Related papers: Sliding vectors, line bivectors, and torque
Whether a string has rotation and shear can be investigated by an anology with the point particle. Rotation and shear involve first covariant spacetime derivatives of a vector field and, because the metric stress tensor for both the point…
We consider the set of pairs of orthogonal vectors in Hilbert space, which is also called the cross because it is the union of the horizontal and vertical axes in the Euclidean plane when the underlying space is the real line. Crosses,…
This is the sixth, concluding part of a series of papers the first five of which have been submitted to the present archive in mid 1998 and published as INR preprints in 1999. The present paper was printed as an INR preprint, too, but for…
This paper extends sliding-mode control theory to nonlinear systems evolving on smooth manifolds. Building on differential geometric methods, we reformulate Filippov's notion of solutions, characterize well-defined vector fields on quotient…
A vector space over a field $\mathbb{F}$ is a set $V$ together with two binary operations, called vector addition and scalar multiplication. It is standard practice to think of a Euclidean space $\mathbb{R}^n$ as an $n$-dimensional real…
At present, whenever we work in newtonian mechanics we consider momentum to be a three-dimensional vector or a 4-dimensional one when we work in relativistic mechanics. However, this mathematical vector model has barely 200 years and its…
Without the use of cameras to record 2D motion and an appropriate analysis tool, creating a laboratory activity for students to experience the vector nature of momentum can be challenging. Even with appropriate measurement tools, it is…
The displacement and deviation vectors in spaces (manifolds), the tangent bundle of which is endowed with a transport along paths, are introduced. In case these spaces are equipped with a linear connection, the deviation equations (between…
We introduce a linear algebraic object called a bidiagonal triad. A bidiagonal triad is a modification of the previously studied and similarly defined concept of bidiagonal triple. A bidiagonal triad and a bidiagonal triple both consist of…
We consider an inverse variational problem for the lines of constant curvature in (pseudo-)Euclidean two-, three-, and four-dimensional spaces. The accumulated results are physically meaningful in the case of relativistic mechanics of…
In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line $x=0$. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector…
In these lectures the relations between symmetries, Lie algebras, Killing vectors and Noether's theorem are reviewed. A generalisation of the basic ideas to include velocity-dependend co-ordinate transformations naturally leads to the…
The concept of a physical space, which actualizes Euclidean geometry, is not confined to the statics of solids but extensible to the phenomena where Newtonian mechanics is valid, defining its concept of time. The laws of propagation of…
We study some field representations of vector supersymmetry with superspin Y=0 and Y=1/2 and nonvanishing central charges. For Y=0, we present two multiplets composed of four spinor fields, two even and two odd, and we provide a free action…
The theory of screws clarifies many analogies between apparently unrelated notions in mechanics, including the duality between forces and angular velocities. It is known that the real 6-dimensional space of screws can be endowed with an…
Gauge-invariant twistor variables are found for the massive spinning particle with N-extended local worldline supersymmetry, in spacetime dimensions D=3,4,6. The twistor action is manifestly Lorentz invariant but the anticommuting spin…
In this paper we consider a non-smooth vector field $Z=(X,Y)$, where $X,Y$ are linear vector fields in dimension 3 and the discontinuity manifold $\Sigma$ of $Z$ is or the usual embedded torus or the unitary sphere at origin. We suppose…
Torsion appears in literature in quite different forms. Generally, spin is considered to be the source of torsion, but there are several other possibilities in which torsion emerges in different contexts. In some cases a phenomenological…
The main objective of this article is part of a research program to link the dynamics of fluid flows with the structure and its transitions in the physical spaces. As a prototype of problem and to demonstrate the main ideas, we study the…
The standard formulation of a massive Abelian vector field in $2+1$ dimensions involves a Maxwell kinetic term plus a Chern-Simons mass term; in its place we consider a Chern-Simons kinetic term plus a Stuekelberg mass term. In this latter…