Integrability of Homogeneous Exact Magnetic Flows on Spheres

• Published in: Regular & chaotic dynamics Vol. 30; no. 4; pp. 582 - 597
• Main Authors: Dragović, Vladimir, Gajić, Borislav, Jovanović, Božidar
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.08.2025; Springer Nature B.V
• Subjects: Dynamical Systems and Ergodic Theory; Mathematics; Mathematics and Statistics; Polynomials; magnetic geodesic flows; gauge Noether symmetries; Liouville integrability; noncommutative integrability; Dirac magnetic Poisson bracket
• ISSN: 1560-3547; 1468-4845
• DOI: 10.1134/S1560354725040082

We consider motion of a material point placed in a constant homogeneous magnetic field in and also motion restricted to the sphere . While there is an obvious integrability of the magnetic system in , the integrability of the system restricted to the sphere is highly nontrivial. We prove complete integrability of the obtained restricted magnetic systems for . The first integrals of motion of the magnetic flows on the spheres , for and , are polynomials of degree , , and in momenta. We prove noncommutative integrability of the obtained magnetic flows for any when the systems allow a reduction to the cases with . We conjecture that the restricted magnetic systems on are integrable for all .