Variational Methods in Nonconservative Phenomena (eBook)

S. E. Jones, B. D. Vujanovic (Autoren)

eBook Download: EPUB | PDF

1989 | 1. Auflage
371 Seiten
Elsevier Science (Verlag)
978-0-08-092642-1 (ISBN)

Infos

This book provides a comprehensive survey of analytic and approximate solutions of problems of applied mechanics, with particular emphasis on nonconservative phenomena. Include

Inhalt

Front Cover 1
Variational Methods in Nonconservative Phenomena 4
Copyright Page 5
Contents 6
Preface 10
Chapter 1. A Brief Account of the Variational Principles of Classical Holonomic Dynamics 12
1.1 Introduction 12
1.2 Constraints and the Forces of Constraint 12
1.3 Actual and Virtual Displacements 15
1.4 D' Alembert's Principle 18
1.5 The Lagrangian Equations with Multipliers 19
1.6 Generalized Coordinates. Lagrangian Equations 23
1.7 A Brief Analysis of the Lagrangian Equations 26
1.8 Hamilton's Principle 37
1.9 Variational Principles Describing the Paths of Conservative Dynamical Systems 45
1.10 Some Elementary Examples Involving Integral Variational Principles 47
1.11 References 54
Chapter 2. Variational Principles and Lagrangians 56
2.1 Introduction 56
2.2 Lagrangians for Systems with One Degree of Freedom 59
2.3 Quadratic Lagrangians for Systems with One Degree of Freedom 63
2.4 Some Other Lagrangians 66
2.5 The Inverse Problem of the Calculus of Variations 71
2.6 Partial Differential Equations 76
2.7 Lagrangians with Vanishing Parameters 79
2.8 Other Variational Principles 80
2.9 References 82
Chapter 3. Conservation Laws 85
3.1 Introduction 85
3.2 Simultaneous and Nonsimultaneous Variations. Infinitesimal Transformations 86
3.3 The Condition of Invariance of Hamilton's Action Integral. Absolute and Gauge Invariance 91
3.4 The Proof of Noether's Theorem. Conservation Laws 94
3.5 The Inertial Motion of a Dynamical System. Killing's Equations 96
3.6 The Generalized Killing Equations 98
3.7 Some Classical Conservation Laws of Dynamical Systems Completely Described by a Lagrangian Function 102
3.8 Examples of Conservation Laws of Dynamical Systems 108
3.9 Some Conservation Laws for the Kepler Problem 116
3.10 Inclusion of Generalized Nonconservative Forces in the Search for Conservation Laws. D'Alembert's Principle 122
3.11 Inclusion of Nonsimultaneous Variations into the Central Lagrangian Equation 128
3.12 The Conditions for Existence of a Conserved Quantity. Conservation Laws of Nonconservative Dynamical Systems 129
3.13 The Generalized Killing Equations for Nonconservative Dynamical Systems 131
3.14 Conservation Laws of Nonconservative Systems Obtained by Means of Variational Principles with Noncommutative Variational Rules 132
3.15 Conservation Laws of Conservative and Nonconservative Dynamical Systems Obtained by Means of the Differential Variational Principles of Gauss and Jourdain 134
3.16 Jourdainian and Gaussian Nonsimultaneous Variations 138
3.17 The Invariance Condition of the Gauss Constraint 140
3.18 An Equivalent Transformation of Jourdain's Principle 143
3.19 The Conservation Laws of Schul'gin and Painlevé 144
3.20 Energy-Like Conservation Laws of Linear Nonconservative Dynamical Systems 146
3.21 Energy-Like Conservation Laws of Linear Dissipative Dynamical Systems 151
3.22 A Special Class of Conservation Laws 155
3.23 References 160
Chapter 4. A Study of the Motion of Conservative and Nonconservative Dynamical Systems by Means of Field Theory 163
4.1 Introduction 163
4.2 Hamilton's Canonical Equations of Motion 164
4.3 Integration of Hamilton's Canonical Equations by Means of the Hamilton–Jacobi Method 173
4.4 Separation of Variables in the Hamilton-Jacobi Equation 186
4.5 Application of the Hamilton–Jacobi Method to Linear Nonconservative Oscillatory Systems 191
4.6 A Field Method for Nonconservative Dynamical Systems 201
4.7 The Complete Solutions of the Basic Field Equation and Their Properties 204
4.8 The Single Solutions of the Basic Field Equation 213
4.9 Illustrative Examples 213
4.10 Applications of the Complete Solutions of the Basic Field Equation to Two-Point Boundary-value Problems 220
4.11 The Potential Method of Arzhanik'h for Nonconservative Dynamical Systems 224
4.12 Applications of the Field Method to Nonlinear Vibration Problems 229
4.13 A Linear Oscillator with Slowly Varying Frequency 246
4.14 References 249
Chapter 5. Variational Principles with Vanishing Parameters and Their Applications 251
5.1 Introduction 251
5.2 A Short Review of Some Variational Formulations Frequently Used in Nonconservative Field Theory 252
5.3 The Variational Principle with Vanishing Parameter 259
5.4 Application of the Direct Method of Partial Integration to the Solution of Linear and Nonlinear Boundary-Value Problems 263
5.5 An Example: A Semi-Infinite Body with a Constant Heat Flux Input 264
5.6 A Semi-Infinite Body with an Arbitrary Heat Flux Input 271
5.7 The Temperature Distribution in a Body Whose End is Kept at Constant Temperature, Temperature-Dependent Thermophysical Coefficients 276
5.8 The Moment–Lagrangian Method 280
5.9 The Temperature Distribution in a Finite Rod with a Nonzero Initial Temperature Distribution 283
5.10 The Temperature Distribution in a Noninsulated Solid 286
5.11 Applications to Laminar Boundary Layer Theory 287
5.12 Applications to Two-Dimensional Boundary Layer Flow of Incompressible, Non-Newtonian Power-Law Fluids 298
5.13 A Variational Solution of the Rayleigh Problem for a Non-Newtonian Power-Law Conducting Fluid 306
5.14 References 313
Chapter 6. Variational Principles with Uncommutative Rules and Their Applications to Nonconservative Phenomena 317
6.1 Introduction 317
6.2 The Variational Principle with Uncommutative Rules 318
6.3 The Connection (Relation) between the Variational Principle with Uncommutative Rules and the Central Lagrangian Equation 320
6.4 The Bogoliubov–Krylov–Mitropolsky Method in Nonlinear Vibration Analysis as a Variational Problem 325
6.5 Applications to Heat Conduction in Solids 328
6.6 References 341
Chapter 7. Applications of Gauss's Principle of Least Constraint to Nonconservative Phenomena 343
7.1 Introduction 343
7.2 Methods of Approximation Based on the Gauss Principle of Least Constraint 344
7.3 Applications to Ordinary Differential Equations 351
7.4 Applications to Transient, Two-Dimensional, Nonlinear Heat Conduction through Prism-Like Infinite Bodies with a Given Cross Section 355
7.5 Melting or Freezing of a Semi-Infinite Solid 359
7.6 A Semi-Infinite Solid with an Arbitrary Heat Flux Input: Gauss's Approach 364
7.7 A Nonconservative Convective Problem 368
7.8 References 371
Author Index 374
Index 378
Mathematics in Science and Engineering 382

Produkt-Details

• Erscheint lt. Verlag: 1.5.1989
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• ISBN-10: 0-08-092642-8 / 0080926428
• ISBN-13: 978-0-08-092642-1 / 9780080926421