Methods of Analytical Dynamics

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ISBN: 9780486432397 | 0486432394
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Copyright: 5/20/2010

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A balanced presentation that encompasses both formalism and structure in analytical dynamics, this graduate-level text discusses fundamentals of Newtonian and analytical mechanics, rigid body dynamics, behavior of dynamical systems, geometric theory, analytical solutions, transformation theory, problems in celestial mechanics and spacecraft dynamics, and more. 128 figures. 1970 edition.

Preface

vii

Chapter 1 Fundamentals of Newtonian Mechanics

(44)

1.1 Historical Survey of Mechanics

(8)

1.2 Newton's Laws

(3)

1.3 Impulse and Momentum

(1)

1.4 Moment of a Force and Angular Momentum

(2)

1.5 Work and Energy

(3)

1.6 Energy Diagrams

(4)

1.7 Systems of Particles

(4)

1.8 The Two-Body Central Force Problem

(5)

1.9 The Inverse Square Law. Orbits of Planets and Satellites

(7)

1.10 Scattering by a Repulsive Central Force

(3)

Problems

(4)

Suggested References

(1)

Chapter 2 Fundamentals of Analytical Mechanics

(56)

2.1 Degrees of Freedom. Generalized Coordinates

(2)

2.2 Systems with Constraints

(5)

2.3 The Stationary Value of a Function

(2)

2.4 The Stationary Value of a Definite Integral

(4)

2.5 The Principle of Virtual Work

(6)

2.6 D'Alembert's Principle

(1)

2.7 Hamilton's Principle

(6)

2.8 Lagrange's Equations of Motion

(7)

2.9 Lagrange's Equations for Impulsive Forces

(3)

2.10 Conservation Laws

(3)

2.11 Routh's Method for the Ignoration of Coordinates

(3)

2.12 Rayleigh's Dissipation Function

(3)

2.13 Hamilton's Equations

(6)

Problems

(3)

Suggested References

100

(1)

Chapter 3 Motion Relative to Rotating Reference Frames

101

(21)

3.1 Transformation of Coordinates

102

(2)

3.2 Rotating Coordinate Systems

104

(6)

3.3 Expressions for the Motion in Terms of Moving Reference Frames

110

(2)

3.4 Motion Relative to the Rotating Earth

112

(2)

3.5 Motion of a Free Particle Relative to the Rotating Earth

114

(2)

3.6 Foucault's Pendulum

116

(3)

Problems

119

(2)

Suggested References

121

(1)

Chapter 4 Rigid Body Dynamics

122

(48)

4.1 Kinematics of a Rigid Body

123

(3)

4.2 The Linear and Angular Momentum of a Rigid Body

126

(4)

4.3 Translation Theorem for the Angular Momentum

130

(2)

4.4 The Kinetic Energy of a Rigid Body

132

(2)

4.5 Principal Axes

134

(3)

4.6 The Equations of Motion for a Rigid Body

137

(1)

4.7 Euler's Equations of Motion

138

(2)

4.8 Euler's Angles

140

(3)

4.9 Moment-Free Inertially Symmetric Body

143

(4)

4.10 General Case of a Moment-Free Body

147

(2)

4.11 Motion of a Symmetric Top

149

(8)

4.12 The Lagrangian Equations for Quasi-Coordinates

157

(3)

4.13 The Equations of Motion Referred to an Arbitrary System of Axes

160

(2)

4.14 The Rolling of a Coin

162

(2)

Problems

164

(5)

Suggested References

169

(1)

Chapter 5 Behavior of Dynamical Systems. Geometric Theory

170

(39)

5.1 Fundamental Concepts

171

(7)

5.2 Motion of Single-Degree-of-Freedom Autonomous Systems about Equilibrium Points

178

(11)

5.3 Conservative Systems. Motion in the Large

189

(6)

5.4 The Index of Poincare

195

(3)

5.5 Limit Cycles of Poincare

198

(8)

Problems

206

(2)

Suggested References

208

(1)

Chapter 6 Stability of Multi-Degree-of-Freedom Autonomous Systems

209

(54)

6.1 General Linear Systems

210

(7)

6.2 Linear Autonomous Systems

217

(5)

6.3 Stability of Linear Autonomous Systems. Routh-Hurwitz Criterion

222

(3)

6.4 The Variational Equations

225

(1)

6.5 Theorem on the First-Approximation Stability

226

(3)

6.6 Variation from Canonical Systems. Constant Coefficients

229

(2)

6.7 The Liapunov Direct Method

231

(8)

6.8 Geometric Interpretation of the Liapunov Direct Method

239

(4)

6.9 Stability of Canonical Systems

243

(9)

