Methods of Analytical Dynamics / Edition 1

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A balanced presentation that encompasses both formalism and structure in analytical dynamics, this text also addresses solution methods. Its remarkably broad and comprehensive exploration of the subject employs an approach as natural as it is logical. In addition to material usually covered in graduate courses in dynamics and nonlinear mechanics, Methods of Analytical Dynamics presents selected modern applications. Contents include discussions of fundamentals of Newtonian and analytical mechanics, motion relative to rotating reference frames, rigid body dynamics, behavior of dynamical systems, geometric theory, stability of multi-degree-of-freedom autonomous and nonautonomous systems, and analytical solutions by perturbation techniques. Later chapters cover transformation theory, the Hamilton-Jacobi equation, theory and applications of the gyroscope, and problems in celestial mechanics and spacecraft dynamics. Two helpful appendixes offer additional information on dyadics and elements of topology and modern analysis.
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Product Details

  • ISBN-13: 9780070414556
  • Publisher: McGraw-Hill Higher Education
  • Publication date: 1/28/1970
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 485

Table of Contents

Preface vii
Chapter 1 Fundamentals of Newtonian Mechanics 1
1.1 Historical Survey of Mechanics 1
1.2 Newton's Laws 9
1.3 Impulse and Momentum 12
1.4 Moment of a Force and Angular Momentum 12
1.5 Work and Energy 14
1.6 Energy Diagrams 17
1.7 Systems of Particles 21
1.8 The Two-Body Central Force Problem 25
1.9 The Inverse Square Law. Orbits of Planets and Satellites 30
1.10 Scattering by a Repulsive Central Force 37
Problems 40
Suggested References 44
Chapter 2 Fundamentals of Analytical Mechanics 45
2.1 Degrees of Freedom. Generalized Coordinates 46
2.2 Systems with Constraints 48
2.3 The Stationary Value of a Function 53
2.4 The Stationary Value of a Definite Integral 55
2.5 The Principle of Virtual Work 59
2.6 D'Alembert's Principle 65
2.7 Hamilton's Principle 66
2.8 Lagrange's Equations of Motion 72
2.9 Lagrange's Equations for Impulsive Forces 79
2.10 Conservation Laws 82
2.11 Routh's Method for the Ignoration of Coordinates 85
2.12 Rayleigh's Dissipation Function 88
2.13 Hamilton's Equations 91
Problems 97
Suggested References 100
Chapter 3 Motion Relative to Rotating Reference Frames 101
3.1 Transformation of Coordinates 102
3.2 Rotating Coordinate Systems 104
3.3 Expressions for the Motion in Terms of Moving Reference Frames 110
3.4 Motion Relative to the Rotating Earth 112
3.5 Motion of a Free Particle Relative to the Rotating Earth 114
3.6 Foucault's Pendulum 116
Problems 119
Suggested References 121
Chapter 4 Rigid Body Dynamics 122
4.1 Kinematics of a Rigid Body 123
4.2 The Linear and Angular Momentum of a Rigid Body 126
4.3 Translation Theorem for the Angular Momentum 130
4.4 The Kinetic Energy of a Rigid Body 132
4.5 Principal Axes 134
4.6 The Equations of Motion for a Rigid Body 137
4.7 Euler's Equations of Motion 138
4.8 Euler's Angles 140
4.9 Moment-Free Inertially Symmetric Body 143
4.10 General Case of a Moment-Free Body 147
4.11 Motion of a Symmetric Top 149
4.12 The Lagrangian Equations for Quasi-Coordinates 157
4.13 The Equations of Motion Referred to an Arbitrary System of Axes 160
4.14 The Rolling of a Coin 162
Problems 164
Suggested References 169
Chapter 5 Behavior of Dynamical Systems. Geometric Theory 170
5.1 Fundamental Concepts 171
5.2 Motion of Single-Degree-of-Freedom Autonomous Systems about Equilibrium Points 178
5.3 Conservative Systems. Motion in the Large 189
5.4 The Index of Poincare 195
5.5 Limit Cycles of Poincare 198
Problems 206
Suggested References 208
