From Lagrangian mechanics to nonequilibrium thermodynamics: a variational perspective

In this paper, we survey our recent results on the variational formulation of nonequilibrium thermodynamics for the finite dimensional case of discrete systems as well as for the infinite dimensional case of continuum systems. Starting with the fundamental variational principle of classical mechanics, namely, Hamilton's principle, we show, with the help of thermodynamic systems with gradually increasing level complexity, how to systematically extend it to include irreversible processes. In the finite dimensional cases, we treat systems experiencing the irreversible processes of mechanical friction, heat and mass transfer, both in the adiabatically closed and in the open cases. On the continuum side, we illustrate our theory with the example of multicomponent Navier-Stokes-Fourier systems.Comment: 7 figure

Representations of Dirac Structures and Implicit Port-Controlled Lagrangian Systems

In this paper, we will develop two different representations for induced Dirac structures and their associated IPCL systems; namely, (1) a standard representation with using Lagrange multipliers; and (2) a representation without using Lagrange multipliers. Those representations are consistent with those developed by Courant and Weinstein. Specifically, the second representation without using Lagrange multipliers may be crucial in formulation of constrained mechanical systems since it systematically enables one to eliminate unnecessary constraint forces. In mechanics, it is known that the elimination of constraint forces can be done by the orthogonal complement method or the null space method, although the link with Dirac structures has not been clarified. The present paper fills this gap to show that the orthogonal complement method can be incorporated into the context of Dirac structures and the associated IPCL systems and we will further show the link with the topological method in electrical network theory using the so-called fundamental cut-set and loop matrices. In the paper, we shall illustrate out ideas by an example of L-C circuits

Dirac structures and Lagrangian systems on tangent bundles (Symmetry and Singularity of Geometric Structures and Differential Equations)

In mechanics, a Dirac structure, which is the unified notion of symplectic and Poisson structures, has been widely used to formulate mechanical systems with nonholonomic constraints, electric circuits as well as thermodynamic systems. In particular, the induced Dirac structure on the cotangent bundle from a given constraint distribution plays an essential role in the context of implicit Lagrangian and Hamiltonian systems. However, there has been almost no research on the Dirac geometry associated to the tangent bundle TQ, although it may be relevant with regular Lagrangian systems. In this paper, we introduce an induced Dirac structure on TQ, called a Lagrangian Dirac structure. For the regular case, we finally show that one can define a Lagrange-Dirac system on TQ

Interconnection of Dirac Structures and Lagrange-Dirac Dynamical Systems

In the paper, we develop an idea of interconnection of Dirac structures and their associated Lagrange- Dirac dynamical systems. First, we briefly review the Lagrange-Dirac dynamical systems (namely, implicit Lagrangian systems) associated to induced Dirac structures. Second, we describe an idea of interconnection of Dirac structures; namely, we show how two distinct Lagrange-Dirac systems can be interconnected through a Dirac structure on the product of configuration spaces. Third, we also show the variational structure of the interconnected Lagrange-Dirac dynamical system in the context of the Hamilton-Pontryagin-d’Alembert principle. Finally, we demonstrate our theory by an example of mass-spring mechanical systems

Multi-Dirac Structures and Hamilton-Pontryagin Principles for Lagrange-Dirac Field Theories

The purpose of this paper is to define the concept of multi-Dirac structures and to describe their role in the description of classical field theories. We begin by outlining a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are the Euler-Lagrange equations in implicit form. Secondly, we introduce multi-Dirac structures as a graded analog of standard Dirac structures, and we show that the graph of a multisymplectic form determines a multi-Dirac structure. We then discuss the role of multi-Dirac structures in field theory by showing that the implicit field equations obtained from the Hamilton-Pontryagin principle can be described intrinsically using multi-Dirac structures. Furthermore, we show that any multi-Dirac structure naturally gives rise to a multi-Poisson bracket. We treat the case of field theories with nonholonomic constraints, showing that the integrability of the constraints is equivalent to the integrability of the underlying multi-Dirac structure. We finish with a number of illustrative examples, including time-dependent mechanics, nonlinear scalar fields and the electromagnetic field.Comment: 50 pages, v2: correction to prop. 6.1, typographical change