Arnold's seminar, Moscow University,
April 25, 2006

On higher analog of current helicity integral in MHD
P.M.Akhmetiev, IZMIRAN

A magnetic field $B$ in a perfect conductive medium satisfies the equation
$$
  \frac {d\chi^m}{dt} = 0, \quad \chi^m=\int (\A,\B) dx, \quad rot
  \A=\B,
$$
where $\chi^m$ is called magnetic helicity. In a medium with
non-vanishing small magnetic viscosity $\lambda$ the magnetic helicity
satisfies the equation
$$ 
  \frac{d\chi^m}{dt}=\lambda \chi^{c}, \quad \chi^c=\int (\B, rot \B) dx.
$$
The integral on the right hand side is called the current
(magnetic) helicity integral. This equation can be proved
analytically and admits a topological interpretation.

We shall discuss a higher analog (i.e. an analog related to higher
algebraic invariants of links and knots) of the currant helicity
integral. The new integral is defined by means of an integral formula for
the generalized Sato-Levine invariant, which was introduced by L.Traldi
(1988), M.Polyak and O.Ya.Viro (1995), P.Kirk and C.Livingston (1997),
D.Repovs and the author (1998), Nakanishi and Ohyama (2002)
independently.

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