arXiv subjects
Publications and source records attributed to D. H. Delphenich.
Some key notions of line geometry are recalled, along with their application to mechanics. It is then shown that most of the basic structures that one introduces in the pre-metric formulation of electromagnetism can be interpreted directly in terms of corresponding concepts in line geometry. The results are summarized in a table.
The geometry of parallelizable manifolds (i.e., teleparallelism) is summarized in the language of local frame fields. Some problems in continuum mechanics that relate to the couple-stresses that are produced in the bending and twisting of prismatic beams and wires are then discussed. It is then shown that by going to a higher-dimensional analogue of the geometry that one used for one-dimensional deformable objects, one is basically using the methods of teleparallelism in the context of the Cosserat approach to deformable bodies.
Ordinarily, the stress tensor that one derives for a Madelung fluid is not regarded as being coupled to a strain tensor, which is consistent with the fluid hypothesis. However, based upon earlier work regarding the geometric nature of the quantum potential, one can, in fact, define a strain tensor, which is not, however, due to a deformation of a spatial region, but to a deformation of a frame field on that region. When one expresses the Madelung stress tensor as a function of the strain tensor and its derivatives, one then defines a constitutive law for the Madelung medium that might lead to a more detailed picture of its elementary structure. It is pointed out that the resulting constitutive law is strongly analogous to laws that were presented by Kelvin and Tait for the bending and torsion of elastic wires and plates, as well as the Einstein equations for gravitation if one takes the viewpoint of metric elasticity.
The Cosserat equations for equilibrium are derived by starting from the action of the group of smooth functions with values in the Lie group of rigid spatial motions on rigid frames in Euclidian space. The method of virtual work is employed, instead of the method of action functionals, and is defined using the methods of of jet manifolds. It is shown how the restriction to fundamental forms (which define the dynamical state of the Cosserat object) that are Euclidian is essential to obtaining the Cosserat equations.
It is shown that the groups of Euclidian rotations, rigid motions, proper, orthochronous Lorentz transformations, and the complex rigid motions can be represented by the groups of unit-norm elements in the algebras of real, dual, complex, and complex dual quaternions, respectively. It is shown how someof the physically-useful tensors and spinors can be represented by the various kinds of quaternions. The basic notions of kinematical states are described in each case, except complex dual quaternions, where their possible role in describing the symmetries of the Maxwell equations is discussed.
It is proposed that the mathematical models for any physical systems that are based in first principles, such as conservation laws or balance principles, have some common elements, namely, a space of kinematical states, a space of dynamical states, a constitutive law that associates dynamical states with kinematical states, as well as a duality principle. The equations of motion or statics then come about from, on the one hand, specifying the integrability of the kinematical state, and on the other hand, specifying a statement that is dual to it for the dynamical states. Examples are given from various fundamental physical systems.
It is first observed that the original formulation of the Volterra construction for dislocations and disclinations was related to the role that homotopy plays in strain compatibility, whereas the modern discussions are chiefly concerned with how it relates to the holonomy groups of connections that have non-vanishing torsion and curvature. However, the Saint Venant conditions that follow from assuming infinitesimal strain compatibility imply that both torsion and curvature must vanish. The resolution of the confusion is in the fact that when a manifold is multiply connected a flat connection might still have non-trivial discrete holonomy.
In a series of previous articles by the author, it was shown that one could effectively give a variational formulation to non-conservative mechanical systems, as well as ones that subject to non-holonomic constraints by starting with the virtual work functional instead of an action functional. In this article, it is shown that when one starts with a virtual work functional that is not exact, so it does not admit an action functional, instead of conservation laws the extension of Noether's theorem gives balance principles for non-conserved currents. Examples of this generalized Noether identity are given in the mechanics of points, rigid bodies, and deformable bodies.
The problem of extending fields that are defined on lattices to fields defined on the continua that they become in the continuum limit is basically one of continuous extension from the 0-skeleton of a simplicial complex to its higher-dimensional skeletons. If the lattice in question has defects, as well as the order parameter space of the field, then this process might be obstructed by characteristic cohomology classes on the lattice with values in the homotopy groups of the order parameter space. The examples from solid-state physics that are discussed are quantum spin fields on planar lattices with point defects or orientable space lattices, vorticial flows or director fields on lattices with dislocations or disclinations, and monopole fields on lattices with point defects.
In a previous article by the author, it was shown that one could effectively give a variational formulation to non-conservative mechanical systems by starting with the first variation functional instead of an action functional. In this article, it is shown that this same approach will also allow one to give a variational formulation to systems with non-holonomic constraints. The key is to use an adapted anholonomic local frame field in the formulation, which then implies the replacement of ordinary derivatives with covariant ones. The method is then applied to the case of a vertical disc rolling without slipping or friction on a plane.
