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

空间机器人和大型柔性空间结构等多体系统的出现，给传统分析力学中如何理解复杂系统结构提出了一个重要问题。几何力学理论为我们直观地认识复杂的系统结构提供了有效的工具；例如，李泊松结构和欧拉庞加莱化简是刚体动力学和流体动力学中的重要例子，它们本质上是从几何的角度发展起来的。本研究的主要目的是建立与Dirac结构相关的隐式拉格朗日系统理论，使我们能够处理具有完整和非完整约束的退化拉格朗日系统，并说明隐式拉格朗日系统理论可以应用于L-C电路，即：具有完整约束的退化拉格朗日系统的一个典型例子，以及非完整系统，如具有滚动接触的多体系统。这些例子可能是理解多体系统复杂结构的基础。在本研究中，我们首先证明了由给定分布导出的位形流形的余切束上的狄拉克结构，并证明了隐式拉格朗日系统可以由拉格朗日、余切束上的部分向量场和给定分布的三倍组合来构造，满足向量场和狄拉克微分的一对成为诱导狄拉克结构的一部分的条件。其次，我们给出了L-C电路和非完整系统的例子，它们可以表示为隐式拉格朗日系统。第三，通过引入Hamilton-Pontryagin原理，阐明了隐式拉格朗日系统的变分结构。进一步，我们发展了狄拉克结构的约简理论，称为李-狄拉克约简，它减少了李群的余切束上的标准狄拉克结构，我们将其纳入隐式拉格朗日系统的对称约简理论，称为欧拉-庞加莱-狄拉克约简。最后，通过将欧拉-庞加莱-狄拉克约简推广到非完整约束的情况，证明了非完整刚体系统中的所谓Suslov问题可以用欧拉-庞加莱-狄拉克约简来表述。本研究成果已以期刊论文、国际会议论文等形式发表。我们正在继续进行目前的研究，以进一步发展理论，并为今后的工作提供详细的应用。少

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