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Integrability of homogeneous exact magnetic flows on spheres

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arxiv 2504.20515 v2 pith:DD2LAYXI submitted 2025-04-29 math.DG nlin.SI

Vladimir Dragovic , Borislav Gajic , Bozidar Jovanovic This is my paper · ORCID
classification math.DGnlin.SI
keywords magneticintegrabilityrestrictedflowsmotionsystemshomogeneousmathbb
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abstract

We consider motion of a material point placed in a constant homogeneous magnetic field in $\mathbb R^n$ and also motion restricted to the sphere $S^{n-1}$. While there is an obvious integrability of the magnetic system in $\mathbb R^n$, the integrability of the system restricted to the sphere $S^{n-1}$ is highly non-trivial. We prove complete integrability of the obtained restricted magnetic systems for $n\le 6$. The first integrals of motion of the magnetic flows on the spheres $S^{n-1}$, 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 $n\ge 7$ when the systems allow a reduction to the cases with $n\le 6$. We conjecture that the restricted magnetic systems on $S^{n-1}$ are integrable for all $n$.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Integrable systems from Poisson reductions of generalized Hamiltonian torus actions

    math-ph 2025-07 unverdicted novelty 7.0 of 10

    Develops sufficient conditions for Poisson reduction of generalized Hamiltonian torus actions to preserve integrability and applies them to open problems on Lie group doubles and flat-connection moduli spaces.

  2. A Lax representation and integrability of homogeneous exact magnetic flows on spheres in all dimensions

    nlin.SI 2025-06 conditional novelty 7.0 of 10

    The paper proves complete integrability of homogeneous exact magnetic flows on spheres in all dimensions via a Lax representation.

  3. Heavy rigid body with a gyroscope in $\mathbb R^n$

    math-ph 2026-01 accept novelty 6.0 of 10

    Multidimensional Lagrange, Euler, and totally symmetric heavy tops remain Liouville integrable after adding a gyroscope with angular momentum in the symmetry subalgebra, with new polynomial Lax representations.

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