2020/03/12 by Boris Khesin, Daniel Peralta‐Salas, Khesin, Boris +3 · 1 citation
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2003.06008
openalex publication_date 2020/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that the helicity is the only regular Casimir function for the coadjoint action of the volume-preserving diffeomorphism group SDiff(M) on smooth exact divergence-free vector fields on a closed three-dimensional manifold M. More precisely, any regular C1 functional defined on the space of C^∞ (more generally, Ck, k≥ 4) exact divergence-free vector fields and invariant under arbitrary volume-preserving diffeomorphisms can be expressed as a C1 function of the helicity. This gives a complete description of Casimirs for adjoint and coadjoint actions of SDiff(M) in 3D and completes the proof of Arnold-Khesin's 1998 conjecture for a manifold M with trivial first homology group. Our proofs make use of different tools from the theory of dynamical systems, including normal forms for divergence-free vector fields, the Poincaré-Birkhoff theorem, and a division lemma for vector fields with hyperbolic zeros.