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Lagrangian and Hamiltonian two-scale reduction

Please always quote using this URN:urn:nbn:de:0296-matheon-4891
  • Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.

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Metadaten
Author:Johannes Giannoulis, Michael Herrmann, Alexander Mielke
URN:urn:nbn:de:0296-matheon-4891
Referee:Christof Schütte
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2008/03/13
Release Date:2008/03/13
Tag:
Institute:Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
Preprint Number:485
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