Contents -- Dedication -- Preface -- Acknowledgments -- PART I: INTRODUCTION: THE TRADITIONAL THEORY -- 1 Basic Dynamics of Point Particles and Collections -- 1.1 Newton's Space and Time -- 1.2 Single Point Particle -- 1.3 Collective Variables -- 1.4 The Law of Momentum for Collections -- 1.5 The Law of Angular Momentum for Collections -- 1.6 "Derivations" of the Axioms -- 1.7 The Work-Energy Theorem for Collections -- 1.8 Potential and Total Energy for Collections -- 1.9 The Center of Mass -- 1.10 Center of Mass and Momentum -- 1.11 Center of Mass and Angular Momentum -- 1.12 Center of Mass and Torque -- 1.13 Change of Angular Momentum -- 1.14 Center of Mass and the Work-Energy Theorems -- 1.15 Center of Mass as a Point Particle -- 1.16 Special Results for Rigid Bodies -- 1.17 Exercises -- 2 Introduction to Lagrangian Mechanics -- 2.1 Configuration Space -- 2.2 Newton's Second Law in Lagrangian Form -- 2.3 A Simple Example -- 2.4 Arbitrary Generalized Coordinates -- 2.5 Generalized Velocities in the q-System -- 2.6 Generalized Forces in the q-System -- 2.7 The Lagrangian Expressed in the q-System -- 2.8 Two Important Identities -- 2.9 Invariance of the Lagrange Equations -- 2.10 Relation Between Any Two Systems -- 2.11 More of the Simple Example -- 2.12 Generalized Momenta in the q-System -- 2.13 Ignorable Coordinates -- 2.14 Some Remarks About Units -- 2.15 The Generalized Energy Function -- 2.16 The Generalized Energy and the Total Energy -- 2.17 Velocity Dependent Potentials -- 2.18 Exercises -- 3 Lagrangian Theory of Constraints -- 3.1 Constraints Defined -- 3.2 Virtual Displacement -- 3.3 Virtual Work -- 3.4 Form of the Forces of Constraint -- 3.5 General Lagrange Equations with Constraints -- 3.6 An Alternate Notation for Holonomic Constraints -- 3.7 Example of the General Method -- 3.8 Reduction of Degrees of Freedom.

3.9 Example of a Reduction -- 3.10 Example of a Simpler Reduction Method -- 3.11 Recovery of the Forces of Constraint -- 3.12 Example of a Recovery -- 3.13 Generalized Energy Theorem with Constraints -- 3.14 Tractable Non-Holonomic Constraints -- 3.15 Exercises -- 4 Introduction to Hamiltonian Mechanics -- 4.1 Phase Space -- 4.2 Hamilton Equations -- 4.3 An Example of the Hamilton Equations -- 4.4 Non-Potential and Constraint Forces -- 4.5 Reduced Hamiltonian -- 4.6 Poisson Brackets -- 4.7 The Schroedinger Equation -- 4.8 The Ehrenfest Theorem -- 4.9 Exercises -- 5 The Calculus of Variations -- 5.1 Paths in an N-Dimensional Space -- 5.2 Variations of Coordinates -- 5.3 Variations of Functions -- 5.4 Variation of a Line Integral -- 5.5 Finding Extremum Paths -- 5.6 Example of an Extremum Path Calculation -- 5.7 Invariance and Homogeneity -- 5.8 The Brachistochrone Problem -- 5.9 Calculus of Variations with Constraints -- 5.10 An Example with Constraints -- 5.11 Reduction of Degrees of Freedom -- 5.12 Example of a Reduction -- 5.13 Example of a Better Reduction -- 5.14 The Coordinate Parametric Method -- 5.15 Comparison of the Methods -- 5.16 Exercises -- 6 Hamilton's Principle -- 6.1 Hamilton's Principle in Lagrangian Form -- 6.2 Hamilton's Principle with Constraints -- 6.3 Comments on Hamilton's Principle -- 6.4 Phase-Space Hamilton's Principle -- 6.5 Exercises -- 7 Linear Operators and Dyadics -- 7.1 Definition of Operators -- 7.2 Operators and Matrices -- 7.3 Addition and Multiplication -- 7.4 Determinant, Trace, and Inverse -- 7.5 Special Operators -- 7.6 Dyadics -- 7.7 Resolution of Unity -- 7.8 Operators, Components, Matrices, and Dyadics -- 7.9 Complex Vectors and Operators -- 7.10 Real and Complex Inner Products -- 7.11 Eigenvectors and Eigenvalues -- 7.12 Eigenvectors of Real Symmetric Operator.

