Mathematics – Symplectic Geometry
Scientific paper
2011-04-30
Mathematics
Symplectic Geometry
33 pages, 5 figures
Scientific paper
Dirac structures unify both presymplectic and Poisson structures, which serve an implicit generalization of Lagrangian and Hamiltonian systems including the case of non-holonomic mechanics as well as the case of degenerate Lagrangians. It is known that a Dirac structure represents a power conserving interconnection structure between physical systems. In this paper, we investigate the interconnection of distinct Dirac structures and associated physical systems. First, we make a brief review on induced Dirac structures and Lagrange-Dirac dynamical systems. Second, we consider how distinct Dirac structures D_1,...,D_n can be interconnected through an interconnection Dirac structure D_int. To do this, we introduce a tensor product called the bowtie product of Dirac structures and then show how the interconnection of Dirac structures can be given by the bowtie product of D_1+ ...+ D_n and D_int. We also explore variational structures associated to the interconnection of Lagrange-Dirac systems. Lastly, we demonstrate the theory of interconnection of Dirac structures and associated Lagrange-Dirac dynamical systems by some examples including electric circuits, nonholonomic mechanical systems, and simple mass-spring mechanical systems.
Jacobs Henry
Yoshimura Hiroaki
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