Analytical Mechanics: Classical, Lagrangian and Hamiltonian Mechanics, Stability Theory, Special Relativity

Preface
Prerequisites and Reference Textbooks
Notations and Conventions
Acknowledgements
General Framework of Analytical Mechanics
Contents
1 The Space and Time of Classical Physics
1.1 The Mathematical Description of Space and Time in Classical Physics
1.1.1 Affine Spaces
1.1.2 Euclidean Spaces and Isometries
1.1.3 The Isometry Group of En and the Active and Passive Interpretation
1.1.4 Invariant Arclengths, Areas and Volumes Under the Isometry Group
1.1.5 Orientation of Euclidean Spaces and Cross Product
1.2 Space and Time for an Observer: Physical Correspondences
1.2.1 Rigid Rulers and Ideal Clocks
1.2.2 Existence of Physical Geometry
1.3 Introduction to the Notion of Differentiable Manifold
1.3.1 Classes of Differentiable Maps
1.3.2 Local Charts and Differentiable Manifolds
1.3.3 Differentiable Functions and Curves on a Manifold and Diffeomorphisms
2 The Spacetime of Classical Physics and Classical Kinematics
2.1 The Spacetime of Classical Physics and Its Geometric Structures
2.1.1 Multiple Rest Spaces and Absolute Metric Structure of Space and Time
2.1.2 The Spacetime of Classical Physics and the World Lines
2.2 Reference Frames
2.2.1 Rest Orthonormal Coordinate Systems of Moving Frames
2.2.2 [0.72cm][l]ACAn Alternative But Equivalent Definition of Frame
2.3 Absolute Point-Particle Kinematics
2.3.1 Differentiating Curves in Affine Spaces
2.3.2 Elementary Kinematic Quantities
2.3.3 Kinematics for Point Particles Constrained to Stationary Curves and Surfaces
2.4 Relative Point-Particle Kinematics
2.4.1 The ω Vector and the Poisson Formulas
2.4.2 Velocity and Acceleration as the Frame Varies
3 Newtonian Dynamics: A Conceptual Critical Review
3.1 Newton's First Law of Motion
3.1.1 Inertial Frames
3.1.2 Galilean Transformations
3.1.3 Relative Motion of Inertial Frames
3.1.4 [0.72cm][l]ACThe Affine Galilean Structure of V4
3.2 General Formulation of the Classical Dynamics of Systems of Point Particles
3.2.1 Masses, Impulses and Forces
3.2.2 Superposition of Forces
3.2.3 The Determinism of Classical Mechanics
3.3 More General Dynamical Situations
3.3.1 Case 1: Prescribed Motion of a Subsystem and Time-Dependent Forces
3.3.2 Case 2: Geometric Constraints and Constraint Forces
3.3.3 Case 3: Dynamics in Non-inertial Frames and the Notion of Inertial Force
3.4 Comments on the General Formulation of Newtonian Dynamics
3.4.1 Galilean Invariance
3.4.2 The Failure of the Newtonian Programme
3.4.3 What Remains Today of ``Mach's Principle''?
4 Balance Equations and First Integrals in Mechanics
4.1 Governing Equations, Conservation of the Impulse and the Angular Momentum
4.1.1 Total Quantities of Systems of Point Particles
4.1.2 Governing Equations
4.1.3 Balance/Conservation Laws of Impulse and Angular Momentum
4.2 Mechanical Energy
4.2.1 Kinetic Energy Theorem
4.2.2 Conservative Forces
4.2.3 Balance and Conservation of the Mechanical Energy
4.3 Two Conservation Laws Arising from Invariance Properties of the Potential Energy
4.4 The Necessity of the Description in Terms of Continua and Fields in Classical Mechanics
5 Introduction to Rigid Body Mechanics
5.1 The Rigidity Constraint for Discrete and Continuous Systems
5.1.1 Generic Rigid Bodies
5.1.2 Continuous Rigid Bodies
5.2 The Inertia Tensor and Its Properties
5.2.1 The Inertia Tensor
5.2.2 Principal Triples of Inertia
5.2.3 Huygens-Steiner Formula
5.3 Rigid Body Dynamics: Introduction to the Theory of Euler Equations
5.3.1 Euler Equations
