A Unified Approach to Multibody Dynamics by Dirac Structures and Implicit Lagrangian Systems

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The advent of multibody systems such as space robots and large flexible space structures throws us an essential problem of how to understand complicated system structures in the traditional analytical mechanics. The theory of geometric mechanics provides us an efficient tool that enables to intuitively understand the complicated system structures ; for instance, the Lie-Poisson structure and Euler-Poincare reduction are crucial examples in rigid body dynamics and fluid dynamics, which were essentially developed from the geometric point of view.The main purposes of this study are to establish theory of implicit Lagrangian systems associated with Dirac structures that enables us to treat degenerate Lagrangian systems with holonomic and nonholonomic constraints, and also to illustrate that the implicit Lagrangian system theory can be applied to L-C circuits, namely, a typical example of degenerate Lagrangian systems with holonomic constraints as well as to nonholonomic systems such as rig … More id body systems with rolling contacts. Those examples may be fundamental to understand complicated structures of multibody systems.In this study, we first showed that a Dirac structure on the cotangent bundle of a configuration manifold induced from a given distribution and also that implicit Lagrangian systems can be constructed by a triple of a Lagrangian, a partial vector field on the cotangent bundle and a given distribution that satisfy the condition that a pair of the vector field and the Dirac differential becomes a section of the induced Dirac structure. Second, we illustrate examples of L-C circuits and nonholonomic systems, which can be represented as implicit Lagrangian systems. Third, we clarified variational structures underlying the implicit Lagrangian systems by introducing Hamilton-Pontryagin principle. Further, we developed a reduction theory of Dirac structure called Lie-Dirac reduction, which reduces the canonical Dirac structure on the cotangent bundle of a Lie group and we incorporate this into a theory of symmetry reduction of implicit Lagrangian systems called Euler-Poincare-Dirac reduction. Finally, we showed that the so-called Suslov problem in nonholonomic rigid body systems can be formulated by Euler-Poincare-Dirac reduction by extending the Euler-Poincare-Dirac reduction to the case of nonholonomic constraints.The results of this research have been published as journal papers, international conference papers and so on. We are further keeping on the current study to develop the theory and with its application in details for future works. Less