[en] In this note we report on some recent developments in geometric knot theory which aims at finding links between geometric properties of a given knotted curve and its knot type. The central object of this field are so-called knot energies which are defined on closed embedded curves. First we present three important examples of two-parameter knot energy families, namely O'Hara's energies, the (generalized) integral Menger curvature, and the (generalized) tangent- point energies. Subsequently we outline the main steps that lead to C^∞-regularity of stationary points- especially minimizers-in the non-degenerate sub-critical range of parameters. Particular attention is devoted to the appearing parallels between these energies which, surprisingly at first glance, are quite similar from an analyst's perspective

[en] Magnetic (flux) coordinates are discussed where two solenoidal vector fields are represented by straight lines. It is shown how to take advantage of this selection rule within a general method of constructing magnetic coordinates based on straightening the magnetic lines and implicit prescription of the functional dependence of the Jacobian. The particular applications for MHD models describing the equilibrium of isotropic plasma, equilibrium of anisotropic plasma, and equilibrium of plasma with stationary flows are considered. In all these cases, there are at least two suitable divergence-free vector fields in the initial equations. If no additional equations are attracted, the straightening of the lines of these two vectors appears to be the only natural requirement of the optimal choice of flux coordinates. Then, an admissible freedom in the choice of such coordinates is completely used, and the two mentioned vectors are expressed in the simplest form

[en] Guiding center equations of particle motion under a toroidal magnetic configuration were derived using a new defined magnetic flux coordinates system. A transformation method to Hamiltonian canonical variables is presented in this work. It was found that the newly defined flux coordinates are simple self-consistent, and could be applied to any magnetostatic equilibrium with a nested flux configuration.

[en] We found a large class of analytic equilibria in which 1/B² is a function of only the flux and one helical coordinate. Such quasi-helical equilibria have identical drift orbits and associated transport in magnetic coordinates to that of a symmetric torus. (author). 6 refs

[en] One of the main issues of the open ended plasma confining devices is to provide a magnetohydrodynamic stable plasma equilibrium. A nonaxisymmetric, marginally stable confining magnetic field for a conventional, single mirror device has been studied for a long thin flux tube. The marginally stable minimum B magnetic field is proven to have quadrupolar symmetry under the assumptions of a straight paraxial central flux line and zero plasma β. The quadrupolar symmetry geometrically means that the flux lines are not twisted. The perpendicular drift of the gyro centers is zero. A possible external current distribution which can generate the magnetic field is determined. The most convenient set of flux coordinates for this magnetic field are (x₀,y₀,s), where x₀ and y₀ are Cartesian-like Clebsch coordinates and s is the arclength of the magnetic field line. Analytical expressions for the magnetic flux coordinates are derived for this magnetic field

[en] In the case of nonlocal dependence of plasma in MHD calculations it is convenient to use flux coordinates. All known up to now methods are based on the assumption of nested magnetic surfaces. In the present paper we consider a principally new method of calculation MHD equilibria with large magnetic islands using flux coordinates. (author). 2 refs, 2 figs

[en] The bootstrap current is the surface-averaged parallel current that is driven by the pressure gradient. It is determined by the variation of the magnetic field strength in the magnetic surface: the shape of the magnetic surfaces is irrelevant. The device dependence of the bootstrap current comes through a single quantity Δ₀, which can be determined by a set of field line integrations. Δ₀ can also be determined using a Monte Carlo code, and it is not necessary in such a code to make the collision operator momentum conserving. (author). 2 figs

[en] The stellarator equilibrium with a steady flow is investigated. A set of averaged, axially symmetric scalar equations is obtained, equations for the shift of the magnetic surfaces and the surfaces of equal density, ρ = const, are derived, and the changes in the topology of a given vacuum magnetic configuration are determined. It is shown that the topology of the ρ = const surfaces both in the cases of tokamak and stellarator can change while the topology of the magnetic surfaces does not. (author). 11 refs, 2 figs

[en] Three-dimensional magnetic structures of pi-vertical-Bloch line (VBL) and 2pi-VBL are investigated in 80 - 320-nm-thick Permalloy films using direct integration of the Landau - Lifshitz - Gilbert equation in an 128x128x80 point Cartesian lattice. A pi-VBL reflects the shapes of its adjacent walls, which change with film thickness. The pi-VBL conducts the flux between walls of opposite chirality by letting the magnetization rotate out of the plane of the walls via a vortex structure. The Neel caps switch chirality via a 'converging point,' or cross-tie, flux at one surface and a vortex flux, or a swirl, at the other surface. The pi-VBL energy per unit area is 0.44, 0.20, and 0.30 erg/cm2 in 80, 160, and 320 nm films, respectively. The corresponding pi-VBL widths are 88, 60, and 86 nm. A stable winding 2pi-VBL structure was also computed by combining two pi-VBL structures of appropriate chirality. The Neel caps intersect via the pair-(swirl, converging point) like flux at both surfaces. The width of the 2pi-VBL is 140 nm and its energy per unit area is 0.57 erg/cm2. [copyright] 2001 American Institute of Physics

[en] The problem of constructing flux coordinates for a tokamak is discussed. Different coordinate systems are considered using a unified approach. The problem is formulated in terms of two magnetic differential equations with known right-hand sides. The construction of tokamak flux coordinates with straightened field lines reduces to the problem of calculating one or two integrals around a poloidal contour on a magnetic surface. Three coordinate systems are analyzed, specifically, coordinates with a prescribed toroidal angle and Hamada and Boozer coordinates. It is shown how one coordinate system can be transformed into another and how the required calculations can be performed without a complete change of variables. The results obtained are intended to be used in practical manipulations with flux coordinates