The Characterization of Affine Symplectic Curves in ℝ4

Mehmet Bektaş,Esra Çiçek Çetin

doi:10.3390/math7010110

Abstract

Symplectic geometry arises as the natural geometry of phase-space in the equations of classical mechanics. In this study, we obtain new characterizations of regular symplectic curves with respect to the Frenet frame in four-dimensional symplectic space. We also give the characterizations of the symplectic circular helices as the third- and fourth-order differential equations involving the symplectic curvatures.

Highlights

As the Riemannian geometry involves the length as the fundamental quantity, symplectic geometry involves the directed area, and contact geometry involves the twisting behavior as the fundamental quantities
It arises as the natural geometry of phase-space in the equations of classical mechanics, which are called Hamilton’s equations, and treating mechanical problems in phase-space greatly simplifies the problem [1]
The aim of this paper is to study some characterization for a special class of symplectic curves called affine symplectic helices, which are a very important tool for both physics and geometric optics

Summary

Introduction

As the Riemannian geometry involves the length as the fundamental quantity, symplectic geometry involves the directed area, and contact geometry involves the twisting behavior as the fundamental quantities. Studying the twisting behavior in symplectic geometry helps us to obtain connections between these two geometries. The even-dimensional symplectic geometry has been found in numerous areas of mathematics and physics. It arises as the natural geometry of phase-space in the equations of classical mechanics, which are called Hamilton’s equations, and treating mechanical problems in phase-space greatly simplifies the problem [1]. The aim of this paper is to study some characterization for a special class of symplectic curves called affine symplectic helices, which are a very important tool for both physics and geometric optics.

Preliminaries

General Properties of Affine Symplectic Curves in R4

The Characterizations of Symplectic Curves in R4

Conclusions