THE WONDERS OF MODERN MECHANISM
Part I Fundamentals 1 Variational Principles 3 11 Prehistory 3 12 Mopertuis Variational Principle 11 13 Euler’s Calculus of Variations 15 14 Lagrange Variational Principle 20 15 Jacobi Variational Principle 26 16 Hamilton Variational Principle 26 17 Hamiltonian Equations 32 18 Physical Meaning of the Least Action Principle 36 2 Thermodynamics 45 21 Thermodynamic Description 45 22 Temperature 47 23 Entropy 51 24 Entropy and Probability 59 25 Gibbs Principles 59 26 Nonequilibrium Processes 60 27 Secondary Thermodynamics and Higher Order Thermodynamics 64 3 Continuum Mechanics 67 31 Continuum Kinematics 67 32 Basic Laws of Continuum Mechanics 93 33 Classical Continuum Models 98 34 Thermodynamic Formalism 112 4 Principle of Least Action in Continuum Mechanics 117 41 Variation of Integral Functionals 117 42 Variations of Kinematic Parameters 121 43 Principle of Least Action 125 44 Variational Equations 128 45 Models with High Derivatives 134 46 Tensor Variations 136 5 Direct Methods of Calculus of Variations 149 51 Introductory Remarks 150 52 Quadratic Functionals 163 53 Existence of the Minimizing Element 167 54 Uniqueness of the Minimizing Element 168 55 Upper and Lower Estimates 172 56 Dual Variational Principles 178 57 Legendre and Young-Fenchel Transformations 181 58 Examples of Dual Variational Principles 201 59 Hashin-Strikman Variational Principle 216 510 Variational Problems with Constraints 224 511 Variational-Asymptotic Method 243 512 Variational Problems and Functional Integrals 270 513 Miscellaneous 278 Part II Variational Features of Classical Continuum Models 6 Statics of a Geometrically Linear Elastic Body 285 61 Gibbs Principle 285 62 Boundedness from Below 289 63 Complementary Energy 293 64 Reissner Variational Principle 294 65 Physically Linear Elastic Body 294 66 Castigliano Variational Principle 298 67 Hashin-Strikman Variational Principle 306 68 Internal Stresses 318 69 Thermoelasticity 321 610 Dislocations 323 611 Continuously Distributed Dislocations 328 7 Statics of a Geometrically Nonlinear Elastic Body 341 71 Energy Functional 341 72 Gibbs Principle 350 73 Dual Variational Principle 355 74 Phase Equilibrium of Elastic Bodies 369 8 Dynamics of Elastic Bodies 375 81 Least Action vs Stationary Action 375 82 Nonlinear Eigenvibrations 377 83 Linear Vibrations: The Rayleigh Principle 379 84 The Principle of Least Action in Eulerian Coordinates 381 9 Ideal Incompressible Fluid 389 91 Least Action Principle 389 92 General Features of Solutions of Momentum Equations 392 93 Variational Principles in Eulerian Coordinates 396 94 Potential Flows 405 95 Variational Features of Kinetic Energy in Vortex Flows 408 96 Dynamics of Vortex Lines 414 97 Quasi-Two-Dimensional and Two-Dimensional Vortex Flows 427 98 Dynamics of Vortex Filaments in Unbounded Space 433 99 Vortex Sheets 444 910 Symmetry of the Action Functional and the Integrals ofMotion 446 911 Variational Principles for Open Flows 453 10 Ideal Compressible Fluid 455 101 Variational Principles in Lagrangian Coordinates 455 102 General Features of Dynamics of Compressible Fluid 457 103 Variational Principles in Eulerian Coordinates 461 104 Potential Flows 468 105 Incompressible Fluid as a Limit Case of Compressible Fluid 470 11 Steady Motion of Ideal Fluid and Elastic Body 473 111 The Kinematics of Steady Flow 473 112 Steady Motion with Impenetrable Boundaries 475 113 Open Steady Flows of Ideal Fluid 479 114 Two-Dimensional Flows 483 115 Variational Principles on the Set of Equivortical Flows 484 116 Potential Flows 490 117 Regularization of Functionals in Unbounded Domains 493 12 Principle of Least Dissipation 495 121 Heat Conduction 495 122 Creeping Motion of Viscous Fluid 498 123 Ideal Plasticity 502 124 Fluctuations and Variations in Steady Non-Equilibrium Processes 505 13 Motion of Rigid Bodies in Fluids 509 131 Motion of a Rigid Body in Creeping Flow of Viscous Fluid 509 132 Motion of a Body in Ideal Incompressible Fluid 514 133 Motion of a Body in a Viscous Fluid 521