Geodesic
Vassiliev invariants and finite-dimensional approximations of the Euler equation in magnetohydrodynamics
Informatics and Automation, Differential equations and dynamical systems, Tome 270 (2010), pp. 161-169.

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We consider Hamiltonian systems that correspond to Vassiliev invariants defined by Chen's iterated integrals of logarithmic differential forms. We show that Hamiltonian systems generated by first-order Vassiliev invariants are related to the classical problem of motion of vortices on the plane. Using second-order Vassiliev invariants, we construct perturbations of Hamiltonian systems for the classical problem of $n$ vortices on the plane. We study some dynamical properties of these systems. 
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     title = {Vassiliev invariants and finite-dimensional approximations of the {Euler} equation in magnetohydrodynamics},
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     url = {http://geodesic.mathdoc.fr/item/TRSPY_2010_270_a10/}
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N. A. Kirin. Vassiliev invariants and finite-dimensional approximations of the Euler equation in magnetohydrodynamics. Informatics and Automation, Differential equations and dynamical systems, Tome 270 (2010), pp. 161-169. http://geodesic.mathdoc.fr/item/TRSPY_2010_270_a10/

[1] Arnold V.I., Khesin B.A., Topologicheskie metody v gidrodinamike, MTsNMO, M., 2007

[2] Borisov A.V., Mamaev I.S., Matematicheskie metody dinamiki vikhrevykh struktur, In-t kompyut. issled., Moskva; Izhevsk, 2005 | MR

[3] Kozlov V.V., Obschaya teoriya vikhrei, Izd-vo Udm. un-ta, Izhevsk, 1998 | MR | Zbl

[4] Khein R.M., Iterirovannye integraly i problema gomotopicheskikh periodov, Nauka, M., 1988 | MR

[5] Berger M.A., “Hamiltonian dynamics generated by Vassiliev invariants”, J. Phys. A: Math. and Gen., 34 (2001), 1363–1374 | DOI | MR | Zbl

[6] Kohno T., “Vassiliev invariants of braids and iterated integrals”, Arrangements — Tokyo 1998, Proc. Workshop on Mathematics Related to Arrangements of Hyperplanes, Adv. Stud. Pure Math., 27, Kinokuniya, Tokyo, 2000, 157–168 | MR | Zbl

[7] Kontsevich M., “Vassiliev's knot invariants”, I.M. Gelfand seminar. Pt. 2: Papers of the Gelfand seminar in functional analysis held at Moscow University 1993, Adv. Sov. Math., 16, no. 2, Amer. Math. Soc., Providence (RI), 1993, 137–150 | MR | Zbl