Lagrangian and Hamiltonian two-scale reduction

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Studying spatially extended Hamiltonian systems with coherent microstructure, it is an important and challenging problem to identify reduced Hamiltonian models that describe the effective dynamics on large spatial and temporal scales. Such models require macroscopically varying, deterministic initial data which can possess a well-organized microstructure. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can be derived from the microscopic system. In the first part we develop a general approach by considering noncanonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices [or Hamiltonian partial differential equations (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, and (iii) an elementary model reduction that is based on a principle of consistent expansions. The result is always a reduced Hamiltonian PDE that governs the macroscopic dynamics, provided that the initial data are chosen consistently with the underlying two-scale ansatz. 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 motion or describe macroscopic modulations of oscillatory microstructures.