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Magnetic Killing tensors and first integrals of the magnetic flow

Andrei Moroianu, Gabriela Ovando

Abstract

In this work we introduce a new family of symmetric tensors generalizing Killing tensors, that we call magnetic Killing symmetric tensors. We make use of them to construct first integrals for the magnetic flow associated to a given magnetic field. We apply the results to prove integrability of some invariant magnetic flows (either exact or non-exact) on some 2-step nilmanifolds: the Kodaira-Thurston manifold and Heisenberg nilmanifolds of higher dimensions.

Magnetic Killing tensors and first integrals of the magnetic flow

Abstract

In this work we introduce a new family of symmetric tensors generalizing Killing tensors, that we call magnetic Killing symmetric tensors. We make use of them to construct first integrals for the magnetic flow associated to a given magnetic field. We apply the results to prove integrability of some invariant magnetic flows (either exact or non-exact) on some 2-step nilmanifolds: the Kodaira-Thurston manifold and Heisenberg nilmanifolds of higher dimensions.
Paper Structure (7 sections, 15 theorems, 91 equations)

This paper contains 7 sections, 15 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Magnetic Killing tensors
  3. The magnetic flow on Lie groups
  4. Examples of Liouville Integrable systems on 2-step nilmanifolds
  5. Complete integrability on compact spaces
  6. Magnetic flows on 4-dimensional nilmanifolds
  7. Magnetic flows on Heisenberg nilmanifolds of any dimension

Key Result

Lemma 2.1

For every $K\in \Gamma(\operatorname{Sym}^l\operatorname{T} M)$ and for every $X\in \chi(M)$, the following relation holds:

Theorems & Definitions (41)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Proposition 2.7
  • proof
  • ...and 31 more