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

François Gay-Balmaz, Hiroaki Yoshimura*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

55 Citations (Scopus)


In this paper, we present a Lagrangian variational formulation for nonequilibrium thermodynamics. This formulation is an extension of Hamilton's principle of classical mechanics that allows the inclusion of irreversible phenomena. The irreversibility is encoded into a nonlinear phenomenological constraint given by the expression of the entropy production associated to all the irreversible processes involved. From a mathematical point of view, our variational formulation may be regarded as a generalization to nonequilibrium thermodynamics of the Lagrange–d'Alembert principle used in nonlinear nonholonomic mechanics, where the conventional Lagrange–d'Alembert principle cannot be applied since the nonlinear phenomenological constraint and its associated variational constraint must be treated separately. 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 to systematically 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 formulation of discrete systems to the case of continuum systems.

Original languageEnglish
Pages (from-to)169-193
Number of pages25
JournalJournal of Geometry and Physics
Publication statusPublished - 2017 Jan 1


  • Discrete systems
  • Irreversible processes
  • Lagrangian formulation
  • Nonequilibrium thermodynamics
  • Nonholonomic constraints
  • Variational formulation

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Geometry and Topology


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