Tensor products of Dirac structures and interconnection in Lagrangian mechanics
Tensor products of Dirac structures and interconnection in Lagrangian mechanics
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DOI：
10.3934/jgm.2014.6.67
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发表时间：
2011-04
期刊：
影响因子：
--
通讯作者：
H. Jacobs;Hiroaki Yoshimura
中科院分区：
文献类型：
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作者：
H. Jacobs;Hiroaki Yoshimura
许多机械系统虽然由简单的子系统组成，但都是大型而复杂的。为了理解这样的大系统，自然要把系统分解成这些子系统。相反地，我们必须了解如何颠倒这种撕裂过程。换句话说，我们必须了解子系统的互连。这样的理解已经通过端口-哈密顿系统程序在向量空间上的哈密顿系统的上下文中得到了展示，其中可以通过识别共享变量来实现互连，于是狄拉克结构的组成概念允许人们互连两个系统。本文试图将向量空间上的port-Hamiltonian系统的程序推广到流形上的Lagrangian系统的情形，并适当地推广了Dirac结构的复合概念。特别是，我们将通过修改所涉及的子系统的各自的狄拉克结构来互连拉格朗日-狄拉克系统。我们通过相互作用狄拉克结构和狄拉克结构的张量积定义了狄拉克结构的互连。我们将展示如何将互联系统的动力学公式化为子系统的函数，并阐明相关的变分原理。然后，我们将说明这个理论如何扩展理论的端口哈密顿系统和概念的组成狄拉克结构的流形与耦合，不需要确定的共享变量。最后，我们将展示一些例子：质量弹簧力学系统，电路和非完整力学系统。
Many mechanical systems are large and complex, despite being composed of simple subsystems. In order to understand such large systems it is natural to tear the system into these subsystems. Conversely we must understand how to invert this tearing procedure. In other words, we must understand interconnection of subsystems. Such an understanding has been already shown in the context of Hamiltonian systems on vector spaces via the port-Hamiltonian systems program, in which an interconnection may be achieved through the identification of shared variables, whereupon the notion of composition of Dirac structures allows one to interconnect two systems. In this paper, we seek to extend the program of the port-Hamiltonian systems on vector spaces to the case of Lagrangian systems on manifolds and also extend the notion of composition of Dirac structures appropriately. In particular, we will interconnect Lagrange-Dirac systems by modifying the respective Dirac structures of the involved subsystems. We define the interconnection of Dirac structures via an interaction Dirac structure and a tensor product of Dirac structures. We will show how the dynamics of the interconnected system is formulated as a function of the subsystems, and we will elucidate the associated variational principles. We will then illustrate how this theory extends the theory of port-Hamiltonian systems and the notion of composition of Dirac structures to manifolds with couplings which do not require the identification of shared variables. Lastly, we will show some examples: a mass-spring mechanical systems, an electric circuit, and a nonholonomic mechanical system.