On alternative Poisson brackets for fluid dynamical systems and their extension to Stokes-Dirac structures

Authors

B. Maschke, A.J. van der Schaft

Abstract

In this paper we shall consider the Hamiltonian formulation of one-dimensional models of fluid dynamical systems such as the Korteweg de Vries equation and the Boussinesq equation and their extension to port-Hamiltonian systems. We consider the Korteweg de Vries equation and recall its bi-Hamiltonian structure, either formulated as a single conservation law or formulated with respect to Magri’s bracket. In both cases we give an extension of the associated Hamiltonian operators to a Stokes-Dirac structure. Then we consider the Boussinesq equation and recall the two Hamiltonian representations either with respect to the canonical Hamiltonian operator associated with a system of two coupled conservation laws or with respect to a third order Hamiltonian operator. In this case we shall suggest a third Hamiltonian operator for which an extension to a Dirac structure is derived.

Keywords

Five to ten keywords; preferably chosen from the IFAC keyword list

Citation

• Journal: IFAC Proceedings Volumes
• Year: 2013
• Volume: 46
• Issue: 26
• Pages: 109–114
• Publisher: Elsevier BV
• DOI: 10.3182/20130925-3-fr-4043.00083
• Note: 1st IFAC Workshop on Control of Systems Governed by Partial Differential Equations

BibTeX

@article{Maschke_2013,
  title={{On alternative Poisson brackets for fluid dynamical systems and their extension to Stokes-Dirac structures}},
  volume={46},
  ISSN={1474-6670},
  DOI={10.3182/20130925-3-fr-4043.00083},
  number={26},
  journal={IFAC Proceedings Volumes},
  publisher={Elsevier BV},
  author={Maschke, B. and van der Schaft, A.J.},
  year={2013},
  pages={109--114}
}

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