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Simulation of the dynamics of three point-like magnetic vortices in a magnetic film is considered. These are vortices of magnetization in a magnetic material, with sizes ranging from 10 to 50 nanometers. They are mathematically similar to ordinary vortices in fluids.

The equations of motion for the vortex positions are given in [1, Eq. (4)]. They are a generalization of the Helmholtz–Kirchhoff equations for point vortices in ordinary fluids. The equations of motion describe a Hamiltonian system that is completely integrable; that is, all vortex trajectories can be formally found by integration based on the conserved quantities of energy, linear momentum and angular momentum. A complete solution for three fluid vortices was given in the 1877 dissertation of Gröbli [2], reviewed in [3]. A complete study for the system of magnetic vortices is given in [1].

Each vortex is characterized by two integers: is the vortex number (, or for vortices and antivortices, respectively), and specifies the direction of the magnetic vortex ( or for magnetization pointing up or down, respectively), which does not pertain in the case of ordinary vortices. We focus on the cases of vortices with parameters , , These cannot be realized in ordinary fluids, but only for magnetic vortices (see cases 2 through 4 following).

We consider four special cases:

1. Three identical vortices forming an equilateral triangle, rotating around the center.

2. A vortex-antivortex pair colliding on a third vortex, scattering at an angle while the third vortex merely changes its position.

In this case a transmutation of linear momentum into position occurs; that is, a vortex-antivortex pair changes its motion while another vortex changes its position in such a way that the conservation law for momentum/position remains satisfied.

3. Three vortices collide at a common point in finite time, thus creating a singularity in the solution.

4. Three vortices are confined in motions bounded in space.

References

[1] S. Komineas and N. Papanicolaou, "Gro¨bli Solution for Three Magnetic Vortices," Journal of Mathematical Physics, 51, 2010 042705. doi:10.1063/1.3393506.

[2] W. Gröbli, Spezielle Probleme über die Bewegung geradliniger paralleler Wirbelfäden, Zürich: Zürcher and Furrer, 1877. Reprinted in Vierteljahrsschrift der Natforschenden Gesellschaft Zurich, 22, 1877 p. 37–81; 22, 1877 p. 129–165.

[3] H. Aref, N. Rott and H. Thomann, "Gro¨bli's Solution of the Three-Vortex Problem," Annual Review of Fluid Mechanics, 24(1), 1992 pp. 1–21. doi:10.1146/annurev.fl.24.010192.000245.