Tensor products of Dirac structures and interconnection in Lagrangian mechanics

Jacobs, Henry O.; Yoshimura, Hidetoshi

doi:10.3934/jgm.2014.6.67

Cited by 15 publications

(14 citation statements)

References 39 publications

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“…3.4. This class of non-simple interconnected systems extends the class of interconnected mechanical systems (see, Jacobs and Yoshimura [2014]) to include the irreversible processes. In addition to the irreversible processes of friction and mass transfer described earlier, these systems can also involve the process of heat conduction.…”

Section: Adiabatically Closed Non-simple Thermodynamic Systemsmentioning

confidence: 98%

“…We now consider a general finite-dimensional system

composed of interconnected simple thermodynamic systems

, as illustrated in Figure 4 . This class of non-simple interconnected systems extends the class of interconnected mechanical systems (see [ 62 ]) to include the irreversible processes. In addition to the irreversible processes of friction and mass transfer described earlier, these systems can also involve the process of heat conduction.…”

Section: Variational Formulation For Discrete Thermodynamic Systemmentioning

confidence: 99%

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From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective

Gay–Balmaz

Yoshimura

2018

Entropy

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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.1 The tangent bundle of a manifold Q is the manifold T Q given by the collection of all tangent vectors in Q. As a set it is given by the disjoint union of the tangent spaces of Q, that is, T Q = q∈Q TqQ, where TqQ is the tangent space to Q at q. The elements in TqQ are denoted (q, v).

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Section: Adiabatically Closed Non-simple Thermodynamic Systemsmentioning

confidence: 98%

“…We now consider a general finite-dimensional system

composed of interconnected simple thermodynamic systems

Section: Variational Formulation For Discrete Thermodynamic Systemmentioning

confidence: 99%

From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective

Gay–Balmaz

Yoshimura

2018

Entropy

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“…There are more general cases of discrete systems in which nonholonomic constraints due to the mechanical interconnection have to be considered. For purely mechanical systems this issue has been considered from a variational perspective in Jacobs and Yoshimura [2014]. The extension of this approach to the setting of nonequilibrium thermodynamics will be explored in a future work.…”

Section: Nonequilibrium Thermodynamics Of General Discrete Systemsmentioning

confidence: 99%

A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems

Gay–Balmaz

Yoshimura

2017

Journal of Geometry and Physics

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120

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In this paper, we present a Lagrangian variational formulation for nonequilibrium thermodynamics. This formulation is an extension of the Hamilton principle in classical mechanics that allows the inclusion of irreversible phenomena. The irreversibility is encoded into a nonlinear phenomenological constraint given by the expression of entropy production associated to all the irreversible processes involved. Hence from a mathematical point of view, our variational formulation may be regarded as a generalization of the Lagrange-d'Alembert principle used in nonlinear nonholonomic mechanics to the nonequilibrium thermodynamics, where the conventional Lagrange-d'Alembert principle cannot be applied since the nonlinear phenomenological constraint and its associated variational constraint must be separately treated. In our approach, to deal with the nonlinear nonholonomic constraint, we introduce a variable called the thermodynamic displacement associated to each irreversible process. This allows us systematically to define the corresponding variational constraint. In Part I, our variational theory is illustrated with various examples of discrete systems such as mechanical systems with friction, matter transfer, electric circuits, chemical reactions, and diffusion across membranes. In Part II of the present paper, we will extend our variational theory of discrete systems to the case of of continuum systems. Contents 1 Introduction 2 2 Some preliminaries 5 3 Variational formulation for nonequilibrium thermodynamics of simple systems 9 3.

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“…The integrability condition is too restrictive to describe, for instance, nonholonomic systems, and for this reason we don't include the integrability in the general definition of a Dirac structure. This is the same convention as in [van der Schaft and Jeltsema, 2014] or [Jacobs and Yoshimura, 2014].…”

Section: Dirac Structures On a Manifoldmentioning

confidence: 99%

“…This construction is also known by some authors as the bowtie product of D 1 and D 2 (denoted D 1 ⊲⊳ D 2 ). For applications of the tensor product in the interconnection of Dirac structures, we refer to [Jacobs and Yoshimura, 2014] and references therein.…”

Section: Backward Input-output Port-hamiltonian Systemsmentioning

confidence: 99%

New insights in the geometry and interconnection of port-Hamiltonian systems

Barbero-Lin͂án¹,

Cendra²,

Andrés³

et al. 2018

J. Phys. A: Math. Theor.

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We discuss a new geometric construction of port-Hamiltonian systems. Using this framework, we revisit the notion of interconnection providing it with an intrinsic description. Special emphasis on theoretical and applied examples is given throughout the paper to show the applicability and the novel contributions of the proposed framework.2 Dirac structures: forward and backward 3 Conventions and terminology. All objects in this paper are assumed to be smooth, unless otherwise stated. The maps between vector spaces and vector bundles are assumed to be linear. If π E : E → M is a vector bundle, a distribution D is an assignment of a subspace x → D x ⊂ E x for each x ∈ M . We will say that the distribution D is regular if dim(D x ) is independent of x. Regular (smooth) distributions are called subbundles. In general, our conventions agree with those on [Abraham and Marsden, 1978].Acknowledgements.

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Tensor products of Dirac structures and interconnection in Lagrangian mechanics

Cited by 15 publications

References 39 publications

From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective

From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective

A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems

New insights in the geometry and interconnection of port-Hamiltonian systems

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