Variational and Extremum Principles in Macroscopic Systems


  • Stanislaw Sieniutycz, Warsaw University of Technology, Faculty of Chemical and Process Engineering, Poland
  • Henrik Farkas, Budapest University of Technology and Economics

Recent years have seen a growing trend to derive models of macroscopic phenomena encountered in the fields of engineering, physics, chemistry, ecology, self-organisation theory and econophysics from various variational or extremum principles. Through the link between the integral extremum of a functional and the local extremum of a function (explicit, for example, in the Pontryagin’s maximum principle variational and extremum principles are mutually related. Thus it makes sense to consider them within a common context. The main goal of the present book is to collect various mathematical formulations and examples of physical reasoning that involve both basic theoretical aspects and applications of variational and extremum approaches to systems of the macroscopic world.The first part of the book is focused on the theory, whereas the second focuses on applications. The unifying variational approach is used to derive the balance or conservation equations, phenomenological equations linking fluxes and forces, equations of change for processes with coupled transfer of energy and substance, and optimal conditions for energy management.
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Readership is extremely broad and includes applied mathematicians, mathematical physicists, applied physicists, chemists, geologists, ecologists, mechanical engineers, chemical engineers, economists and system theorists, undergraduates, graduates and instructors, both from academia and industry.


Book information

  • Published: March 2005
  • Imprint: ELSEVIER
  • ISBN: 978-0-08-044488-8


"This book will be valuable for mathematicians, physicists, chemists, and engineers, in particular those involoved in the application of the mathematical and thermodynamic knowledge to systems with energy generation and transport, solar radiation, chemical waves, liquid crystals, thermo-elastic media, composites, multiphase flows, porous media, membrane transfer, microeconomics, etc." -INTERNATIONAL JOURNAL OF NONLINEAR SCIENCES AND NUMERICAL SIMULATION, 2006

Table of Contents

List of contributorsPreface
Part I: Theory
I.1. Progress in Variational Formulations for Macroscopic ProcessesI.2. Lagrange-Formalism and Thermodynamics of Irreversible Processes: The 2nd Law of Thermodynamics and the Principle of Least Entropy Production as Straightforward Structures in Lagrange-Formalism I.3. Fundamental Problems of Variational Principles: Objectivity, Symmetries and ConstructionI.4. Semi-Inverse Method for Establishment of Variational Principles for Incremental Thermoelasticity with VoidsI.5. Variational Formulations of Relativistic Elasticity and Thermo-Elasticity I.6. The Geometric Variational Framework for Entropy in General Relativity I.7. Translational and Rotational Motion of a Unaxial Liquid Crystal as Derived Using Hamilton’s Principle of Least ActionI.8. An Introduction to Variational Derivation of the Pseudo-Momentum Conservation in Thermo-Hydrodynamics I.9. Towards a Variational Mechanics of Dissipative Continua?I.10. On the Principle of Least Action and its Role in the Alternative Theory of Non-Equilibrium ProcessesI.11. Variational Principles for the Linearly Damped Flow of Barotropic and Madelung-Type Fluids I.12. Least Action Principle for Dissipative Processes I.13. Hamiltonian Formulation as a Basis of Quantized Thermal Processes I.14. Conservation Laws and Variational Conditions for Wave Propagation in Planarly-Stratified MediaI.15. Master Equations and Path-Integral Formulation of Variational Principles for ReactionsI.16. Variational Principles for the Speed of Traveling Fronts of Reaction-Diffusion EquationsI.17. The Fermat Principle and Chemical WavesPart II: ApplicationsStatistical Physics and ThermodynamicsII.1. Fisher Variational Principle and Thermodynamics II.2. Generalized Entropy and the Hamiltonian Structure of Statistical MechanicsHydrodynamics and Continuum MechanicsII.3. Some Observations of Entropy Extrema in Physical ProcessesII.4. A Variational Principle for the Drag in Linear HydrodynamicsII.5. A Variational Principle for the Impinging Streams Problem II.6. Variational Principles in Stability Analysis of Composite StructuresTransport Phenomena and Energy ConversionII.7. Field Variational Principles for Irreversible Energy and Mass Transfer II.8. Variational Principles for Irreversible Hyperbolic TransportII.9. A Variational Principle for Transport Processes in Continuous Systems: Derivation and Application II.10. Do the Navier-Stokes Equations Admit a Variational Formulation?II.11. Entropy Generation Minimization in Steady State Heat Conduction II.12. The Nonequilibrium Thermodynamics of Radiation InteractionII.13. Optimal Finite-Time Endoreversible Processes- General Theory and Applications II.14. Evolutionary Energy Method (EEM) – An AerothermoservoelectrostaticApplicationEcologyII.15. Maximization of Eco-Exergy in EcosystemsSelforganization and EconophysicsII.16. Self-Organized Criticality within the Framework of Variational PrincipleII.17. Extremum Criteria for Nonequilibrium States of Dissipative Macroeconomic systems II.18. Extremal Principles and Limiting Possibilities of Open Thermodynamic and Economic Systems Glossary of principal symbolsIndex