7.13 Eigenvectors of Real Anti-Symmetric Operator -- 7.14 Normal Operators -- 7.15 Determinant and Trace of Normal Operator -- 7.16 Eigen-Dyadic Expansion of Normal Operator -- 7.17 Functions of Normal Operators -- 7.18 The Exponential Function -- 7.19 The Dirac Notation -- 7.20 Exercises -- 8 Kinematics of Rotation -- 8.1 Characterization of Rigid Bodies -- 8.2 The Center of Mass of a Rigid Body -- 8.3 General Definition of Rotation Operator -- 8.4 Rotation Matrices -- 8.5 Some Properties of Rotation Operators -- 8.6 Proper and Improper Rotation Operators -- 8.7 The Rotation Group -- 8.8 Kinematics of a Rigid Body -- 8.9 Rotation Operators and Rigid Bodies -- 8.10 Differentiation of a Rotation Operator -- 8.11 Meaning of the Angular Velocity Vector -- 8.12 Velocities of the Masses of a Rigid Body -- 8.13 Savio's Theorem -- 8.14 Infinitesimal Rotation -- 8.15 Addition of Angular Velocities -- 8.16 Fundamental Generators of Rotations -- 8.17 Rotation with a Fixed Axis -- 8.18 Expansion of Fixed-Axis Rotation -- 8.19 Eigenvectors of the Fixed-Axis Rotation Operator -- 8.20 The Euler Theorem -- 8.21 Rotation of Operators -- 8.22 Rotation of the Fundamental Generators -- 8.23 Rotation of a Fixed-Axis Rotation -- 8.24 Parameterization of Rotation Operators -- 8.25 Differentiation of Parameterized Operator -- 8.26 Euler Angles -- 8.27 Fixed-Axis Rotation from Euler Angles -- 8.28 Time Derivative of a Product -- 8.29 Angular Velocity from Euler Angles -- 8.30 Active and Passive Rotations -- 8.31 Passive Transformation of Vector Components -- 8.32 Passive Transformation of Matrix Elements -- 8.33 The Body Derivative -- 8.34 Passive Rotations and Rigid Bodies -- 8.35 Passive Use of Euler Angles -- 8.36 Exercises -- 9 Rotational Dynamics -- 9.1 Basic Facts of Rigid-Body Motion -- 9.2 The Inertia Operator and the Spin -- 9.3 The Inertia Dyadic.

