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Advances in Condensed Matter Physics
Volume 2012, Article ID 765709, 16 pages
Review Article

Hamiltonian and Lagrangian Dynamical Matrix Approaches Applied to Magnetic Nanostructures

1CNISM Unit of Ferrara and Department of Physics, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy
2Department of Sciences for Engineering and Architecture, University of Messina, C.da di Dio, Villaggio S.Agata, 98166 Messina, Italy

Received 30 March 2012; Accepted 23 April 2012

Academic Editor: Eduardo Martinez Vecino

Copyright © 2012 Roberto Zivieri and Giancarlo Consolo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Two micromagnetic tools to study the spin dynamics are reviewed. Both approaches are based upon the so-called dynamical matrix method, a hybrid micromagnetic framework used to investigate the spin-wave normal modes of confined magnetic systems. The approach which was formulated first is the Hamiltonian-based dynamical matrix method. This method, used to investigate dynamic magnetic properties of conservative systems, was originally developed for studying spin excitations in isolated magnetic nanoparticles and it has been recently generalized to study the dynamics of periodic magnetic nanoparticles. The other one, the Lagrangian-based dynamical matrix method, was formulated as an extension of the previous one in order to include also dissipative effects. Such dissipative phenomena are associated not only to intrinsic but also to extrinsic damping caused by injection of a spin current in the form of spin-transfer torque. This method is very accurate in identifying spin modes that become unstable under the action of a spin current. The analytical development of the system of the linearized equations of motion leads to a complex generalized Hermitian eigenvalue problem in the Hamiltonian dynamical matrix method and to a non-Hermitian one in the Lagrangian approach. In both cases, such systems have to be solved numerically.