Methods of Analytical Dynamics / Edition 1

Methods of Analytical Dynamics / Edition 1

by Leonard Meirovitch

ISBN-10: 0070414556

ISBN-13: 9780070414556

Pub. Date: 01/28/1970

Publisher: McGraw-Hill Higher Education

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…  See more details below


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

McGraw-Hill Higher Education
Publication date:
Edition description:
New Edition
Product dimensions:
6.30(w) x 9.45(h) x (d)

Table of Contents

Chapter 1Fundamentals of Newtonian Mechanics1
1.1Historical Survey of Mechanics1
1.2Newton's Laws9
1.3Impulse and Momentum12
1.4Moment of a Force and Angular Momentum12
1.5Work and Energy14
1.6Energy Diagrams17
1.7Systems of Particles21
1.8The Two-Body Central Force Problem25
1.9The Inverse Square Law. Orbits of Planets and Satellites30
1.10Scattering by a Repulsive Central Force37
Suggested References44
Chapter 2Fundamentals of Analytical Mechanics45
2.1Degrees of Freedom. Generalized Coordinates46
2.2Systems with Constraints48
2.3The Stationary Value of a Function53
2.4The Stationary Value of a Definite Integral55
2.5The Principle of Virtual Work59
2.6D'Alembert's Principle65
2.7Hamilton's Principle66
2.8Lagrange's Equations of Motion72
2.9Lagrange's Equations for Impulsive Forces79
2.10Conservation Laws82
2.11Routh's Method for the Ignoration of Coordinates85
2.12Rayleigh's Dissipation Function88
2.13Hamilton's Equations91
Suggested References100
Chapter 3Motion Relative to Rotating Reference Frames101
3.1Transformation of Coordinates102
3.2Rotating Coordinate Systems104
3.3Expressions for the Motion in Terms of Moving Reference Frames110
3.4Motion Relative to the Rotating Earth112
3.5Motion of a Free Particle Relative to the Rotating Earth114
3.6Foucault's Pendulum116
Suggested References121
Chapter 4Rigid Body Dynamics122
4.1Kinematics of a Rigid Body123
4.2The Linear and Angular Momentum of a Rigid Body126
4.3Translation Theorem for the Angular Momentum130
4.4The Kinetic Energy of a Rigid Body132
4.5Principal Axes134
4.6The Equations of Motion for a Rigid Body137
4.7Euler's Equations of Motion138
4.8Euler's Angles140
4.9Moment-Free Inertially Symmetric Body143
4.10General Case of a Moment-Free Body147
4.11Motion of a Symmetric Top149
4.12The Lagrangian Equations for Quasi-Coordinates157
4.13The Equations of Motion Referred to an Arbitrary System of Axes160
4.14The Rolling of a Coin162
Suggested References169
Chapter 5Behavior of Dynamical Systems. Geometric Theory170
5.1Fundamental Concepts171
5.2Motion of Single-Degree-of-Freedom Autonomous Systems about Equilibrium Points178
5.3Conservative Systems. Motion in the Large189
5.4The Index of Poincare195
5.5Limit Cycles of Poincare198
Suggested References208
Chapter 6Stability of Multi-Degree-of-Freedom Autonomous Systems209
6.1General Linear Systems210
6.2Linear Autonomous Systems217
6.3Stability of Linear Autonomous Systems. Routh-Hurwitz Criterion222
6.4The Variational Equations225
6.5Theorem on the First-Approximation Stability226
6.6Variation from Canonical Systems. Constant Coefficients229
6.7The Liapunov Direct Method231
6.8Geometric Interpretation of the Liapunov Direct Method239
6.9Stability of Canonical Systems243
6.10Stability in the Presence of Gyroscopic and Dissipative Forces252
6.11Construction of Liapunov Functions for Linear Autonomous Systems258
Suggested References262
Chapter 7Nonautonomous Systems263
7.1Linear Systems with Periodic Coefficients. Floquet's Theory264
7.2Stability of Variational Equations with Periodic Coefficients271
7.3Orbital Stability272
7.4Variation from Canonical Systems. Periodic Coefficients273
7.5Second-Order Systems with Periodic Coefficients277
7.6Hill's Infinite Determinant280
7.7Mathieu's Equation282
7.8The Liapunov Direct Method288
Suggested References292
Chapter 8Analytical Solutions by Perturbation Techniques293
8.1The Fundamental Perturbation Technique294
8.2Secular Terms297
8.3Lindstedt's Method299
8.4The Krylov-Bogoliubov-Mitropolsky (KBM) Method302
8.5A Perturbation Technique Based on Hill's Determinants309
8.6Periodic Solutions of Nonautonomous Systems. Duffing's Equation313
8.7The Method of Averaging322
Suggested References328
Chapter 9Transformation Theory. The Hamilton-Jacobi Equation329
9.1The Principle of Least Action330
9.2Contact Transformations334
9.3Further Extensions of the Concept of Contact Transformations339
9.4Integral Invariants346
9.5The Lagrange and Poisson Brackets349
9.6Infinitesimal Contact Transformations352
9.7The Hamilton-Jacobi Equation355
9.8Separable Systems361
9.9Action and Angle Variables365
9.10Perturbation Theory372
Suggested References380
Chapter 10The Gyroscope: Theory and Applications381
10.1Oscillations of a Symmetric Gyroscope382
10.2Effect of Gimbal Inertia on the Motion of a Free Gyroscope386
10.3Effect of Rotor Shaft Flexibility on the Frequency of Oscillation of a Free Gyroscope389
10.4The Gyrocompass393
10.5The Gyropendulum. Schuler Tuning398
10.6Rate and Integrating Gyroscopes403
Suggested References407
Chapter 11Problems in Celestial Mechanics408
11.1Kepler's Equation. Orbit Determination409
11.2The Many-Body Problem413
11.3The Three-Body Problem416
11.4The Restricted Three-Body Problem420
11.5Stability of Motion Near the Lagrangian Points425
11.6The Equations of Relative Motion. Disturbing Function428
11.7Gravitational Potential and Torques for an Arbitrary Body430
11.8Precession and Nutation of the Earth's Polar Axis438
11.9Variation of the Orbital Elements442
11.10The Resolution of the Disturbing Function447
Suggested References451
Chapter 12Problems in Spacecraft Dynamics452
12.1Transfer Orbits. Changes in the Orbital Elements Due to a Small Impulse453
12.2Perturbations of a Satellite Orbit in the Gravitational Field of an Oblate Planet457
12.3The Effect of Atmospheric Drag on Satellite Orbits463
12.4The Attitude Motion of Orbiting Satellites. General Considerations466
12.5The Attitude Stability of Earth-Pointing Satellites470
12.6The Attitude Stability of Spinning Symmetrical Satellites475
12.7Variable-Mass Systems483
12.8Rocket Dynamics487
Suggested References492
Appendix ADyadics494
Appendix BElements of Topology and Modern Analysis497
B.1Sets and Functions498
B.2Metric Spaces501
B.3Topological Spaces504
Suggested References506
Name Index507
Subject Index511

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