Polynomial rigidity of strong-field magnetic billiards
Abstract
Let Ω⊂ℝ2 be a bounded domain whose boundary γ is a smooth simple closed curve, and let r=|B|−1 be the Larmor radius of a constant magnetic field. Assume 0<r<r0(γ)/2, where r0(γ) is the maximal embedded tubular radius of γ. We prove that if the corresponding magnetic billiard admits a nonconstant first integral polynomial in the velocity, then γ is a circle and Ω is a disk. No convexity assumption is imposed. This eliminates the finite set of possible exceptional strong-field values left by the earlier polynomial nonintegrability theory. The proof uses the two highest reflection modes of the polynomial induced on the space of Larmor centers. A boundary index identity shows that the highest coefficient has degree equal to the true Fourier degree and that all of its roots lie inside the table. After complexification and passage to the normalization of the boundary curve, comparison of the two leading modes at infinity excludes simultaneous poles of the coordinate functions and forces all of these roots to coincide. The reflection identity then implies that the boundary has constant distance from their common root and hence is circular.
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BibTeX
@misc{bhandari2026polynomial,
title = {{Polynomial rigidity of strong-field magnetic billiards}},
author = {Dipesh Bhandari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.18937},
eprint = {2609.18937},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18937},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Polynomial rigidity of strong-field magnetic billiards
AU - Dipesh Bhandari
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.18937
UR - https://arxiv.org/abs/2609.18937
AB - Let Ω⊂ℝ^2 be a bounded domain whose boundary γ is a smooth simple closed curve, and let r=|B|^−1 be the Larmor radius of a constant magnetic field. Assume 0<r<r_0(γ)/2, where r_0(γ) is the maximal embedded tubular radius of γ. We prove that if the corresponding magnetic billiard admits a nonconstant first integral polynomial in the velocity, then γ is a circle and Ω is a disk. No convexity assumption is imposed. This eliminates the finite set of possible exceptional strong-field values left by the earlier polynomial nonintegrability theory. The proof uses the two highest reflection modes of the polynomial induced on the space of Larmor centers. A boundary index identity shows that the highest coefficient has degree equal to the true Fourier degree and that all of its roots lie inside the table. After complexification and passage to the normalization of the boundary curve, comparison of the two leading modes at infinity excludes simultaneous poles of the coordinate functions and forces all of these roots to coincide. The reflection identity then implies that the boundary has constant distance from their common root and hence is circular.
ER -