9.4 Kinetic Energy of a Rigid Body -- 9.5 Meaning of the Inertia Operator -- 9.6 Principal Axes -- 9.7 Guessing the Principal Axes -- 9.8 Time Evolution of the Spin -- 9.9 Torque-Free Motion of a Symmetric Body -- 9.10 Euler Angles of the Torque-Free Motion -- 9.11 Body with One Point Fixed -- 9.12 Preserving the Principal Axes -- 9.13 Time Evolution with One Point Fixed -- 9.14 Body with One Point Fixed, Alternate Derivation -- 9.15 Work-Energy Theorems -- 9.16 Rotation with a Fixed Axis -- 9.17 The Symmetric Top with One Point Fixed -- 9.18 The Initially Clamped Symmetric Top -- 9.19 Approximate Treatment of the Symmetric Top -- 9.20 Inertial Forces -- 9.21 Laboratory on the Surface of the Earth -- 9.22 Coriolis Force Calculations -- 9.23 The Magnetic - Coriolis Analogy -- 9.24 Exercises -- 10 Small Vibrations About Equilibrium -- 10.1 Equilibrium Defined -- 10.2 Finding Equilibrium Points -- 10.3 Small Coordinates -- 10.4 Normal Modes -- 10.5 Generalized Eigenvalue Problem -- 10.6 Stability -- 10.7 Initial Conditions -- 10.8 The Energy of Small Vibrations -- 10.9 Single Mode Excitations -- 10.10 A Simple Example -- 10.11 Zero-Frequency Modes -- 10.12 Exercises -- PART II: MECHANICS WITH TIME AS A COORDINATE -- 11 Lagrangian Mechanics with Time as a Coordinate -- 11.1 Time as a Coordinate -- 11.2 A Change of Notation -- 11.3 Extended Lagrangian -- 11.4 Extended Momenta -- 11.5 Extended Lagrange Equations -- 11.6 A Simple Example -- 11.7 Invariance Under Change of Parameter -- 11.8 Change of Generalized Coordinates -- 11.9 Redundancy of the Extended Lagrange Equations -- 11.10 Forces of Constraint -- 11.11 Reduced Lagrangians with Time as a Coordinate -- 11.12 Exercises -- 12 Hamiltonian Mechanics with Time as a Coordinate -- 12.1 Extended Phase Space -- 12.2 Dependency Relation -- 12.3 Only One Dependency Relation.

12.4 From Traditional to Extended Hamiltonian Mechanics -- 12.5 Equivalence to Traditional Hamilton Equations -- 12.6 Example of Extended Hamilton Equations -- 12.7 Equivalent Extended Hamiltonians -- 12.8 Alternate Hamiltonians -- 12.9 Alternate Traditional Hamiltonians -- 12.10 Not a Legendre Transformation -- 12.11 Dirac's Theory of Phase-Space Constraints -- 12.12 Poisson Brackets with Time as a Coordinate -- 12.13 Poisson Brackets and Quantum Commutators -- 12.14 Exercises -- 13 Hamilton's Principle and Noether's Theorem -- 13.1 Extended Hamilton's Principle -- 13.2 Noether's Theorem -- 13.3 Examples of Noether's Theorem -- 13.4 Hamilton's Principle in an Extended Phase Space -- 13.5 Exercises -- 14 Relativity and Spacetime -- 14.1 Galilean Relativity -- 14.2 Conflict with the Aether -- 14.3 Einsteinian Relativity -- 14.4 What Is a Coordinate System? -- 14.5 A Survey of Spacetime -- 14.6 The Lorentz Transformation -- 14.7 The Principle of Relativity -- 14.8 Lorentzian Relativity -- 14.9 Mechanism and Relativity -- 14.10 Exercises -- 15 Fourvectors and Operators -- 15.1 Fourvectors -- 15.2 Inner Product -- 15.3 Choice of Metric -- 15.4 Relativistic Interval -- 15.5 Spacetime Diagram -- 15.6 General Fourvectors -- 15.7 Construction of New Fourvectors -- 15.8 Covariant and Contravariant Components -- 15.9 General Lorentz Transformations -- 15.10 Transformation of Components -- 15.11 Examples of Lorentz Transformations -- 15.12 Gradient Fourvector -- 15.13 Manifest Covariance -- 15.14 Formal Covariance -- 15.15 The Lorentz Group -- 15.16 Proper Lorentz Transformations and the Little Group -- 15.17 Parameterization -- 15.18 Fourvector Operators -- 15.19 Fourvector Dyadics -- 15.20 Wedge Products -- 15.21 Scalar, Fourvector, and Operator Fields -- 15.22 Manifestly Covariant Form of Maxwell's Equations -- 15.23 Exercises -- 16 Relativistic Mechanics.

16.1 Modification of Newton's Laws.