Integral formula for a generalized Sato-Levine invariant in magnetic hydrodynamics. (English. Russian original) Zbl 1190.53071

Math. Notes 85, No. 4, 503-514 (2009); translation from Mat. Zametki 85, No. 4, 524-537 (2009).
The authors investigate the problem of finding a higher analog of the helicity integral for divergence-free vector fields. They propose the construction of a new fourth-order integral \(\mathbf{W}\) which is defined for the magnetic field represented by two tubes, and extend a previous integral defined for two tubes with zero linking number [J. Geom. Phys. 15, No. 2, 95–101 (1995; Zbl 0836.57005)].
It is shown that \(\mathbf{W}\) is preserved under the motion of tubes in an ideal medium.
Also, the topological meaning of \(\mathbf{W}\) is discussed and compared with the generalized Sato-Levine invariant.


53C65 Integral geometry
76W05 Magnetohydrodynamics and electrohydrodynamics


Zbl 0836.57005
Full Text: DOI


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