Discrete Embeddings for Lagrangian and Hamiltonian Systems

Abstract

The topic of this paper is to study the conservation of variational properties for a given problem when discretising it. Precisely, we are interested in Lagrangian or Hamiltonian structures and thus with variational problems attached to a least action principle. Consider a partial differential equation (PDE) deriving from a variational principle. A natural question is to know whether this structure is preserved at the discrete level when discretising the PDE. To address this question, a concept of coherence is introduced. Both the differential equation (the PDE translating the least action principle) and the variational structure can be embedded at the discrete level. This provides two discrete embeddings for the original problem. If these procedures finally provide the same discrete problem, we will say that the discretisation is coherent. Our purpose is illustrated with the Poisson problem. Coherence for discrete embeddings of Lagrangian structures is studied for various classical discretisations. For Hamiltonian structures, we show the coherence between a discrete Hamiltonian and the discretisation of the mixed formulation of the Poisson problem.

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Correspondence to Charles Pierre.

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Cresson, J., Greff, I. & Pierre, C. Discrete Embeddings for Lagrangian and Hamiltonian Systems. Acta Math Vietnam 43, 391–413 (2018). https://doi.org/10.1007/s40306-018-0257-0

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Keywords

  • PDE discretisation
  • Variational integrators
  • Lagrangian and Hamiltonian systems
  • Discrete embeddings

Mathematics Subject Classification (2010)

  • 65P10
  • 65M06
  • 65M08