Variational Principles of Continuum Mechanics

Cover13; -- Preface -- Acknowledgements -- Contents -- I. Fundamentals -- Contents -- II. Applications -- Part I Fundamentals -- to 1 Variational Principles -- Prehistory -- Mopertuis Variational Principle -- Euler's Calculus of Variations -- Lagrange Variational Principle -- Jacobi Variational Principle -- Hamilton Variational Principle -- Hamiltonian Equations -- Physical Meaning of the Least Action Principle -- to 2 Thermodynamics -- Thermodynamic Description -- Temperature -- Entropy -- Entropy and Probability -- Gibbs Principles -- Nonequilibrium Processes -- Secondary Thermodynamics and Higher Order Thermodynamics -- to 3 Continuum Mechanics -- Continuum Kinematics -- Basic Laws of Continuum Mechanics -- Classical Continuum Models -- Thermodynamic Formalism -- to 4 Principle of Least Action in Continuum Mechanics -- Variation of Integral Functionals -- Variations of Kinematic Parameters -- Principle of Least Action -- Variational Equations -- Models with High Derivatives -- Tensor Variations -- to 5 Direct Methods of Calculus of Variations -- Introductory Remarks -- Quadratic Functionals -- Existence of the Minimizing Element -- Uniqueness of the Minimizing Element -- Upper and Lower Estimates -- Dual Variational Principles -- Legendre and Young-Fenchel Transformations -- Examples of Dual Variational Principles -- Hashin-Strikman Variational Principle -- Variational Problems with Constraints -- Variational-Asymptotic Method -- Variational Problems and Functional Integrals -- Miscellaneous -- Part II Variational Features of Classical Continuum Models -- to 6 Statics of a Geometrically Linear Elastic Body -- Gibbs Principle -- Boundedness from Below -- Complementary Energy -- Reissner Variational Principle -- Physically Linear Elastic Body -- Castigliano Variational Principle -- Hashin-Strikman Variational Principle -- Internal Stresses -- Thermoelasticity -- Dislocations -- Continuously Distributed Dislocations -- to 7 Statics of a Geometrically Nonlinear Elastic Body -- Energy Functional -- Gibbs Principle -- Dual Variational Principle -- Phase Equilibrium of Elastic Bodies -- to 8 Dynamics of Elastic Bodies -- Least Action vs Stationary Action -- Nonlinear Eigenvibrations -- Linear Vibrations: The Rayleigh Principle -- The Principle of Least Action in Eulerian Coordinates -- to 9 Ideal Incompressible Fluid -- Least Action Principle -- General Features of Solutions of Momentum Equations -- Variational Principles in Eulerian Coordinates -- Potential Flows -- Variational Features of Kinetic Energy in Vortex Flows -- Dynamics of Vortex Lines -- Quasi-Two-Dimensional and Two-Dimensional Vortex Flows -- Dynamics of Vortex Filaments in Unbounded Space -- Vortex Sheets -- Symmetry of the Action Functional and the Integrals of Motion -- Variational Principles for Open Flows -- to 10 Ideal Compressible Fluid -- Variational Principles in Lagrangian Coordinates -- General Features of Dynamics of Compressible Fluid -- Variational Principles in Eulerian Coordinates -- Potential Flows -- Incompressible Fluid as a Limit Case of Compressible Fluid -- to 11 Steady Motion of Ideal Fluid and Elastic Body -- The Kinematics of Steady Flow -- Steady Motion with Impenetrable Boundaries -- O.