5.3.2 Poinsot Motions
5.3.3 Permanent Rotations
5.3.4 Poinsot Motions for Gyroscopic Bodies
5.3.5 Poinsot Motions for Non-gyroscopic Bodies
6 Introduction to Stability Theory with Applications to Mechanics
6.1 Singular Points and Equilibrium Configurations
6.1.1 Stable and Unstable Equilibria
6.1.2 Introduction to Lyapunov's Methods for Studying Stability
6.1.3 More on Asymptotic Stability
6.1.4 An Instability Criterion Based on Linearisation
6.2 Applications to Physical Systems in Classical Mechanics
6.2.1 The Lagrange-Dirichlet Theorem
6.2.2 An Instability Criterion
6.2.3 Stability of Permanent Rotations for Non-gyroscopic Rigid Bodies
7 Foundations of Lagrangian Mechanics
7.1 An Introductory Example
7.2 The General Case: Holonomic Systems and Euler-Lagrange Equations
7.2.1 Spacetime of Configurations Vn+1 in Presence of Holonomic Constraints
7.2.2 Tangent Vectors to the Space of Configurations Qt
7.2.3 Ideal Constraints
7.2.4 Kinematic Quantities and Kinetic Energy
7.2.5 Euler-Lagrange Equations for Systems of Point Particles
7.3 Extension to Systems of Continuous Rigid Bodies and Point Particles
7.3.1 Articulated Systems
7.3.2 Computing the Tangent Vectors δP(k)i and the Kinetic Energy of Rigid Bodies
7.3.3 Generalisation of Identity (7.40) to Continuous Rigid Bodies
7.3.4 Euler-Lagrange Equations for Articulated Systems
7.4 Elementary Properties of the Euler-Lagrange Equations
7.4.1 Normality of Euler-Lagrange Equations and Existence and Uniqueness Theorem
7.4.2 Spacetime of Kinetic States A(Vn+1)
7.4.3 Non-dependency of the Euler-Lagrange Solutions on Coordinates
7.4.4 Maximal Solutions of the Euler-Lagrange Equations Defined Globally on A(Vn+1)
7.4.5 The Notion of Lagrangian
7.4.6 Regularity of Lagrangians in Standard Form
7.4.7 Change of Inertial Frame and Lagrangian Non-uniqueness
7.5 [0.72cm][l]AC Global Differential-Geometric Formulation of the Euler-Lagrange Equations
7.5.1 The Bundle Structures of Vn+1 and A(Vn+1)
7.5.2 The Dynamic Vector Field Associated with the Euler-Lagrange Equations
7.5.3 Contact Forms, Poincaré-Cartan Form and Intrinsic Formulation of the Euler-Lagrange Equations Induced by a Lagrangian
7.5.4 Dynamic Vector Field on A(Vn+1) Without Global Lagrangian
8 Symmetries and Conservation Laws in Lagrangian Mechanics
8.1 The Relationship Between Symmetry and Conservation Laws: Cyclic Coordinates
8.1.1 Cyclic Coordinates and Constancy of Conjugate Momenta on the Motion
8.1.2 Translation-Invariance and Conservation of the Impulse
8.1.3 Rotation-Invariance and Conservation of the Angular Momentum
8.2 The Relationship Between Symmetries and Conservation Laws: Emmy Noether's Theorem
8.2.1 Transformations on A(Vn+1)
8.2.2 Noether's Theorem in Elementary Local Form
8.2.3 Noether's First Integral's Independence of the Coordinate System
8.2.4 Action of the (Weak) Symmetries on the Solutions of the Euler-Lagrange Equations
8.3 Jacobi's First Integral, Invariance Under ``Temporal Displacements'' and Conservation of the Mechanical Energy
8.4 Comments on the Relationship Between Symmetries and Constant of Motion
8.4.1 Galilean Invariance in Classical Lagrangian Mechanics
8.4.2 The Noether and Jacobi Theorems Beyond Classical Mechanics
8.5 [0.72cm][l]AC: General and Global Formulation of Noether's Theorem
8.5.1 Symmetries and First Integrals in Terms of Vector Fields on A(Vn+1)
8.5.2 Noether's Theorem in General Global Form
8.5.3 Properties of Vector Fields X Generating Symmetries
8.5.4 Jacobi's First Integral as Consequence of Noether's Theorem
