Variational Methods in Nonconservative Phenomena

Vujanovic’, B. D.; Jones, S. E.0-12-728450-8

Table of contents :
Variational Methods in Nonconservative Phenomena……Page 4
Copyright Page……Page 5
Contents……Page 6
Preface……Page 10
1.2 Constraints and the Forces of Constraint……Page 12
1.3 Actual and Virtual Displacements……Page 15
1.4 D’ Alembert’s Principle……Page 18
1.5 The Lagrangian Equations with Multipliers……Page 19
1.6 Generalized Coordinates. Lagrangian Equations……Page 23
1.7 A Brief Analysis of the Lagrangian Equations……Page 26
1.8 Hamilton’s Principle……Page 37
1.9 Variational Principles Describing the Paths of Conservative Dynamical Systems……Page 45
1.10 Some Elementary Examples Involving Integral Variational Principles……Page 47
1.11 References……Page 54
2.1 Introduction……Page 56
2.2 Lagrangians for Systems with One Degree of Freedom……Page 59
2.3 Quadratic Lagrangians for Systems with One Degree of Freedom……Page 63
2.4 Some Other Lagrangians……Page 66
2.5 The Inverse Problem of the Calculus of Variations……Page 71
2.6 Partial Differential Equations……Page 76
2.7 Lagrangians with Vanishing Parameters……Page 79
2.8 Other Variational Principles……Page 80
2.9 References……Page 82
3.1 Introduction……Page 85
3.2 Simultaneous and Nonsimultaneous Variations. Infinitesimal Transformations……Page 86
3.3 The Condition of Invariance of Hamilton’s Action Integral. Absolute and Gauge Invariance……Page 91
3.4 The Proof of Noether’s Theorem. Conservation Laws……Page 94
3.5 The Inertial Motion of a Dynamical System. Killing’s Equations……Page 96
3.6 The Generalized Killing Equations……Page 98
3.7 Some Classical Conservation Laws of Dynamical Systems Completely Described by a Lagrangian Function……Page 102
3.8 Examples of Conservation Laws of Dynamical Systems……Page 108
3.9 Some Conservation Laws for the Kepler Problem……Page 116
3.10 Inclusion of Generalized Nonconservative Forces in the Search for Conservation Laws. D’Alembert’s Principle……Page 122
3.11 Inclusion of Nonsimultaneous Variations into the Central Lagrangian Equation……Page 128
3.12 The Conditions for Existence of a Conserved Quantity. Conservation Laws of Nonconservative Dynamical Systems……Page 129
3.13 The Generalized Killing Equations for Nonconservative Dynamical Systems……Page 131
3.14 Conservation Laws of Nonconservative Systems Obtained by Means of Variational Principles with Noncommutative Variational Rules……Page 132
3.15 Conservation Laws of Conservative and Nonconservative Dynamical Systems Obtained by Means of the Differential Variational Principles of Gauss and Jourdain……Page 134
3.16 Jourdainian and Gaussian Nonsimultaneous Variations……Page 138
3.17 The Invariance Condition of the Gauss Constraint……Page 140
3.18 An Equivalent Transformation of Jourdain’s Principle……Page 143
3.19 The Conservation Laws of Schul’gin and Painlevé……Page 144
3.20 Energy-Like Conservation Laws of Linear Nonconservative Dynamical Systems……Page 146
3.21 Energy-Like Conservation Laws of Linear Dissipative Dynamical Systems……Page 151
3.22 A Special Class of Conservation Laws……Page 155
3.23 References……Page 160
4.1 Introduction……Page 163
4.2 Hamilton’s Canonical Equations of Motion……Page 164
4.3 Integration of Hamilton’s Canonical Equations by Means of the Hamilton–Jacobi Method……Page 173
4.4 Separation of Variables in the Hamilton-Jacobi Equation……Page 186
4.5 Application of the Hamilton–Jacobi Method to Linear Nonconservative Oscillatory Systems……Page 191
4.6 A Field Method for Nonconservative Dynamical Systems……Page 201
4.7 The Complete Solutions of the Basic Field Equation and Their Properties……Page 204
4.9 Illustrative Examples……Page 213
4.10 Applications of the Complete Solutions of the Basic Field Equation to Two-Point Boundary-value Problems……Page 220
4.11 The Potential Method of Arzhanik’h for Nonconservative Dynamical Systems……Page 224
4.12 Applications of the Field Method to Nonlinear Vibration Problems……Page 229
4.13 A Linear Oscillator with Slowly Varying Frequency……Page 246
4.14 References……Page 249
5.1 Introduction……Page 251
5.2 A Short Review of Some Variational Formulations Frequently Used in Nonconservative Field Theory……Page 252
5.3 The Variational Principle with Vanishing Parameter……Page 259
5.4 Application of the Direct Method of Partial Integration to the Solution of Linear and Nonlinear Boundary-Value Problems……Page 263
5.5 An Example: A Semi-Infinite Body with a Constant Heat Flux Input……Page 264
5.6 A Semi-Infinite Body with an Arbitrary Heat Flux Input……Page 271
5.7 The Temperature Distribution in a Body Whose End is Kept at Constant Temperature, Temperature-Dependent Thermophysical Coefficients……Page 276
5.8 The Moment–Lagrangian Method……Page 280
5.9 The Temperature Distribution in a Finite Rod with a Nonzero Initial Temperature Distribution……Page 283
5.10 The Temperature Distribution in a Noninsulated Solid……Page 286
5.11 Applications to Laminar Boundary Layer Theory……Page 287
5.12 Applications to Two-Dimensional Boundary Layer Flow of Incompressible, Non-Newtonian Power-Law Fluids……Page 298
5.13 A Variational Solution of the Rayleigh Problem for a Non-Newtonian Power-Law Conducting Fluid……Page 306
5.14 References……Page 313
6.1 Introduction……Page 317
6.2 The Variational Principle with Uncommutative Rules……Page 318
6.3 The Connection (Relation) between the Variational Principle with Uncommutative Rules and the Central Lagrangian Equation……Page 320
6.4 The Bogoliubov–Krylov–Mitropolsky Method in Nonlinear Vibration Analysis as a Variational Problem……Page 325
6.5 Applications to Heat Conduction in Solids……Page 328
6.6 References……Page 341
7.1 Introduction……Page 343
7.2 Methods of Approximation Based on the Gauss Principle of Least Constraint……Page 344
7.3 Applications to Ordinary Differential Equations……Page 351
7.4 Applications to Transient, Two-Dimensional, Nonlinear Heat Conduction through Prism-Like Infinite Bodies with a Given Cross Section……Page 355
7.5 Melting or Freezing of a Semi-Infinite Solid……Page 359
7.6 A Semi-Infinite Solid with an Arbitrary Heat Flux Input: Gauss’s Approach……Page 364
7.7 A Nonconservative Convective Problem……Page 368
7.8 References……Page 371
Author Index……Page 374
Index……Page 378
Mathematics in Science and Engineering……Page 382