Variational and Extremum Principles in Macroscopic Systems

1st Edition

Print ISBN: 9780080972299
eBook ISBN: 9780080456140
Imprint: Elsevier Science
Published Date: 30th March 2005
Page Count: 810
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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.

Key Features

A unique multidisciplinary synthesis of variational and extremum principles in theory and application.
A comprehensive review of current and past achievements in variational formulations for macroscopic processes.
*Uses Lagrangian and Hamiltonian formalisms as a basis for the exposition of novel approaches to transfer and conversion of thermal, solar and chemical energy.


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.

Table of Contents

List of contributors Preface Part I: Theory

I.1. Progress in Variational Formulations for Macroscopic Processes I.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 Construction I.4. Semi-Inverse Method for Establishment of Variational Principles for Incremental Thermoelasticity with Voids I.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 Action I.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 Processes I.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 Media I.15. Master Equations and Path-Integral Formulation of Variational Principles for Reactions I.16. Variational Principles for the Speed of Traveling Fronts of Reaction-Diffusion Equations I.17. The Fermat Principle and C


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"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