The mechanics of wave motion in a medium are founded in conservation laws for the physical quantities that the waves carry, combined with the constitutive laws of the medium, and define Lorentzian structures only in degenerate cases of the dispersion laws that follow from the field equations. It is suggested that the transition from wave motion to point motion is best factored into an intermediate step of extended matter motion, which then makes the dimension-codimension duality of waves and trajectories a natural consequence of the bicharacteristic (geodesic) foliation associated with the dispersion law. This process is illustrated in the conventional case of quadratic dispersion, as well as quartic ones, which include the Heisenberg-Euler dispersion law. It is suggested that the contributions to geodesic motion from the non-quadratic nature of a dispersion law might represent another source of quantum fluctuations about classical extremals, in addition to the diffraction effects that are left out by the geometrical optics approximation.
The equations of electrostatics are presented in pre-metric form, and it is pointed out that if the origin of the nonlinearity is the constitutive law for the medium then the differential equations themselves remain linear, while the nonlinearity is confined to an algebraic equation. These equations are solved for a general class of electric fields that include the common textbook examples, namely, fields that are adapted to a coordinate vector field. The special forms that they then take for particular electric constitutive laws of quantum origin, namely, the constitutive laws derived from the Born-Infeld and Heisenberg-Euler Lagrangians, are then discussed. Finally, the classical problem of modeling the electron is redefined in light of the established facts of quantum physics.
It is shown that physical mechanics for pointlike bodies can be effectively modeled in terms of the action of transformation groups that act as symmetries of the solutions of systems of differential equations that describe the integrability of dynamical states. The equations of motion are then obtained from these integrability equations for the dynamical states. It is also observed that the functions that define the components of a dynamical state represent a set of mechanical constitutive laws. The variational formulation of mechanics is shown to be a specialization of these principles.
The analogy between 1+3 splittings of the spacetime tangent bundle and the splitting of the tangent bundle to the bundle of linear frames into vertical and horizontal sub-bundles is described from the unifying standpoint of the geometry of foliations. The physical nature of the line field on spacetime that plays the role of vertical sub-bundle is discussed in some detail. The notion that the complementary spatial bundle is most fundamentally a representation of the normal bundle to the foliation of spacetime by the integral curves of the line field is proposed, such that the geometry of space becomes the transverse geometry of the flow and the integration of the spatial bundle into proper time simultaneity hypersurfaces becomes a secondary issue to the geometry of space as the geometry of the leaf space of the foliation. The concept of an adapted convected frame field is introduced as a means of locally representing the transversal geometry of a flow in terms of the information that is contained in the flow and its derivatives, and some discussion is given to the role of the Bott connection.
Some of the basic notions of nonlinear optics are summarized and then applied to the case of the Dirac vacuum, as described by the Heisenberg-Euler effective one-loop Lagrangian. The theoretical and experimental basis for the appearance of nonlinear optical phenomena, such as the Kerr effect, Cotton-Mouton effect, and four-wave mixing are discussed. Further effects due to more exotic assumptions on the structure of spacetime, such as gravitational curvature and the topology of the Casimir vacuum are also presented.
It is shown that not all linear electromagnetic constitutive laws will define almost-complex structure on the bundle of 2-forms on the spacetime manifold when composed with the Poincare duality isomorphism, but only a restricted class of them that includes linear spatially isotropic and some bi-isotropic laws. Although this does not trivialize the formulation of the basic equations equations of pre-metric electromagnetism, it does affect their reduction to metric electromagnetism by its effect on the types of media that are reducible, and possibly its effect on the way that such media support the propagation of electromagnetic waves.
The intimate link between complex geometry and the problem of the pre-metric formulation of electromagnetism is explored. In particular, the relationship between 3+1 decompositions of R4 and the decompositions of the vector space of bivectors over R4 into real and imaginary subspaces relative to a choice of complex structure is emphasized. The role of the various scalar products on the space of bivectors that are defined in terms of a volume element on R4 and a complex structure on the space of bivectors that makes it C-linear isomorphic to C3 is discussed in the context of formulation of a theory of electromagnetism in which the Lorentzian metric on spacetime follows as a consequence of the existence of electromagnetic waves, not a prior assumption.
The axioms of topological electromagnetism are refined by the introduction of the de Rham homology of k-vector fields on orientable manifolds and the use of Poincare duality in place of Hodge duality. The central problem of defining the electromagnetic constitutive law is elaborated upon in the linear and nonlinear cases. The manner by which the spacetime metric might follow from the constitutive law is examined in the linear case. The possibility that the intersection form of the spacetime manifold might play a role in defining a topological basis for the constitutive law is explored. The manner by which wave motion might follow from the electromagnetic structure is also discussed.