Motions of Curves in the Pseudo-Galilean SpaceG31

Suleyman Cengiz,Ufuk Ozturk,Esra Betul Koc Ozturk

doi:10.1155/2015/150685

Suleyman Cengiz, Ufuk Ozturk + Show 1 more

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https://doi.org/10.1155/2015/150685

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Abstract

We study the flows of curves in the pseudo-Galilean 3-space and its equiform geometry without any constraints. We find that the Frenet equations and intrinsic quantities of the inelastic flows of curves are independent of time. We show that the motions of curves in the pseudo-Galilean 3-space and its equiform geometry are described by the inviscid and viscous Burgers’ equations.

Highlights

In mathematical modeling of many nonlinear events of the natural and the applied sciences such as dynamics of vortex filaments, motions of interfaces, shape control of robot arms, propagation of flame fronts, image processing, supercoiled DNAs, magnetic fluxes, deformation of membranes, and dynamics of proteins, the motions of space curves are being used
The motions of curves have been widely investigated by many authors in different geometries
In 1992 Nakayama and others explained that the close relation between the integrable evolution equations and the motions of curves is based on the equivalence of Frenet equations and the inverse scattering problem at zero eigenvalue [1], so that they identified the evolution equations that govern the 2D and 3D motions of the curves

Summary

Introduction

In mathematical modeling of many nonlinear events of the natural and the applied sciences such as dynamics of vortex filaments, motions of interfaces, shape control of robot arms, propagation of flame fronts, image processing, supercoiled DNAs, magnetic fluxes, deformation of membranes, and dynamics of proteins, the motions of space curves are being used. The evolutions of these nonlinear phenomena are described by the differential equations which characterize the motions of curves as a family. We study the curve evolution in the equiform geometry of the pseudo-Galilean 3-space regarding the relations between the Frenet vectors of these spaces

The Pseudo-Galilean Space G13

Motions of Curves in the Pseudo-Galilean Space G13

Motions of Curves in the Equiform Geometry of G13