2012/06/17 by Marko Šešlija, Arjan van der Schaft, Seslija, Marko +3
Engineering · Physics and Astronomy · #Control and Stability of Dynamical Systems #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mechanical and Optical Resonators #Optimization and Control (math.OC) #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.1206.3781
openalex publication_date 2012/06/17 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
Stokes-Dirac structures are infinite-dimensional Dirac structures defined in\nterms of differential forms on a smooth manifold with boundary. These Dirac\nstructures lay down a geometric framework for the formulation of Hamiltonian\nsystems with a nonzero boundary energy flow. Simplicial triangulation of the\nunderlaying manifold leads to the so-called simplicial Dirac structures,\ndiscrete analogues of Stokes-Dirac structures, and thus provides a natural\nframework for deriving finite-dimensional port-Hamiltonian systems that emulate\ntheir infinite-dimensional counterparts. The port-Hamiltonian systems defined\nwith respect to Stokes-Dirac and simplicial Dirac structures exhibit gauge and\na discrete gauge symmetry, respectively. In this paper, employing Poisson\nreduction we offer a unified technique for the symmetry reduction of a\ngeneralized canonical infinite-dimensional Dirac structure to the Poisson\nstructure associated with Stokes-Dirac structures and of a fine-dimensional\nDirac structure to simplicial Dirac structures. We demonstrate this Poisson\nscheme on a physical example of the vibrating string.\n