Circular rigidity of totally integrable strong-field magnetic billiards
Let $D \subset \mathbb{R}^2$ be a bounded domain with $C^\infty$ boundary of strictly positive curvature $\kappa$, and let $B > \max_{\partial D} \kappa$. Between collisions with the boundary a particle of unit speed obeys $\ddot x = B J \dot x$, where $J(a,b) = (-b,a)$, so that it moves along arcs of circles of radius $1/B$ (Larmor circles); at a collision it is reflected across the tangent line of the boundary. Let $T$ be the collision map on the annulus of inward-pointing unit velocities at boundary points, grazing velocities excluded. Larmor circles inside $D$ that never meet the boundary are not part of this annulus.
For the disk, the distance from its centre to the centre of the Larmor circle is a first integral. Bialy proved that when $0 \le B < \min \kappa$, a magnetic billiard whose phase cylinder has an invariant closed non-contractible curve through every point is circular (2012). When the Larmor radius is less than half the maximal radius of a tubular neighbourhood of $\partial D$, which forces $B > 2 \max \kappa$, Bialy, Mironov and Shalom proved that for a non-circular domain the billiard has no nonconstant first integral polynomial in the velocities, for all but finitely many values of $B$ (2020).
Problem. Suppose the annulus admits a foliation by continuous $T$-invariant simple closed curves, each winding once around it. Must $D$ be a disk?
Source: Bialy (2026), Question 1 of his contribution to Albach et al., Open problems in billiards and quantitative symplectic geometry. Stated by AIM — Open Applied Problems for B > max κ, with total integrability as a foliation of the phase annulus by invariant curves.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- SG
- Keywords
- AIM Open Applied Problems, integrable billiards, magnetic billiards
References
@misc{albach2026,
author = {Albach, Bernhard and Barraud, Jean-Fran{\c{c}}ois and Bialy, Misha and Bimmermann, Johanna and Ch{\'a}vez C{\'a}liz, Ana and Damian, Mihai and Deschamps, Lina and Hryniewicz, Umberto and Humili{\`e}re, Vincent and Khesin, Boris and Maier, Levin and Moreno, Agustin and Oancea, Alexandru and Paris-Romaskevich, Olga and Sorrentino, Alfonso and Tabachnikov, Serge},
title = {Open problems in billiards and quantitative symplectic geometry},
year = {2026},
eprint = {2602.12896},
archivePrefix = {arXiv},
}
@article{bialy2012,
author = {Bialy, Misha},
title = {On totally integrable magnetic billiards on constant curvature surface},
journal = {Electron. Res. Announc. Math. Sci.},
volume = {19},
year = {2012},
pages = {112--119},
doi = {10.3934/era.2012.19.112},
eprint = {1208.2455},
archivePrefix = {arXiv},
}
@article{bialy2020,
author = {Bialy, Misha and Mironov, Andrey E. and Shalom, Lior},
title = {Magnetic billiards: non-integrability for strong magnetic field; {G}utkin type examples},
journal = {J. Geom. Phys.},
volume = {154},
year = {2020},
pages = {103716},
doi = {10.1016/j.geomphys.2020.103716},
eprint = {2001.02119},
archivePrefix = {arXiv},
}
@misc{aim-strong-magnetic-billiard-rigidity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Circular rigidity of totally integrable strong-field magnetic billiards}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/253-strong-magnetic-billiard-rigidity.md},
note = {Entry 253 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00564, "Circular rigidity of totally integrable strong-field magnetic billiards", clopen.solutions.