On Arnold's problem on higher analogs of the asymptotic Hopf invariant
Let \({\mathbf B}\) be a magnetic field defined in the domain \(U \subset \mathbb{R}^3\) with no zeros inside \(U\) and tangent to \(\partial U\), where \(U\) is called a magnetic tube. Let \(L_1\), \(L_2\) be magnetic lines of \({\mathbf B}\) which are parametrized by the times \(T_1\), \(T_2\) and issued from the points \(x_1\), \(x_2\), respectively. Then the linking number of these two lines is given by \[ \lambda_{\mathbf B} (T_1, T_2; x_1, x_2)=\frac{1}{4\pi T_1T_2} \int_0^{T_1}\int_0^{T_2}\frac{\langle \dot{x_1}, \dot{x_2}, x_1-x_2\rangle} {\parallel x_1-x_2\parallel ^3}dt_1dt_2 \] and then the asymptotic Hopf invariant \(H({\mathbf B})\) of \({\mathbf B}\) can be obtained by integrating the limit of \(\lambda_{\mathbf B} (T_1, T_2; x_1, x_2)\) when \(T_1, T_2\) tend to \(\infty\). This paper presents a generalization of this invariant along with the proposal made by [\textit{V. I. Arnol'd}, Arnold's problems. (Zadachi Arnol'da) (Russian). Moscow: FAZIS (2000; Zbl 1052.00003)]. There, a triple \({\mathbf L}=L_1\cup L_2\cup L_3 \subset \mathbb{R}^3\) of magnetic lines is considered which is parametrized by the disjoint union of three copies of the standard unit circle. Let \(M\) be the integral introduced by the author in [\textit{P. M. Akhmetiev}, J. Geom. Phys. 53, No. 2, 180--196 (2005; Zbl 1070.57007)] as an invariant of a triple of magnetic \textit{tubes}. Using this \(M\) the author constructs an invariant \(M({\mathbf L})\) of \({\mathbf L}\) which assigns an real number to each \({\mathbf L}\) and shows that the principal term in its definitional formula is given by the Gaussian integral of the form \[ W=\frac{1}{4\pi}\int\int \frac{\langle {\mathbf F}(x), {\mathbf F}(y), x-y \rangle}{\parallel x -y\parallel ^3} dxdy, \] \({\mathbf F}(x)\) denoting a vector field with singularities on three-components \(L_i\)s of \({\mathbf L}\). Then the main result (partially omitted) states that the \(W\) above defines an asymptotic functional of magnetic fields on \(U\); and in the case when \({\mathbf B}\) satisfies the return condition, its local functional can be defined by integrating \(W\) over all possible triples of magnetic lines of \({\mathbf B}\).
scientific article; zbMATH DE number 6497498