical medium, and diffraction effects. The nature of the problem of obtaining explicit forms for the diffraction correction in various problems of interest to optics is discussed briefly.

physics.optics↗

arXiv2020

The Foucault pendulum as an example of motion on a pseudo-surface

The Foucault pendulum is shown to be an example of motion on a pseudo-surface, and the consequences of that are explored. In particular, its first and second fundamental forms are obtained, as well as its Gaussian and mean curvatures and the equations of its geodesics. However, a physical consideration that relates to the extension from space to space-time introduces a complication into the discussion of the geometry of the Foucault pseudo-surface that relates to the possible signatures of the space-time metric.

math-ph↗

arXiv2020

On geodesics of gradient-index optical metrics and the optical-mechanical analogy

The geodesic equations for optical media whose refractive indices have a non-vanishing gradient are developed. It is shown that when those media are optically isotropic, the light paths will be mull geodesics of a spatial metric that is conformally related to the metric of the ambient space. Various aspects of the optical-mechanical analogy are discussed as they relate to the geodesics of conformally related metrics. Some applications of the concepts of gradient-index optics to general relativity are examined, such as effective indices of refraction for gravitational lensing and Gordon's optical metrics for optical media that are in a state of relative motion with respect to an observer. The latter topic is approached from the standpoint of pre-metric electromagnetism.

gr-qc↗

arXiv2020

A pre-metric generalization of the Lorentz transformation

The concept of an observer and their associated rest space is defined in a pre-metric (i.e., projective-geometric) context that relates to time+space decompositions of the tangent bundle to space-time. The transformation from one observer to another when the two are in a state of relative motion is then defined, and its relationship to the Lorentz transformation is discussed. The group of all linear transformations that preserve the observer quadric, which generalizes the proper-time hyperboloid in Minkowski space, is defined and the reductions to some of its subgroups are described, as well as its extension to the group that preserves the fundamental quadric, which generalizes the light cone.

physics.class-ph↗

arXiv2019

The role of pseudo-hypersurfaces in non-holonomic motion

The geometry of hypersurfaces is generalized to pseudo-hypersurfaces, which are defined by Pfaff equations. The general methods are then applied to modeling the kinematics of motion constrained by a single linear, non-holonomic constraint. They are then applied to the example of a charge moving in an electromagnetic field, and the Lorentz equation of motion is shown to represent a geodesic that is constrained to lie in a pseudo-hypersurface that is defined by the potential 1-form.

math-ph↗

arXiv2019

Homology, equilibrium, and conservation laws I: Discrete systems of points

The methods of abstract simplicial homology and cohomology are reviewed and applied to the topology of electrical networks. Kirchhoffs laws of electrical circuits are shown to be manifestly homological in their origins. Since they are based in conservation laws, the geometric realization of abstract simplicial complexes is then reviewed and applied to the case of mechanical networks. The equilibrium condition for statics, the conservation laws for closed systems, and the balance principles for open systems are then shown to admit homological formulations.

math-ph↗

arXiv2018

Five-dimensional electromagnetism

The usual pre-metric Maxwell equations of electromagnetism in Minkowski space are extended to five dimensions, and the equations of linear five-dimensional electromagnetic waves are obtained from them by using the simplest electromagnetic constitutive law. The role of the dispersion law for five-dimensional plane waves in dictating the most appropriate extension of the four-dimensional Minkowski space metric to the five-dimensional one is discussed. The fact that the five-dimensional dispersion law includes both massless waves and massive ones is emphasized, as well as the fact that one can obtain the Klein-Gordon equation or the Proca equation by separating the fifth coordinate.

gr-qc↗

arXiv2018

Singular teleparallelism

It is shown that the geometry of parallelizable manifolds can be extended to non-parallelizable ones by extending the connection that a global frame field would define on a parallelizable manifold to a connection that a singular frame field would define on a non-parallelizable one. The resulting connection would typically have non-vanishing curvature in the neighborhood of the singular points of the frame field. The example of a 2-sphere is discussed as a motivating example and later extended to more general suspensions of parallelizable manifolds.

