dorsal/arxiv
View SchemaLocal Lagrangian Formalism and Discretization of the Heisenberg Magnet Model
| Authors | D. Karpeev, C. M. Schober |
|---|---|
| Categories | |
| ArXiv ID | physics/0412082 |
| URL | https://arxiv.org/abs/physics/0412082 |
Abstract
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 N\"other'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.
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"abstract": "In this paper we develop the Lagrangian and multisymplectic structures of the\nHeisenberg magnet (HM) model which are then used as the basis for geometric\ndiscretizations of HM. Despite a topological obstruction to the existence of a\nglobal Lagrangian density, a local variational formulation allows one to derive\nlocal conservation laws using a version of N\\\"other\u0027s theorem from the formal\nvariational calculus of Gelfand-Dikii. Using the local Lagrangian form we\nextend the method of Marsden, Patrick and Schkoller to derive local\nmultisymplectic discretizations directly from the variational principle. We\nemploy a version of the finite element method to discretize the space of\nsections of the trivial magnetic spin bundle $N = M\\times S^2$ over an\nappropriate space-time $M$. Since sections do not form a vector space, the\nusual FEM bases can be used only locally with coordinate transformations\nintervening on element boundaries, and conservation properties are guaranteed\nonly within an element. We discuss possible ways of circumventing this problem,\nincluding the use of a local version of the method of characteristics,\nnon-polynomial FEM bases and Lie-group discretization methods.",
"arxiv_id": "physics/0412082",
"authors": [
"D. Karpeev",
"C. M. Schober"
],
"categories": [
"physics.comp-ph"
],
"title": "Local Lagrangian Formalism and Discretization of the Heisenberg Magnet Model",
"url": "https://arxiv.org/abs/physics/0412082"
},
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