### Abstract

A Hamiltonian formulation of classes of distributed-parameter systems is presented, which incorporates the energy flow through the boundary of the spatial domain of the system, and which allows to represent the system as a boundary control Hamiltonian system. The system is Hamiltonian with respect to an infinite-dimensional Dirac structure associated with the exterior derivative and based on Stokes' theorem. The theory is applied to the telegraph equations for an ideal transmission line, Maxwell's equations on a bounded domain with non-zero Poynting vector at its boundary, and a vibrating string with traction forces at its ends. Furthermore, the framework is extended to cover Euler's equations for an ideal fluid on a domain with permeable boundary. Finally, some properties of the Stokes-Dirac structure are investigated, including the analysis of conservation laws.

Original language | Undefined |
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Article number | 10.1016/S0393-0440(01)00083-3 |

Pages (from-to) | 166-194 |

Number of pages | 29 |

Journal | Journal of geometry and physics |

Volume | 42 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - May 2002 |

### Keywords

- Stokes theorem
- EWI-16720
- METIS-211061
- Boundary variables
- Conservation laws
- IR-69115
- Distributed-parameter systems
- Hamiltonian systems
- Dirac structures

## Cite this

van der Schaft, A., & Maschke, B. M. (2002). Hamiltonian formulation of distributed-parameter systems with boundary energy flow.

*Journal of geometry and physics*,*42*(1-2), 166-194. [10.1016/S0393-0440(01)00083-3]. https://doi.org/10.1016/S0393-0440(01)00083-3