Periodic \(J\)-trajectories on \(\mathbb{R} \times \mathbb{S}^3\)
type: Item
instance of: scholarly article
title: Periodic \(J\)-trajectories on \(\mathbb{R} \times \mathbb{S}^3\)
zbMATH Open document ID: 1401.53013
author: Osman Ateş
author: Marian Ioan Munteanu
published in: Journal of Geometry and Physics
publication date: 2018-10-08T00:00:00Z
review text: If \((M,g)\) is a Riemannian manifold, then the magnetic curves on \(M\) represent, in Physics, trajectories of charged particles moving under the action of a magnetic field. A magnetic field on \(M\) is defined by a closed 2-form \(\Omega\), and equivalently, a skew-symmetric \((1,1)\)-tensor field \(\varphi\), called the Lorentz force on \(M\), can be defined by \(g(\varphi X,Y)=\Omega(X,Y)\). A magnetic trajectory of this dynamical system \((M,g,\varphi)\) is a solution of the Lorentz equation \((*)\) \(\nabla_{\dot\gamma}\dot\gamma=q\,\varphi\dot\gamma\), where \(q\neq 0\) is known as the strength of the magnetic field. If \(\{T=\dot\gamma,\nu_1,\dots,\nu_r\}\) is an orthonormal frame of rank \(r\) along \(\gamma\) such that \(\nabla_TT=\kappa_1\nu_1\), \(\nabla_T\nu_1=-\kappa_1T+\kappa_2\nu_2\), \(\nabla_T\nu_j=-\kappa_{j}\nu_{j-1}+\kappa_{j+1}\nu_{j+1}\) for \(j=2,\dots,r-2\), \(\nabla_T\nu_{r-1}=-\kappa_{r-1}\nu_{r-2}\), where \(\kappa_1\), \(\kappa_2\),\(\dots\),\(\kappa_{r-1}\) are positive \(C^\infty\) functions on the manifold, then \(\gamma\) is a Frenet curve of osculating order \(r\). Moreover, \(\kappa_j\) is called the \(j\)th curvature of \(\gamma\). A geodesic is a Frenet curve of osculating order 1, a Riemannian circle is a Frenet curve of osculating order 2 with constant curvature \(\kappa_1\), and a helix of order \(r\) is defined as a Frenet curve of osculating order \(r\) such that all the curvatures \(\kappa_j\) are constant. If a \((2n+1)\)-dimensional almost contact metric manifold \((\overline M,\varphi,\xi,\eta,\overline g)\) is normal, that is its Nijenhuis tensor \(N_\varphi\) vanishes, then \(M\) is called a Sasakian manifold. If \(\gamma(s)\) is a curve on \(M\) parametrized by an arc-length, then the contact angle \(\theta\) of \(\gamma\) is a function defined as the angle between the tangent vector field and the characteristic vector field \(\xi\), that is, \(\cos\theta=\overline g(\dot\gamma,\xi_{|\gamma(s)})\). In particular, if t…
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 53C80
zbMATH DE Number: 6948935
zbMATH Keywords: \(F\)-planar curve
zbMATH Keywords: magnetic curve
zbMATH Keywords: periodic curve
zbMATH Keywords: Sasakian manifold
MaRDI profile type: Publication
full work available at URL: j.geomphys.2018.07.002
OpenAlex ID: W2884771134
cites work: Q3594209
cites work: F-geodesics on manifolds
cites work: Q4957157
cites work: Trajectories of charged particles in Kähler magnetic fields
cites work: Q5830177
cites work: Q3416926
cites work: Kähler magnetic fields on a complex projective space
cites work: The contact magnetic flow in 3D Sasakian manifolds
cites work: Q4655022
cites work: Periodic magnetic curves in Berger spheres
cites work: Magnetic vortex filament flows
cites work: Contact manifolds in Riemannian geometry
cites work: Riemannian geometry of contact and symplectic manifolds
cites work: Generic submanifolds of Sasakian manifolds
cites work: A note on magnetic curves on image
cites work: Magnetic curves in Sasakian manifolds
cites work: Magnetic curves in cosymplectic manifolds
cites work: Periodic solutions of quaternionic-valued ordinary differential equations
cites work: Solving simple quaternionic differential equations
cites work: Linear Quaternion Differential Equations: Basic Theory and Fundamental Results
DOI: 10.1016/J.GEOMPHYS.2018.07.002
description: scientific article; zbMATH DE number 6948935