Some remarks on high degree polynomial integrals of the magnetic geodesic flow on the two-dimensional torus
type: Item
instance of: scholarly article
title: Some remarks on high degree polynomial integrals of the magnetic geodesic flow on the two-dimensional torus
zbMATH Open document ID: 1483.37071
DOI: 10.1134/S0037446621040017
author: V. V. Shubin
author: Sergey Agapov
author: Alexandr Valyuzhenich
published in: Siberian Mathematical Journal
publication date: 2021-09-27T00:00:00Z
review text: This paper aims at fixing a gap in the proof of the main result from [the first two authors, Discrete Contin. Dyn. Syst. 39, No. 11, 6565--6583 (2019; Zbl 1432.37082)]. Consider conformal coordinates \((x,y)\) on the \(2\)-torus with metric \(ds^2=\Lambda(x,y)(dx^2+dy^2)\), and let \[H=\frac{p_x^2+p_y^2}{2 \Lambda(x,y)}.\] The magnetic gedesic flow is the Hamiltonian flow of \(H\) with respect to the magnetic Poisson bracket \[\{-,-\}_{\text{mg}} = \frac{\partial}{\partial x}\wedge \frac{\partial}{\partial p_x}+ \frac{\partial}{\partial y} \wedge \frac{\partial}{\partial p_y} + \Omega(x,y) \frac{\partial}{\partial p_x} \wedge \frac{\partial}{\partial p_y}\,,\] where \(\Omega(x,y)\, dx\wedge dy\) is a closed \(2\)-form defining a fixed nonzero magnetic field. The main result of [loc. cit.] states that if the magnetic geodesic flow has a first integral \(F\) which is polynomial of degree \(N\) in the momenta \((p_x,p_y)\) and which has periodic analytic coefficients at \(N+2\) energy levels \(\{H=E_j\}\), \(E_1,\ldots, E_{N+2}\in \mathbb{R}\) pairwise distinct, then \(\Omega\) and \(\Lambda\) are functions of a unique variable. Furthermore \(H\) admits a first integral \(F_1\) which is linear in momenta. In Section 2 the authors outline the main steps of the proof given in [loc. cit.] and identify an argument which can fail. It is explained in Section 3 how to fix this issue without altering the statement of the important result that was presented.
reviewed by: Maxime Fairon
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37J39
Mathematics Subject Classification ID: 37E35
Mathematics Subject Classification ID: 53D25
zbMATH DE Number: 7401155
zbMATH Keywords: magnetic geodesic flow
zbMATH Keywords: polynomial first integral
Wikidata QID: Q114075219
MaRDI profile type: Publication
cites work: Polynomial integrals of magnetic geodesic flows on the 2-torus on several energy levels
cites work: On first integrals of geodesic flows on a two-torus
cites work: GEODESIC FLOWS ON TWO-DIMENSIONAL MANIFOLDS WITH AN ADDITIONAL FIRST INTEGRAL THAT IS POLYNOMIAL IN THE VELOCITIES
cites work: On first integrals of two-dimensional geodesic flows
cites work: Integrable Hamiltonian systems with velocity-dependent potentials
cites work: Integrable magnetic geodesic flows on 2-torus: new examples via quasi-linear system of PDEs
full work available at URL: s0037446621040017
OpenAlex ID: W3198188101
Recommended article: Polynomial integrals of magnetic geodesic flows on the 2-torus on several energy levels
Recommended article: On the integrable magnetic geodesic flow on a 2-torus
Recommended article: New semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces
Recommended article: Integrable magnetic geodesic flows on 2-torus: new examples via quasi-linear system of PDEs
Recommended article: On first integrals of two-dimensional geodesic flows
Recommended article: Magnetic geodesic flows on coadjoint orbits
Recommended article: On an integrable magnetic geodesic flow on the two-torus
Recommended article: Integrable magnetic geodesic flows on 2-surfaces *
Recommended article: On the periodic motions of a charged particle in an oscillating magnetic field on the two-torus
Recommended article: Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus
description: scientific article; zbMATH DE number 7401155