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Reuven Segev

Publications and source records attributed to Reuven Segev.

At least 19 recordsLinked to original sources
arXiv2026

Notes on Electrostatics in $\mathbb R^3$

A compact formulation of electrostatics is presented. Without using constitutive relations, including the aether relations, an expression for the potential energy of a charged region under a potential function is proposed, where the charge distribution is specified by the charge potential field. Assuming that during a virtual motion of the charge, the virtual work expended by the field is equal to minus the time derivative of the potential energy, an expression for the mechanical force functional is derived. The force functional contains the action of an asymmetric active stress field D_{i}E_{j}, the skew-symmetric part of which is the mechanical couple density.

math-ph↗
arXiv2026

On Force Interactions for Electrodynamics-Like Theories

A framework for premetric p-form electrodynamics is proposed. Independently of particular constitutive relations, the corresponding Maxwell equations are derived as a special case of stress theory in geometric continuum mechanics. Expressions for the potential energy of a charged region in spacetime, as well as expressions for the force and stress interactions on the region, are presented. The expression for the force distribution is obtained by computing the rate of change of the proposed potential energy under a virtual motion of the region. These expressions differ from those appearing in the standard references. The cases of electrostatics and magnetostatics in R^3 are presented as examples.

math-ph↗
arXiv2025

Multipole Distributions and Hyper-Flux Fields

We outline here a simple mathematical introduction to the notions of multipoles for a general extensive property $Π$ from the point of view of continuum mechanics. Classically, $Π$ is the electric charge, but the theory is not limited to electrostatics. The proposed framework allows a simple computation of the bound "charges" and bound multipoles of lower orders. In addition, if the property $Π$ has a potential function in the sense described below, a general expression for the mechanical force (power) functional acting on bodies containing the property is presented. Finally, using a similar viewpoint, we consider hyper-fluxes -- flux fields of tensorial order greater than one -- and show that moving multipoles (in particular, a moving dielectric) give rise to hyper-fluxes.

math-ph↗
arXiv2024

Notes on Optimal Flux Fields

For a given region, and specified boundary flux and density rate of an extensive property, the optimal flux field that satisfies the balance conditions is considered. The optimization criteria are the $L^{p}$-norm and a Sobolev-like norm of the flux field. Finally, the capacity of the region to accommodate various boundary fluxes and density rates is defined and analyzed.

math.AP↗
arXiv2023

Electrodynamics and Geometric Continuum Mechanics

This paper offers an informal instructive introduction to some of the main notions of geometric continuum mechanics for the case of smooth fields. We use a metric invariant stress theory of continuum mechanics to formulate a simple generalization of the fields of electrodynamics and Maxwell's equations to general differentiable manifolds of any dimension, thus viewing generalized electrodynamics as a special case of continuum mechanics. The basic kinematic variable is the potential, which is represented as a $p$-form in an $n$-dimensional spacetime. The stress for the case of generalized electrodynamics is assumed to be represented by an $(n-p-1)$-form, a generalization of the Maxwell $2$-form.

math-ph↗
arXiv2023

Notes on Smooth and Singular Volumetric Growth

The material structure of bodies undergoing growth is considered. In the geometric framework of a general differential manifold modeling the physical space and a fiber bundle modeling spacetime, body points may be defined for any extensive property for which a smooth flux field exists, even if the property is not conserved. Singular flux fields are considered using the notion of a de Rham current. Writing a generalized balance law using the boundary of the current corresponding to a singular flux field, surface growth is unified with volumetric growth.

math-ph↗
arXiv2023

Continuum Kinematics with Incompatible-Compatible Decomposition

Abstract. We present a framework for the kinematics of a material body undergoing anelastic deformation. For such processes, the material structure of the body, as reflected by the geometric structure given to the set of body points, changes. The setting we propose may be relevant to phenomena such as plasticity, fracture, discontinuities, and non-injectivity of the deformations. In this framework, we construct an unambiguous decomposition into incompatible and compatible factors which includes the standard elastic-plastic decomposition in plasticity.

math-ph↗
arXiv2020

Optimization of Robot Grasping Forces and Worst Case Loading

We consider the optimization of the vector of grasping forces that support a known generalized force acting on the grasped object---a rigid body or a mechanism. Working in the framework of finite-dimensional normed vector spaces and their dual spaces, the cost function to be minimized is assumed to be a norm on the space of grasping forces. We present an expression for the optimum which depends on the external force and the kinematics of the grasping system. Next, assuming that optimal grasping forces are applied using force control, and assuming that there is a bound on the norm of the admissible grasping forces, we characterize the largest norm of an external force that the grasping system may support, that is, the norm of the worst-case loading that may be applied and still be supported. A few simple examples are given for the sake of illustration.

cs.RO↗
arXiv2019

Notes on Global Stress and Hyper-Stress Theories

The fundamental ideas and tools of the global geometric formulation of stress and hyper-stress theory of continuum mechanics are introduced. The proposed framework is the infinite dimensional counterpart of statics of systems having finite number of degrees of freedom, as viewed in the geometric approach to analytical mechanics. For continuum mechanics, the configuration space is the manifold of embeddings of a body manifold into the space manifold. Generalized velocity fields are viewed as elements of the tangent bundle of the configuration space and forces are continuous linear functionals defined on tangent vectors, elements of the cotangent bundle. It is shown, in particular, that a natural choice of topology on the configuration space, implies that force functionals may be represented by objects that generalize the stresses of traditional continuum mechanics.

