Analytical Dynamics of Discrete Systems

This book is to serve as a text for engineering students at the senior or beginning graduate level in a second course in dynamics. It grew out of many years experience in teaching such a course to senior students in mechanical engineering at the University of California, Berkeley. While temperamentally disinclined to engage in textbook writing, I nevertheless wrote the present volume for the usual reason-I was unable to find a satisfactory English-language text with the content covered in my inter­ mediate course in dynamics. Originally, I had intended to fit this text very closely to the content of my dynamics course for seniors. However, it soon became apparent that that course reflects too many of my personal idiosyncracies, and perhaps it also covers too little material to form a suitable basis for a general text. Moreover, as the manuscript grew, so did my interest in certain phases of the subject. As a result, this book contains more material than can be studied in one semester or quarter. My own course covers Chapters 1 to 5 (Chapters 1,2, and 3 lightly) and Chapters 8 to 20 (Chapter 17 lightly).

1. Introduction.- 2. Dynamical Systems.- 2.1. Particles.- 2.2. Systems of Particles.- 2.3. Forces and Laws of Motion.- 2.4. Galilean Transformations.- 2.5. Arguments of the Forces.- 2.6. The Problems of Particle Mechanics.- 3. Representations of the Motion.- 3.1. The Configuration Space.- 3.2. The Event Space.- 3.3. The State Space.- 3.4. The State-Time Space.- 3.5. Notions on the Concept of Stability.- 3.6. Problems.- 4. Constraints.- 4.1. General Observations.- 4.2. Holonomic Constraints.- 4.3. Nonholonomic Constraints.- 4.4. The Pfaffian Forms.- 4.5. When is a System of Constraints Holonomic?.- 4.6. Accessibility (of the Configuration Space).- 4.7. Problems.- 5. The Strictly Newtonian Mechanics Problem.- 5.1. General Remarks.- 5.2. The Given Quantities and Relations.- 5.3. The First Problem.- 5.4. The Second Problem.- 5.5. Other Problems.- 5.6. Concluding Remarks.- 6. Some Rigid Body Kinematics.- 6.1. The Rigid Body.- 6.2. Finite Rotation.- 6.3. The Direction Cosines.- 6.4. Orthogonal Transformations.- 6.5. The Matrix Notation.- 6.6. Properties of the Rotation Matrix.- 6.7. The Composition of Rotations.- 6.8. Applications.- (a) The Euler Angles.- (b) The Rodrigues Formulas.- 6.9. Problems.- 7. Some Rigid Body Kinetics.- 7.1. Introductory Remarks.- 7.2. The Inertial Parameters in Rotated Axes.- 7.3. Angular Momentum and Principal Axes.- 7.4. The Ellipsoids of Cauchy and Poinsot.- 7.5. The General Motion of Rigid Bodies.- 7.6. Problems.- 8. The Nature of Lagrangean Mechanics.- 8.1. General Remarks.- 8.2. The Generalizations by Lagrange.- 9. Virtual Displacement and Virtual Work.- 9.1. General Observations.- 9.2. Classification of Displacements.- 9.3. D’Alembert’s Principle.- 9.4. The Nature of the Forces of Constraint.- 9.5. The Virtual Velocity.- 9.6. The Variation.- 9.7. Possible Velocities and Accelerations.- 9.8. The Fundamental Equation.- 9.9. The Nature of the Given Forces.- 9.10. Given Forces Which Are Functions of Constraint Forces.- 9.11. Problems.- 10. Hamilton’s Principle.- 10.1. The Kinetic Energy.- 10.2. Kinetic Energy in Catastatic Systems.- 10.3. The Energy Relations in Catastatic Systems.- 10.4. The Central Principle.- 10.5. The Principle of Hamilton.- 10.6. Noncontemporaneous Variations.- 10.7. Lagrange’s Principle of Least Action.- 10.8. Jacobi’s Principle of Least Action.- 10.9. Problems.- 11. Generalized Coordinates.- 11.1. Introductory Remarks.- 11.2. The Theory of Generalized Coordinates.- 11.3. The Nature of Generalized Coordinates.- 11.4. The ? Operator for Generalized Coordinates.- 11.5. Exceptional Cases.- 11.6. Problems.- 12. The Fundamental Equation in Generalized Coordinates.- 12.1. The Kinetic Energy.- 12.2. Two Equalities.- 12.3. The Fundamental Equation.- 12.4. Generalized Potential Forces.- 12.5. Velocity-Dependent Potentials.- 12.6. Problems.- 13. Lagrange’s Equations.- 13.1. The Dynamical Problem.- 13.2. The Multiplier Rule.- 13.3. Derivation from the Fundamental Equation.- 13.4. Derivation from the Central Principle.- 13.5. Derivation from Hamilton’s Principle.- 13.6. Dynamic Coupling and Decoupling.- 13.7. Special Forms of Lagrange’s Equations.- (a) Existence of a Potential.- (b) Holonomic Systems.- (c) Rayleigh’s Dissipation Function.- (d) The Dissipation Function of Lur’e.- 13.8. The Principle of LeastAction Reconsidered.- 13.9. Problems.- 14. Embedding Constraints.- 14.1. Introductory Remarks.- 14.2. A Fallacy.- 14.3. Embedding of Nonholonomic Constraints..- 14.4. Problems.- 15. Formulating Problems by Lagrange’s Equations.- 15.1. General Remarks.- 15.2. The Unconstrained Particle.- 15.3. The Holonomically Constrained Particle.- 15.4. The Nonholonomically Constrained Particle.- 15.5. Systems of Particles and Rigid Bodies.- 15.6. Problems.- 16. The Integration.- 16.1. The Meaning of an Integral.- 16.2. Jacobi’s Integral.- 16.3. TheRouthianFunctionandtheMomentumIntegrals.- (a) The Legendre Transformation.- (b) The Routhian Function.- 16.4. Partial and Complete Separation of Variables.- 16.5. Solution in Quadratures.- 16.6. Qualitative Integration.- 16.7. Problems.- 17. Stability.- 17.1. Introductory Remarks.- 17.2. Definition of Stability.- 17.3. The Variational Equations.- 17.4. Some Remarks on Indirect Methods.- 17.5. Some Remarks on Liapunov’s Direct Method.- (a) The Autonomous Case.- (b) The Nonautonomous Case.- 17.6. Problems.- 18. Applications.- 18.1. Introductory Remarks.- 18.2. The Single Particle.- 18.3. Systems of Particles.- 18.4. Nonholonomic Systems.- 18.5. Problems.- 19. About Celestial Problems.- 19.1. Historical Notes.- 19.2. The Central Force Problem.- 19.3. The Central Force Problem Continued—The Apsides.- 19.4. The Central Force Problem Continued—On Bertrand’s Theorem.- 19.5. The n-Body Problem.- 19.6. The Two-Body Problem.- 19.7. Some Information about the Three-Body Problem.- 19.8. Problems.- 20. Topics in Gyrodynamics.- 20.1. Introduction.- 20.2. The Heavy Symmetrical Top.- 20.3. The Gyroscope.- 20.4. The Gyrocompass.- 20.5. Problems.- 21. Impulsive Motion.- 21.1. General Remarks.- 21.2. The Fundamental Equation.- 21.3. Impulsive Constraints.- 21.4. The Fundamental Equation with Impulsive Constraints.- 21.5. Impulsive Motion Theorems.- 21.6. Lagrange’s Equations for Impulsive Motion.- 21.7. Problems.