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Homogeneity of magnetic trajectories in the Berger sphere

Jun-ichi Inoguchi, Marian Ioan Munteanu

TL;DR

This work addresses the homogeneity of contact magnetic trajectories on Berger spheres $\mathscr{M}^3(c)$, a naturally reductive Sasakian family with constant holomorphic sectional curvature $c>-3$. By formulating the dynamics via the magnetized Euler-Arnold framework on the naturally reductive model, the authors derive an explicit representation of magnetic trajectories. They prove that every contact magnetic trajectory is homogeneous and can be written as a product of a homogeneous geodesic and a charged Reeb flow: $\gamma(t)=\exp_{G\times K}(t\widehat X)\exp_{K}\left( t\frac{q}{2}\xi\right)$ for $c\neq 1$ (with a variant for $c=1$). This result preserves the integrability and symmetry of geodesic flows under the standard contact magnetic perturbation and links magnetic dynamics to rigid-body motion via the Euler-Arnold equations.

Abstract

We study the homogeneity of contact magnetic trajectories in naturally reductive Berger spheres. We prove that every contact magnetic trajectory is a product of a homogeneous geodesic and a charged Reeb flow.

Homogeneity of magnetic trajectories in the Berger sphere

TL;DR

This work addresses the homogeneity of contact magnetic trajectories on Berger spheres , a naturally reductive Sasakian family with constant holomorphic sectional curvature . By formulating the dynamics via the magnetized Euler-Arnold framework on the naturally reductive model, the authors derive an explicit representation of magnetic trajectories. They prove that every contact magnetic trajectory is homogeneous and can be written as a product of a homogeneous geodesic and a charged Reeb flow: for (with a variant for ). This result preserves the integrability and symmetry of geodesic flows under the standard contact magnetic perturbation and links magnetic dynamics to rigid-body motion via the Euler-Arnold equations.

Abstract

We study the homogeneity of contact magnetic trajectories in naturally reductive Berger spheres. We prove that every contact magnetic trajectory is a product of a homogeneous geodesic and a charged Reeb flow.
Paper Structure (27 sections, 11 theorems, 233 equations)

This paper contains 27 sections, 11 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homogeneous geometry
  4. The Euler-Arnold equation
  5. Static magnetism
  6. Contact magnetic field
  7. Magnetic fields on homogeneous Riemannian spaces
  8. The Berger 3-sphere
  9. Contact magnetic field of $\mathbb{S}^3$
  10. The unit tangent sphere bundle $\mathrm{U}\mathbb{S}^3$
  11. The Berger $3$-sphere
  12. The Levi-Civita connection
  13. Homogeneous structure
  14. The Euler-Arnold equation
  15. Some remarks on geodesics
  16. ...and 12 more sections

Key Result

Proposition 2.1

Every geodesic $\gamma(s)$ of a naturally reductive homogeneous space $L/H$ with naturally reductive decomposition $\mathfrak{l}= \mathfrak{h}+\mathfrak{p}$ starting at the origin $o$ with initial velocity $X\in \mathfrak{p}$ is homogeneous, that is, it is expressed as where $\exp_{L}:\mathfrak{l}\to L$ is the exponential map.

Theorems & Definitions (39)

  • Proposition 2.1
  • Proposition 2.2
  • Example 2.1: Rigid bodies
  • Remark 1
  • Proposition 2.3
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 29 more