Hamiltonian formulation of distributed-parameter systems with boundary energy flow

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. \u

On the Hamiltonian formulation of nonholonomic mechanical systems

A simple procedure is provided to write the equations of motion of mechanical systems with constraints as Hamiltonian equations with respect to a ¿Poisson¿ bracket on the constrained state space, which does not necessarily satisfy the Jacobi identity. It is shown that the Jacobi identity is satisfied if and only if the constraints are holonomic

Conservation laws and open systems on higher-dimensional networks

We discuss a framework for defining physical open systems on higher-dimensional complexes. We start with the formalization of the dynamics of open electrical circuits and the Kirchhoff behavior of the underlying open graph or 1-complex. It is discussed how the graph can be closed to an ordinary graph, and how this defines a Dirac structure on the extended graph. Then it is shown how this formalism can be extended to arbitrary k-complexes, which is illustrated by a discrete formulation of heat transfer on a two-dimensional spatial domain.

An intrinsic Hamiltonian formulation of the dynamics of LC-circuits

First, the dynamics of LC-circuits are formulated as a Hamiltonian system defined with respect to a Poisson bracket which may be degenerate, i.e., nonsymplectic. This Poisson bracket is deduced from the network graph of the circuit and captures the dynamic invariants due to Kirchhoff's laws. Second, the antisymmetric relations defining the Poisson bracket are realized as a physical network using the gyrator element and partially dualizing the network graph constraints. From the network realization of the Poisson bracket, the reduced standard Hamiltonian system as well as the realization of the embedding standard Hamiltonian system are deduce

Port contact systems for irreversible thermodynamical systems

In this paper we propose a definition of control contact systems, generalizing input-output Hainiltonian systems, to cope with models arising from irreversible Thermodynamics. We exhibit a particular subclass of these systems, called conservative, that leaves invariant some Legendre submanifold (the geometric structures associated with thermodynamic properties). These systems, both energy-preserving and irreversible, are then used to analyze the loss-lessness of these systems with respect to different generating functions

Mathematical structures in the network representation of energy-conserving physical systems

It is shown that network modelling of energy-conserving physical systems naturally leads to the consideration of (nonlinear) implicit generalized Hamiltonian systems. Behavioral systems theory may be invoked to formulate and analyze the system-theoretic properties of these systems.