gr-qc↗

arXiv2018

Teleparallelism as anholonomic geometry

The geometry of parallelizable manifolds is presented from the standpoint of regarding it as conventional (e.g., Euclidian or Minkowskian) geometry, when it is described with respect to an anholonomic frame field that is defined on the space in question.

gr-qc↗

arXiv2017

On the electromagnetic constitutive laws that are equivalent to spacetime metrics

The raising of both indices in the components of the Minkowski electromagnetic field strength 2-form to give the components of the electromagnetic excitation bivector field can be regarded as being equivalent to an electromagnetic constitutive law, as well as being defined by the components of the spacetime metric. This notion is clarified, and the nature of the equivalent dielectric tensors and magnetic permeability tensors that are defined by some common spacetime metrics is discussed. The relationship of the basic construction to effective metrics is discussed, and, in particular, the fact that this effective metric is more general than the Gordon metric.

gr-qc↗

arXiv2016

Line geometry and electromagnetism IV: electromagnetic fields as infinitesimal Lorentz transformations

It is first shown that the scalar product on any orthogonal space (V, g) allows one to define linear isomorphisms of the vector spaces of bivectors and 2-forms on V with the underlying vector spaces of the Lie algebra so(p, q) and its dual, respectively. When those isomorphisms are applied to the electromagnetic excitation bivector and field strength 2-form, resp., one can associate various algebraic constructions that pertain to them as bivector fields and 2-forms with corresponding constructions in terms of so(1, 3) and its dual. The subsequent association with corresponding things in line geometry will then become straightforward. In particular, the fields can be represented by motors, such as screws and wrenches, while the Cartan-Killing form on so(1, 3) is isometric to the scalar product on bivectors that gives the Klein quadric. When the space of bivectors (and therefore the space of 2-forms) is given an almost-complex structure (and therefore, a complex structure), one can also represent most of the constructions on the former and its dual in terms of so(3; C) and its dual.

gr-qc↗

arXiv2016

On the local integrability of almost-product structures defined by space-time metrics

The splitting of the tangent bundle of space-time into temporal and spatial sub-bundles defines an almost-product structure. In particular, any space-time metric can be locally expressed in time-orthogonal form, in such a way that whether or not that almost-product structure is locally generated by a coordinate chart is a matter of the integrability of the Pfaff equation that the temporal 1-form of that expression for the metric defines. When one applies that analysis to the known exact solutions to the Einstein field equations, one finds that many of the common ones are completely-integrable, although some of the physically-interesting ones are not.

gr-qc↗

arXiv2015

Mechanics of Cosserat media: II. relativistic theory

The derivation of the non-relativistic Cosserat equations that was described in Part I of this series of papers is extended from the group of rigid motions in three-dimensional Euclidian space to the Poincaré group of four-dimensional Minkowski space. Examples of relativistic Cosserat media are then given in the form of the free Dirac electron and the Weyssenhoff fluid.

math-ph↗

arXiv2014

The role of homology in fluid vortices I: non-relativistic flow

The methods of singular and de Rham homology and cohomology are reviewed to the extent that they are applicable to the structure and motion of vortices. In particular, they are first applied to the concept of integral invariants. After a brief review of the elements of fluid mechanics, when expressed in the language of exterior differential forms and homology theory, the basic laws of vortex theory are shown to be statements that are rooted in the homology theory of integral invariants.

math-ph↗

arXiv2014

Line geometry and electromagnetism III: groups of transformations

The role of linear and projective groups of transformations in line geometry and electromagnetism is examined in accordance with Klein's Erlanger Programm for geometries. The group of collineations of real projective space is chosen as the most general group, and reductions to some of its various subgroups are then detailed according to their relevance to electromagnetic fields, and especially wave-like ones.

gr-qc↗

arXiv2013

Line geometry and electromagnetism II: wave motion

The fundamental role of line geometry in the study of wave motion is first introduced in the general context by way of the tangent planes to the instantaneous wave surfaces, in which it is first observed that the possible frequency-wave number 1-forms are typically constrained by a dispersion law that is derived from a constitutive law by way of the field equations. After a general review of the basic concepts that relate to quadratic line complexes, these geometric notions are applied to the study of electromagnetic waves, in particular.

gr-qc↗