2025/04/29 by Vladimir Dragović, Dragovic, Vladimir, Borislav Gajić +3 · 3 citations
Mathematics · Physics and Astronomy · #37J35 #53D25 #Advanced Differential Equations and Dynamical Systems #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2504.20515
openalex publication_date 2025/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider motion of a material point placed in a constant homogeneous magnetic field in \mathbb Rn and also motion restricted to the sphere Sn-1. While there is an obvious integrability of the magnetic system in \mathbb Rn, the integrability of the system restricted to the sphere Sn-1 is highly non-trivial. We prove complete integrability of the obtained restricted magnetic systems for n≤ 6. The first integrals of motion of the magnetic flows on the spheres Sn-1, for n=5 and n=6, are polynomials of the degree 1, 2, and 3 in momenta. We prove noncommutative integrability of the obtained magnetic flows for any n≥ 7 when the systems allow a reduction to the cases with n≤ 6. We conjecture that the restricted magnetic systems on Sn-1 are integrable for all n.