拉格朗日力学系统中Dirac结构的张量积与互连
Tensor products of Dirac structures and interconnection in Lagrangian mechanical systems
摘要：
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，我们必须了解如何反转这种撕裂。换句话说，我们必须理解互联。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成功地理解。在端口-哈密顿系统理论中，互连是通过识别共享变量来实现的，因此狄拉克结构的组成概念允许人们互连两个系统。在本文中，我们试图推广端口-哈密顿系统程序的拉格朗日系统的流形上，并适当地扩展狄拉克结构的组成的概念。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Dirac结构和Dirac结构的张量积定义了Dirac结构的互连。我们将展示如何将关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港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，我们将用一些例子来结束：质量弹簧力学系统，电路和非完整力学系统。
英文摘要：
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. In other words, we must understand interconnection. Such an understanding has already successfully understood in the context of Hamiltonian systems on vector spaces via the port-Hamiltonian systems program. In port-Hamiltonian systems theory, interconnection is 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 port-Hamiltonian systems program to Lagrangian systems on manifolds and 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 close with some examples: a mass-spring mechanical systems, an electric circuit, and a nonholonomic mechanical system.
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