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Local Lagrangian formalism and discretization of the Heisenberg magnet model. (English) Zbl 1069.78011

Summary: In this paper we develop the Lagrangian and multisymplectic structures of the Heisenberg magnet (HM) model which are then used as the basis for geometric discretizations of HM. Despite a topological obstruction to the existence of a global Lagrangian density, a local variational formulation allows one to derive local conservation laws using a version of Nöther’s theorem from the formal variational calculus of Gelfand-Dikii. Using the local Lagrangian form we extend the method of Marsden, Patrick and Schkoller to derive local multisymplectic discretizations directly from the variational principle. We employ a version of the finite element method to discretize the space of sections of the trivial magnetic spin bundle \(N=M\times S^2\) over an appropriate space-time \(M\). Since sections do not form a vector space, the usual FEM bases can be used only locally with coordinate transformations intervening on element boundaries, and conservation properties are guaranteed only within an element. We discuss possible ways of circumventing this problem, including the use of a local version of the method of characteristics, non-polynomial FEM bases and Lie-group discretization methods.

MSC:

78A99 General topics in optics and electromagnetic theory
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
82D40 Statistical mechanics of magnetic materials
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References:

[1] T. Bridges, S. Reich, Multi-symplectic integrators; numerical schemes for hamiltonian pdes that conserve symplecticity, Phys. Lett. A 284, 184-193.; T. Bridges, S. Reich, Multi-symplectic integrators; numerical schemes for hamiltonian pdes that conserve symplecticity, Phys. Lett. A 284, 184-193. · Zbl 0984.37104
[2] Christie, I.; Griffiths, D.; Mitchell, A.; Sanz-Serna, J., Product approximation for nonlinear problems in the finite element method, IMA J. Num. Anal., 1, 253-266 (1981) · Zbl 0469.65072
[3] Dickey, L., Soliton Equations and Hamiltonian Systems (1987), World Scientific · Zbl 1140.35012
[4] Deligne, P., Quantum Fields and Strings: A Course for Mathematicians (1999), American Mathematical Society, Institute for Advanced Study
[5] Faddeev, L. D.; Takhtajan, L. A., Hamiltonian Methods in Soliton Theory (1987), Springer-Verlag · Zbl 1327.39013
[6] Karasev, M. V.; Maslov, V. P., Nonlinear Poisson brackets. Geometry and Quantization (1993), American Mathematical Society · Zbl 0731.58002
[7] Manin, Y., Algebraic aspects of nonlinear differential equations, Itogi Nauki i Tekhniki, 11, 5-152 (1978)
[8] Marsden, G.; Patrick, J.; Shkoller, S., Multisymplectic geometry, variational integrators and nonlinear PDEs, Commun. Math. Phys., 199, 351-395 (1998) · Zbl 0951.70002
[9] Novikov, S. P., Hamiltonian formalism and a multivalued analog of morse theory, Russ. Math. Surv., 37, 5, 3-49 (1982)
[10] Palais, R., Foundations of Global Nonlinear Analysis (1968), W.A. Benjamin Inc. · Zbl 0164.11102
[11] Steenrod, N., The Topology of Fibre Bundles (1951), Princeton University Press · Zbl 0054.07103
[12] Strang, G.; Fix, G., An Analysis of the Finite Element Method (1973), Series in Automatic Computation, Prentice-Hall · Zbl 0278.65116
[13] Witten, E., Global aspects of current algebra, Nucl. Phys., 223, 2, 422-432 (1983)
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