ИСТИНА

By an ideal magnetic field in a magnetic tube we mean a closed 2-form B in the full torus V = D^2 x S^1 whose restriction to the boundary torus dV = S^1 x S^1 vanishes, while the restriction to any fibre D^2 x {*} has no zeros, moreover the integral of B over the fibre equals 1. By a topological invariant of magnetic fields in V we mean a real-valued function I=I(B) on the space of such 2-forms such that I(B)=I(h^*B) for any diffeomorphism h of V isotopic to the identity. An important example of topological invariants is helicity, which is the averaged linking number of magnetic lines. By using [1], we prove that a topological invariant I=I(B) is either a function in helicity or has no derivative with C^1-continuous density [2]. Similar properties of the helicity and the closely related Calabi invariant were proved in [3], [4]. Literature: [1] C. Bonatti and S. Crovisier, R\'ecurrence et g\'en\'ericit\'e. Invent.Math. 158 (1), 33 (2004). [2] E. A. Kudryavtseva, Conjugation Invariants on the Group of Area-Preserving Diffeomorphisms of the Disk, Math. Notes, 95:6 (2014), 877–880. [3] D. Serre, Les invariants du premier ordre de l'\'equation d'Euler en dimension trois. Phys. D 13 (1-2), 105 (1984). [4] A. Banyaga, Sur la structure du groupe des diff\'eomorphismes qui pr\'eservent une forme symplectique. Comment.Math. Helv. 53 (2), 174 (1978).