math-ph↗
arXiv2018

De Donder Construction for Higher Jets

In this paper, we generalize De Donder approach to construct boundary forms that depend on the adapted coordinate system used. In continuum mechanics, use of boundary forms leads to splitting of the total force acting on the body into body force and surface traction. Moreover, this splitting is independent of the choice of the boundary form used. In calculus of variations, use of boundary forms leads to equations in exterior differential forms that are equivalent to the Euler-Lagrange equations. Infinitesimal symmetries of the theory lead to conservation laws valid for any choice of the boundary form used. In an example, we show that the boundary conditions lead to independence of constants of motion of the choice of the boundary form.

math-ph↗
arXiv2018

Hyper-Stresses in $k$-Jet Field Theories

For high-order continuum mechanics and classical field theories configurations are modeled as sections of general fiber bundles and generalized velocities are modeled as variations thereof. Smooth stress fields are considered and it is shown that three distinct mathematical stress objects play the roles of the traditional stress tensor of continuum mechanics in Euclidean spaces. These objects are referred to as the variational hyper-stress, the traction hyper-stress and the non-holonomic stress. The properties of these three stress objects and the relations between them are studied.

math-ph↗
arXiv2017

On jets, almost symmetric tensors, and traction hyper-stresses

The paper considers the formulation of higher-order continuum mechanics on differentiable manifolds devoid of any metric or parallelism structure. For generalized velocities modeled as sections of some vector bundle, a variational kth order hyper-stress is an object that acts on jets of generalized velocities to produce power densities. The traction hyper-stress is introduced as an object that induces hyper-traction fields on the boundaries of subbodies. Additional aspects of multilinear algebra relevant to the analysis of these objects are reviewed.

math-ph↗
arXiv2017

Geometric Analysis of Hyper-Stresses

A geometric analysis of high order stresses in continuum mechanics is presented. Virtual velocity fields take their values in a vector bundle \vbts over the n-dimensional space manifold. A stress field of order k is represented mathematically by an n-form valued in the dual of the vector bundle of k-jets of \vbts. While only limited analysis can be performed on high order stresses as such, they may be represented by non-holonomic hyper-stresses, n-forms valued in the duals of iterated jet bundles. For non-holonomic hyper-stresses, the analysis that applies to first order stresses may be iterated. In order to determine a unique value for the tangent surface stress field on the boundary of a body and the corresponding edge interactions, additional geometric structure should be specified, that of a vector field transversal to the boundary.

math-ph↗
arXiv2017

Stress theory for classical fields

Classical field theories together with the Lagrangian and Eulerian approaches to continuum mechanics are embraced under a geometric setting of a fiber bundle. The base manifold can be either the body manifold of continuum mechanics, space manifold, or space-time. Differentiable sections of the fiber bundle represent configurations of the system and the configuration space containing them is given the structure of an infinite dimensional manifold. Elements of the cotangent bundle of the configuration space are interpreted as generalized forces and a representation theorem implies that there exist a stress object representing forces, non-uniquely. The properties of stresses are studies as well as the role of constitutive relations in the present general setting.

math-ph↗
arXiv2016

Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies

This paper is concerned with the dynamics of continua on differentiable manifolds. We present a covariant derivation of equations of motion, viewing motion as a curve in an infinite-dimensional Banach space of embeddings of a body manifold in a space manifold. Our main application is the motion of residually-stressed elastic bodies; residual stress results from a geometric incompatibility between body and space manifolds. We then study a particular example of elastic vibrations of a two- dimensional curved annulus embedded in a sphere.

math-ph↗
arXiv2015

On the role of sharp chains in the transport theorem

A generalized transport theorem for convecting irregular domains is presented in the setting of Federer's geometric measure theory. A prototypical $r$-dimensional domain is viewed as a flat $r$-chain of finite mass in an open set of an $n$-dimensional Euclidean space. The evolution of such a generalized domain in time is assumed to be in accordance to a bi-Lipschitz type map. The induced curve is shown to be continuous with respect to the flat norm and differential with respect to the sharp norm on currents in $\mathbb{R}^{n}$. A time dependent property is naturally assigned to the evolving region via the action of an $r$-cochain on the current associated with the domain. Applying a representation theorem for cochains the properties are shown to be locally represented by an $r$-form. Using these notions a generalized transport theorem is presented.

math-ph↗
arXiv2014

Generalized Electrodynamics as a Special Case of Metric Independent Stress Theory

We use a metric invariant stress theory of continuum mechanics to formulate a simple generalization of the the basic variables of electrodynamics and Maxwell's equations to general differentiable manifolds of any dimension, thus viewing generalized electrodynamics as a special case of continuum mechanics. The basic variable is the potential, or a variation thereof, which is represented as an $r$-form in a $d$-dimensional spacetime. The stress for the case of generalized electrodynamics, is assumed to be represented by an $(d-r-1)$-form, a generalization of the Maxwell $2$-form.

math-ph↗