Interconnection of Dirac Structures and Lagrange-Dirac Dynamical Systems

In the paper, we develop an idea of interconnection of Dirac structures and their associated Lagrange- Dirac dynamical systems. First, we briefly review the Lagrange-Dirac dynamical systems (namely, implicit Lagrangian systems) associated to induced Dirac structures. Second, we describe an idea of interconnection of Dirac structures; namely, we show how two distinct Lagrange-Dirac systems can be interconnected through a Dirac structure on the product of configuration spaces. Third, we also show the variational structure of the interconnected Lagrange-Dirac dynamical system in the context of the Hamilton-Pontryagin-d’Alembert principle. Finally, we demonstrate our theory by an example of mass-spring mechanical systems

Reduction of Stokes-Dirac structures and gauge symmetry in port-Hamiltonian systems

Stokes-Dirac structures are infinite-dimensional Dirac structures defined in terms of differential forms on a smooth manifold with boundary. These Dirac structures lay down a geometric framework for the formulation of Hamiltonian systems with a nonzero boundary energy flow. Simplicial triangulation of the underlaying manifold leads to the so-called simplicial Dirac structures, discrete analogues of Stokes-Dirac structures, and thus provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The port-Hamiltonian systems defined with respect to Stokes-Dirac and simplicial Dirac structures exhibit gauge and a discrete gauge symmetry, respectively. In this paper, employing Poisson reduction we offer a unified technique for the symmetry reduction of a generalized canonical infinite-dimensional Dirac structure to the Poisson structure associated with Stokes-Dirac structures and of a fine-dimensional Dirac structure to simplicial Dirac structures. We demonstrate this Poisson scheme on a physical example of the vibrating string

Dirac groupoids and Dirac bialgebroids

We describe infinitesimally Dirac groupoids via geometric objects that we call Dirac bialgebroids. In the two well-understood special cases of Poisson and presymplectic groupoids, the Dirac bialgebroids are equivalent to the Lie bialgebroids and IM-$2$-forms, respectively. In the case of multiplicative involutive distributions on Lie groupoids, we find new properties of infinitesimal ideal systems.Comment: New expanded version; the construction of the Manin pair associated to an LA-Dirac structure has moved from arXiv:1209.6077 to here. Added background on double vector bundles, VB-algebroids and 2-term representations up to homotop

Dirac-Jacobi Bundles

We show that a suitable notion of Dirac-Jacobi structure on a generic line bundle $L$, is provided by Dirac structures in the omni-Lie algebroid of $L$. Dirac-Jacobi structures on line bundles generalize Wade's $\mathcal E^1 (M)$-Dirac structures and unify generic (i.e.~non-necessarily coorientable) precontact distributions, Dirac structures and local Lie algebras with one dimensional fibers in the sense of Kirillov (in particular, Jacobi structures in the sense of Lichnerowicz). We study the main properties of Dirac-Jacobi structures and prove that integrable Dirac-Jacobi structures on line-bundles integrate to (non-necessarily coorientable) precontact groupoids. This puts in a conceptual framework several results already available in literature for $\mathcal E^1 (M)$-Dirac structures.Comment: v6: 55 pages, corrected some minor mistakes, final version, to appear in J. Sympl. Geom, 16 (2018