6.10 Stability In the Presence of Gyroscopic and Dissipative Forces

252

(6)

6.11 Construction of Liapunov Functions for Linear Autonomous Systems

258

(2)

Problems

260

(2)

Suggested References

262

(1)

Chapter 7 Nonautonomous Systems

263

(30)

7.1 Linear Systems with Periodic Coefficients. Floquet's Theory

264

(7)

7.2 Stability of Variational Equations with Periodic Coefficients

271

(1)

7.3 Orbital Stability

272

(1)

7.4 Variation from Canonical Systems. Periodic Coefficients

273

(4)

7.5 Second-Order Systems with Periodic Coefficients

277

(3)

7.6 Hill's Infinite Determinant

280

(2)

7.7 Mathieu's Equation

282

(6)

7.8 The Liapunov Direct Method

288

(4)

Suggested References

292

(1)

Chapter 8 Analytical Solutions by Perturbation Techniques

293

(36)

8.1 The Fundamental Perturbation Technique

294

(3)

8.2 Secular Terms

297

(2)

8.3 Lindstedt's Method

299

(3)

8.4 The Krylov-Bogoliubov-Mitropolsky (KBM) Method

302

(7)

8.5 A Perturbation Technique Based on Hill's Determinants

309

(4)

8.6 Periodic Solutions of Nonautonomous Systems. Duffing's Equation

313

(9)

8.7 The Method of Averaging

322

(5)

Problems

327

(1)

Suggested References

328

(1)

Chapter 9 Transformation Theory. The Hamilton-Jacobi Equation

329

(52)

9.1 The Principle of Least Action

330

(4)

9.2 Contact Transformations

334

(5)

9.3 Further Extensions of the Concept of Contact Transformations

339

(7)

9.4 Integral Invariants

346

(3)

9.5 The Lagrange and Poisson Brackets

349

(3)

9.6 Infinitesimal Contact Transformations

352

(3)

9.7 The Hamilton-Jacobi Equation

355

(6)

9.8 Separable Systems

361

(4)

9.9 Action and Angle Variables

365

(7)

9.10 Perturbation Theory

372

(6)

Problems

378

(2)

Suggested References

380

(1)

Chapter 10 The Gyroscope: Theory and Applications

381

(27)

10.1 Oscillations of a Symmetric Gyroscope

382

(4)

10.2 Effect of Gimbal Inertia on the Motion of a Free Gyroscope

386

(3)

10.3 Effect of Rotor Shaft Flexibility on the Frequency of Oscillation of a Free Gyroscope

389

(4)

10.4 The Gyrocompass

393

(5)

10.5 The Gyropendulum. Schuler Tuning

398

(5)

10.6 Rate and Integrating Gyroscopes

403

(3)

Problems

406

(1)

Suggested References

407

(1)

Chapter 11 Problems in Celestial Mechanics

408

(44)

11.1 Kepler's Equation. Orbit Determination

409

(4)

11.2 The Many-Body Problem

413

(3)

11.3 The Three-Body Problem

416

(4)

11.4 The Restricted Three-Body Problem

420

(5)

11.5 Stability of Motion Near the Lagrangian Points

425

(3)

11.6 The Equations of Relative Motion. Disturbing Function

428

(2)

11.7 Gravitational Potential and Torques for an Arbitrary Body

430

(8)

11.8 Precession and Nutation of the Earth's Polar Axis

438

(4)

11.9 Variation of the Orbital Elements

442

(5)

11.10 The Resolution of the Disturbing Function

447

(3)

Problems

450

(1)

Suggested References

451

(1)

Chapter 12 Problems in Spacecraft Dynamics

452

(42)

12.1 Transfer Orbits. Changes in the Orbital Elements Due to a Small Impulse

453

(4)

12.2 Perturbations of a Satellite Orbit In the Gravitational Field of an Oblate Planet

457

(6)

12.3 The Effect of Atmospheric Drag on Satellite Orbits

463

(3)

12.4 The Attitude Motion of Orbiting Satellites. General Considerations

466

(4)

12.5 The Attitude Stability of Earth-Pointing Satellites

470

(5)

12.6 The Attitude Stability of Spinning Symmetrical Satellites

475

(8)

12.7 Variable-Mass Systems

483

(4)

12.8 Rocket Dynamics

487

(4)

Problems

491

(1)

Suggested References

492

(2)

Appendix A Dyadics

494

(3)

Appendix B Elements of Topology and Modern Analysis

497

(9)

B.1 Sets and Functions

498

(3)

B.2 Metric Spaces

501

(3)

B.3 Topological Spaces

504

(2)

Suggested References

506

(1)

Name Index

507

(4)

Subject Index

511

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Methods of Analytical Dynamics

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