ON REPRESENTATIONS AND INTEGRABILITY OF MATHEMATICAL STRUCTURES IN ENERGY-CONSERVING PHYSICAL SYSTEMS

In the present paper we elaborate on the underlying Hamiltonian struct ure of interconnected energy-conserving physical systems. It is shown that a power-conserving interconnection of port-controlled generalized Hamiltonian systems leads to an implicit generalized Hamiltonian syst em, and a power-conserving partial interconnection to an implicit port -controlled Hamiltonian system. The crucial concept is the notion of a (generalized) Dirac structure, defined on the space of energy-variabl es or on the product of the space of energy-variables and the space of flow-variables in the port-controlled case. Three natural representat ions of generalized Dirac structures are treated. Necessary and suffic ient conditions for closedness (or integrability) of Dirac structures in all three representations are obtained. The theory is applied to im plicit port-controlled generalized Hamiltonian systems, and it is show n that the closedness condition for the Dirac structure leads to stron g conditions on the input vector fields.