Chapter 6 Stability of Multi-Degree-of-Freedom Autonomous Systems 209
6.1 General Linear Systems 210
6.2 Linear Autonomous Systems 217
6.3 Stability of Linear Autonomous Systems. Routh-Hurwitz Criterion 222
6.4 The Variational Equations 225
6.5 Theorem on the First-Approximation Stability 226
6.6 Variation from Canonical Systems. Constant Coefficients 229
6.7 The Liapunov Direct Method 231
6.8 Geometric Interpretation of the Liapunov Direct Method 239
6.9 Stability of Canonical Systems 243
6.10 Stability in the Presence of Gyroscopic and Dissipative Forces 252
6.11 Construction of Liapunov Functions for Linear Autonomous Systems 258
Problems 260
Suggested References 262
Chapter 7 Nonautonomous Systems 263
7.1 Linear Systems with Periodic Coefficients. Floquet's Theory 264
7.2 Stability of Variational Equations with Periodic Coefficients 271
7.3 Orbital Stability 272
7.4 Variation from Canonical Systems. Periodic Coefficients 273
7.5 Second-Order Systems with Periodic Coefficients 277
7.6 Hill's Infinite Determinant 280
7.7 Mathieu's Equation 282
7.8 The Liapunov Direct Method 288
Suggested References 292
Chapter 8 Analytical Solutions by Perturbation Techniques 293
8.1 The Fundamental Perturbation Technique 294
8.2 Secular Terms 297
8.3 Lindstedt's Method 299
8.4 The Krylov-Bogoliubov-Mitropolsky (KBM) Method 302
8.5 A Perturbation Technique Based on Hill's Determinants 309
8.6 Periodic Solutions of Nonautonomous Systems. Duffing's Equation 313
8.7 The Method of Averaging 322
Problems 327
Suggested References 328
Chapter 9 Transformation Theory. The Hamilton-Jacobi Equation 329
9.1 The Principle of Least Action 330
9.2 Contact Transformations 334
9.3 Further Extensions of the Concept of Contact Transformations 339
9.4 Integral Invariants 346
9.5 The Lagrange and Poisson Brackets 349
9.6 Infinitesimal Contact Transformations 352
9.7 The Hamilton-Jacobi Equation 355
9.8 Separable Systems 361
9.9 Action and Angle Variables 365
9.10 Perturbation Theory 372
Problems 378
Suggested References 380
Chapter 10 The Gyroscope: Theory and Applications 381
10.1 Oscillations of a Symmetric Gyroscope 382
10.2 Effect of Gimbal Inertia on the Motion of a Free Gyroscope 386
10.3 Effect of Rotor Shaft Flexibility on the Frequency of Oscillation of a Free Gyroscope 389
10.4 The Gyrocompass 393
10.5 The Gyropendulum. Schuler Tuning 398
10.6 Rate and Integrating Gyroscopes 403
Problems 406
Suggested References 407
Chapter 11 Problems in Celestial Mechanics 408
11.1 Kepler's Equation. Orbit Determination 409
11.2 The Many-Body Problem 413
11.3 The Three-Body Problem 416
11.4 The Restricted Three-Body Problem 420
11.5 Stability of Motion Near the Lagrangian Points 425
11.6 The Equations of Relative Motion. Disturbing Function 428
11.7 Gravitational Potential and Torques for an Arbitrary Body 430
11.8 Precession and Nutation of the Earth's Polar Axis 438
11.9 Variation of the Orbital Elements 442
11.10 The Resolution of the Disturbing Function 447
Problems 450
Suggested References 451
Chapter 12 Problems in Spacecraft Dynamics 452
12.1 Transfer Orbits. Changes in the Orbital Elements Due to a Small Impulse 453
12.2 Perturbations of a Satellite Orbit in the Gravitational Field of an Oblate Planet 457
12.3 The Effect of Atmospheric Drag on Satellite Orbits 463
12.4 The Attitude Motion of Orbiting Satellites. General Considerations 466
12.5 The Attitude Stability of Earth-Pointing Satellites 470
12.6 The Attitude Stability of Spinning Symmetrical Satellites 475
12.7 Variable-Mass Systems 483
12.8 Rocket Dynamics 487
Problems 491
Suggested References 492
Appendix A Dyadics 494
Appendix B Elements of Topology and Modern Analysis 497
B.1 Sets and Functions 498
B.2 Metric Spaces 501
B.3 Topological Spaces 504
Suggested References 506
Name Index 507
Subject Index 511
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