Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems; For Engineers, Physicists, and Mathematicians (Hardback)

• 1 Introduction to Analytical Mechanics
• 2 History of Theoretical Mechanics: A Bird's Eye View
• 3 Suggestions to the Reader
• 4 Abbreviations, Symbols, Notations, Formulae

• 1.1 Vector and (Cartesian) Tensor Algebra
• 1.2 Space-Time Axioms: Particle Kinematrics
• 1.3 Bodies and their Masses
• 1.4 Force; Law of Newton—Euler
• 1.5 Space-Time and the Principle of Galilean Relativity
• 1.6 The Fundamental Principles (or Balance Laws) of General System Mechanics
• 1.7 Accelerated (Noninertial) Frames of Reference (or Relative Motion, or Moving Axes); Angular Velocity and Acceleration
• 1.8 The Rigid Body: Introduction
• 1.9 The Rigid Body: Geometry of Motions and Kinematics (Summary of Basic Theorems)
• 1.10 The Rigid Body: Geometry of Rotational Motion; Finite Rotation
• 1.11 The Rigid Body: Active and Passive Interpretations of a Proper Orthogonal Tensor; Successive Finite Rotations
• 1.12 The Rigid Body: Eulerian Angles
• 1.13 The Rigid Body: Transformation Matrices (Direction Cosines) Between Space-Fixed and Body-Fixed Triads; and Angular Velocity Components along Body-Fixed Axes, for All-Sequences of Eulerian Angles
• 1.14 The Rigid Body: An Introduction to Quasi Coordinates
• 1.15 The Rigid Body: Tensor of Inertia, Kinetic Energy
• 1.16 The Rigid Body: Linear and Angular Momentum
• 1.17 The Rigid Body: Kinetic Energy and Kinetics of Translation and Rotation (Eulerian "Gyro Equations")
• 1.18 The Rigid Body: Contact Forces, Friction

• 2.1 Introduction
• 2.2 Introduction to Constraints and their Classifications
• 2.3 Quantitative Introduction to Nonholonomicity
• 2.4 System Positional Coordinates and System Forms of the Holonomic Constraints
• 2.5 Velocity, Acceleration, Admissible and Virtual Displacements; in System Variables
• 2.6 System Forms of Linear Velocity (Pfaffian) Constraints
• 2.7 Geometrical Interpretations of Constraints
• 2.8 Noncommutativity versus Nonholonomicity; Introduction to the Theorem of Frobenius
• 2.9 Quasi Coordinates, and their Calculus
• 2.10 Transitivity, or Transpositional, Rotations; Hamel Coefficients
• 2.11 Pfaffian (Velocity) Constraints via Quasi Variables, and their Geometrical Interpretation
• 2.12 Constrained Transitivity Equations, and Hamel's Form of Frobenus' Theorem
• 2.13 General Examples and Problems

• 3.1 Introduction
• 3.2 The Principle of Lagrange
• 3.3 Virtual Work of Inertial Forces (δI), and Related Kinematico-Inertial Identites
• 3.4 Virtual Work of Forces (δW), and Constraint Reactions (δW_R)
• 3.5 Equations of Motion via Lagrange's Principle: General Forms
• 3.6 The Central Equation (The Zentralgleichung of Heun and Hamel)
• 3.7 The Principle of Relaxation of Constraints (The Lagrange-Hamel Befreiungsprinzip)
• 3.8 Equations of Motion: Special Forms
• 3.9 Kinetic and Potential Energies: Energy Rate, or Power, Theorems
• 3.10 Lagrange's Equations: Explicit Forms; and Linear Variational Equations (or Method of Small Oscillations)
• 3.11 Appell's Equations: Explicit Forms
• 3.12 Equations of Motion: Integration and Conservation Theorems
• 3.13 The Rigid Body: Lagrangean-Eulerian Kinematico-Inertial Identities
• 3.14 The Rigid Body: Appellian Kinematico-Inertial Identities
• 3.15 The Rigid Body: Virtual Work of Forces
• 3.16 Relative Motion (or Moving Axes/Frames via Lagrange's Method)
• 3.17 Servo (or Control) Constraints
• 3.18 General Examples and Problems

• APPENDIX 3.A1 Remarks on the History of the Hamel-type Equations of Analytical Mechanics

• APPENDIX 3.A2 Critical Comments on Virtual Displacements/Work and Lagrange's Principle

• 4.1 Introduction
• 4.2 Brief Overview of the Newton-Euler Impulsive Theory
• 4.3 The Lagrangean Impulsive Theory; Namely, Constrained Discontinuous Motion
• 4.4 The Appellian Classification of Impulsive Constraints, and Corresponding Equations of Impulsive Motion
• 4.5 Impulsive Motion via Quasi Variables
• 4.6 Extremum Theorems of Impulsive Motion (of Carnot, Kelvin, Bertrand, Robin, et al.)

• 5.1 Introduction
• 5.2 Kinematics: The Nonlinear Transitivity Equations
• 5.3 Kinetics: Variational Equations/Principles; General and Specific Equations of Motion (Johnson, Hamel, et al.)
• 5.4 Second- and Higher-Order Constraints

• 6.1 Introduction
• 6.2 The General Theory
• 6.3 Principle of Jourdain, and Equations of Nielsen
• 6.4 Introduction to the Principle of Gauss and the Equations of Tsenov
• 6.5 Additional Forms of the Equations of Nielsen and Tsenov
• 6.6 The Principle of Gauss (Extensive Treatment)
• 6.7 The Principle of Hertz

• 7.1 Introduction

TIME-INTEGRAL THEOREMS

• 7.2 Time-Integral Theorems: Pfaffian Constraints, Holonomic Variables
• 7.3 Time-Integral Theorems: Pfaffian Constraints, Linear Nonholonomic Variables
• 7.4 Time-Integral Theorems: Nonlinear Velocity Constraints, Holonomic Variables
• 7.5 Time-Integral Theorems: Nonlinear Velocity Constraints, Nonlinear Nonholonomic Variables

TIME-INTEGRAL VARIATIONAL PRINCIPLES (IVP)

• 7.6 Hamilton's Principle versus Calculus of Variations
• 7.7 Integral Variational Equations of Mechanics
• 7.8 Special Integral Variational Principles (of Suslov, Voronets, et al.)
• 7.9 Noncontemporaneous Variations; Additional IVP Forms

• APPENDIX 7.A Extremal Properties of the Hamiltonian Action (Is the Action Really a Minimum; Namely Least?)

• 8.1 Introduction
• 8.2 The Hamiltonian, or Canonical, Central Equation and Hamilton's Canonical Equation of Motion
• 8.3 The Routhian Central Equations and Routh's Equations of Motion
• 8.4 Cyclic Systems; Equations of Kelvin-Tait
• 8.5 Steady Motion (of Cyclic Systems)
• 8.6 Stability of Steady Motion (of Cyclic Systems)
• 8.7 Variations of Constraints (or Parameters)
• 8.8 Canonical Transformations
• 8.9 Canonicity Conditions via Poisson's Brackets (PB)
• 8.10 The Hamilton-Jacobi Theory
• 8.11 Hamilton's Principal and Characteristic Functions, and Associated Variational Principles/Differential Equations
• 8.12 Intergal Variants
• 8.13 Noether's Theorem
• 8.14 Periodic Motions; Action-Angle Variables
• 8.15 Adiabatic Invariants
• 8.16 Canonical Perturbation Theory in Action-Angle Variables

• References and Suggested Reading
• Index