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A geometric perspective on the inextensible flows and energy of curves in 4-dimensional pseudo-Galilean space

Fatma Almaz, Handan Oztekin

TL;DR

This work develops a geometric-energy framework for inextensible curve flows in the 4-dimensional pseudo-Galilean space $G_{1}^{4}$. It establishes directional derivatives and extended Serret-Frenet relations for the Frenet-Serret frame $(T,N,B_{1},B_{2})$ in $G_{1}^{4}$ and derives necessary and sufficient PDEs governing inextensible flows, linking the curvatures $(\kappa,\tau,\sigma)$ to the flow components $f_i$. The paper also computes explicit energy functionals for unit tangent vectors along both $s$-lines and $t$-lines using the Sasaki metric, including pseudo-angle considerations between Frenet vectors, thereby quantifying the bending energy of a moving curve in this geometry. These results provide a rigorous geometric and energetic description of elastic-like curve dynamics in $G_{1}^{4}$ with potential applications to physical models in pseudo-Galilean contexts.

Abstract

In this study, inextensible flows of curves in four-dimensional pseudo-Galilean space are expressed, and the necessary and sufficient conditions of these curve flows are given as partial differential equations. Also, the directional derivatives are defined in accordance with the Serret-Frenet frame in G41, the extended Serret-Frenet relations are expressed by using Frenet formulas in G41. Furthermore, the bending elastic energy functions are expressed for the same particle according to curve a(s,t).

A geometric perspective on the inextensible flows and energy of curves in 4-dimensional pseudo-Galilean space

TL;DR

This work develops a geometric-energy framework for inextensible curve flows in the 4-dimensional pseudo-Galilean space . It establishes directional derivatives and extended Serret-Frenet relations for the Frenet-Serret frame in and derives necessary and sufficient PDEs governing inextensible flows, linking the curvatures to the flow components . The paper also computes explicit energy functionals for unit tangent vectors along both -lines and -lines using the Sasaki metric, including pseudo-angle considerations between Frenet vectors, thereby quantifying the bending energy of a moving curve in this geometry. These results provide a rigorous geometric and energetic description of elastic-like curve dynamics in with potential applications to physical models in pseudo-Galilean contexts.

Abstract

In this study, inextensible flows of curves in four-dimensional pseudo-Galilean space are expressed, and the necessary and sufficient conditions of these curve flows are given as partial differential equations. Also, the directional derivatives are defined in accordance with the Serret-Frenet frame in G41, the extended Serret-Frenet relations are expressed by using Frenet formulas in G41. Furthermore, the bending elastic energy functions are expressed for the same particle according to curve a(s,t).

Paper Structure

This paper contains 6 sections, 10 theorems, 117 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some characterizations of inextensible flows curves in $G_{1}^{4}$
  4. The Energy and Pseudo-angle of vector fields in $G_{1}^{4}$
  5. The energy of unit tangent vector of $s-$lines on a moving particle in $G_{1}^{4}$
  6. The energy of unit tangent vector of $t-$lines on a moving particle in $G_{1}^{4}$

Key Result

Theorem 1

The flow of $\Omega (u,t)$ in $G_{1}^{4}$ is inextensible if and only if $f_{1}$ is constant.

Figures (3)

  • Figure 1: Extended Serret-Frenet functions and the rotational surfaces generated by extended Serret-Frenet functions of $t-$lines of inextensible curve $\alpha .$
  • Figure 2: Energy functions and rotational surfaces generated by energy functions of inextensible curve $\alpha .$
  • Figure 3: Pseudo angle functions and rotational surfaces generated by pseudo angle functions of inextensible curve $\alpha .$

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 13 more