Abstract

The notion of implicit port-Lagrangian systems for nonholonomic mechanics was proposed in Yoshimura & Marsden (2006a, J. Geom. Phys., 57, 133–156; 2006b, J. Geom. Phys., 57, 209–250; 2006c, Proc. of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto) as a Lagrangian analogue of implicit port-Hamiltonian systems. Such port-systems have an interconnection structure with ports through which power is exchanged with the exterior and which can be modeled by Dirac structures. In this paper, we present the notions of implicit port-Lagrangian systems and port-Dirac dynamical systems in nonequilibrium thermodynamics by generalizing the Dirac formulation to the case allowing irreversible processes, both for closed and open systems. Port-Dirac systems in nonequilibrium thermodynamics can be also deduced from a variational formulation of nonequilibrium thermodynamics for closed and open systems introduced in Gay-Balmaz & Yoshimura (2017a, J. Geom. Phys., 111, 169–193; 2018a, Entropy, 163, 1–26). This is a type of Lagrange–d’Alembert principle for the specific class of nonholonomic systems with nonlinear constraints of thermodynamic type, which are associated to the entropy production equation of the system. We illustrate our theory with some examples such as a cylinder-piston with ideal gas, an electric circuit with entropy production due to a resistor and an open piston with heat and matter exchange with the exterior.

中文翻译：

---

非平衡热力学中端口-狄拉克系统的狄拉克结构和变分公式

摘要

非完整力学隐含端口拉格朗日系统的概念在吉村＆马斯登（2006年，提出了。J.的Geom物理学2006年b;，57，133-156 。J.的Geom物理学。，57，209-250; 2006年c，作为隐含端口-哈密尔顿系统的拉格朗日模拟，第17届国际网络和系统数学理论国际研讨会论文集，京都。这种端口系统具有带有端口的互连结构，通过该端口可以与外部交换功率，并且可以用Dirac结构进行建模。在本文中，我们提出的概念隐含的端口-拉格朗日系统和端口狄拉克动力系统在非平衡态热力学通过将Dirac公式推广到允许封闭和开放系统不可逆过程的情况。在非平衡热力学端口-狄拉克系统可以从非平衡热力学的用于在盖伊-Balmaz＆吉村（2017A，引入封闭和开放系统变分制剂中还推断。J.的Geom物理学。，111，169-193; 2018A，熵，163，1–26）。这是一类Lagrange-d'Alembert原理，用于特定类的具有热力学类型非线性约束的非完整系统，该系统与系统的熵产生方程相关联。我们以一些示例来说明我们的理论，例如具有理想气体的气缸活塞，由于电阻器而产生熵的电路以及与外界进行热和物质交换的开放式活塞。