Local Lagrangian formalism and discretization of the Heisenberg magnet model
Abbreviated Journal Title
Math. Comput. Simul.
multisymplectic structure; geometric integrators; finite element methods; Computer Science, Interdisciplinary Applications; Computer Science, ; Software Engineering; Mathematics, Applied
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 Nother'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 x 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. (c) 2005 IMACS. Published by Elsevier B.V. All rights reserved.
Mathematics and Computers in Simulation
Article; Proceedings Paper
"Local Lagrangian formalism and discretization of the Heisenberg magnet model" (2005). Faculty Bibliography 2000s. 5325.