8.5.5 Jacobi's Global First Integral from the Global Noether Theorem
8.5.6 The Runge-Lenz Vector from Noether's Theorem
9 Advanced Topics in Lagrangian Mechanics
9.1 The Stationary-Action Principle for Systems that Admit a Lagrangian
9.1.1 Rudiments of Calculus of Variations
9.1.2 Hamilton's Stationary-Action Principle
9.2 Generalised Potentials
9.2.1 The Case of the Lorentz Force
9.2.2 Generalisation of the Notion of Potential
9.2.3 Conditions for the Existence of the Generalised Potential
9.2.4 Generalised Potentials of Inertial Forces
9.3 Equilibrium and Stability in the Lagrangian Formulation
9.3.1 Equilibrium Configurations with Respect to a Frame
9.3.2 Stability and the Lagrange-Dirichlet Theorem
9.4 Introduction to the Theory of Small Vibrations and Normal Coordinates
9.4.1 Linearised and Decoupled Equations: Normal Coordinates
9.4.2 Natural Frequencies (or Eigenfrequencies) and Normal Vibration Modes
10 [1.3cm][l]AC Mathematical Introduction to Special Relativity and the Relativistic Lagrangian Formulation
10.1 Linear Algebra Preliminaries
10.1.1 The Dual of a Finite-Dimensional Real Vector Space
10.1.2 Indefinite Inner Products, Covariant and Contravariant Components
10.1.3 Applied Vectors
10.2 The Geometry of Special Relativity
10.2.1 Minkowski Spacetime, Light Cone and Time Orientation
10.2.2 Physical Correspondences: Proper Time, Four-Velocity and Causality
10.2.3 Minkowski Coordinates and Minkowski Frames
10.2.4 Physical and Kinematic Properties of Minkowski Coordinates and Minkowski Frames
10.3 Introduction to Relativistic Dynamics
10.3.1 Mass, Four-Momentum and Their Elementary Properties
10.3.2 The So-Called Mass-Energy Equivalence Principle
10.3.3 Relativistic Equation of Motion and Identification Between Minkowski and Inertial Frames
10.3.4 The Geometry of the So-Called Twin Paradox
10.4 The Lorentz and Poincaré Groups
10.4.1 The Lorentz and Poincaré Groups and Their Orthochronous Subgroups
10.4.2 Special and Special Orthochronous Subgroups, Discrete Transformations
10.4.3 Elementary Properties of O(1,3)+ and IO(1,3)+
10.4.4 Relevance of Pure Lorentz Transformations
10.4.5 Two Decomposition Results for the Lorentz Group
10.5 Introduction to the Lagrangian Formalism in Special Relativity
10.5.1 The Covariant Quadratic Lagrangian for the Charged Relativistic Particle
10.5.2 First Integrals of the Covariant Quadratic Lagrangian
10.5.3 The Non-quadratic Covariant Lagrangian for the Charged Relativistic Particle
10.5.4 First Integrals of the Non-quadratic Covariant Lagrangian
10.5.5 The Non-quadratic and Non-covariant Lagrangian for the Charged Relativistic Particle
10.5.6 First Integrals of the Non-quadratic, Non-covariant Lagrangian
10.5.7 Extension of the Formalism to N Point-Particles
11 Fundamentals of Hamiltonian Mechanics
11.1 The Phase Spacetime and Hamilton's Equations
11.1.1 The Phase Spacetime F(Vn+1)
11.1.2 The Legendre Transform
11.1.3 Hamilton's Equations and the Local Uniqueness of the Hamiltonian
11.1.4 The Hamiltonian's Dependency on the Local Chart
11.1.5 Independence of the Solutions to Hamilton's Equations of Local Charts
11.1.6 Global Solutions to Hamilton's Equations on F(Vn+1)
11.2 Hamilton's Equations from a Variational Principle
11.3 Hamiltonian Formulation on RR2n
11.3.1 Hamiltonian Systems on RR2n and the Symplectic Matrix S
11.3.2 Hamiltonian Matrices and Hamiltonian Dynamical Systems
11.3.3 Liouville's Theorem in R R2n
11.4 [0.72cm][l]ACThe Bundle F(Vn+1) and Hamilton's Equations as Global Equations
11.4.1 The Bundle F(Vn+1)
11.4.2 Global Legendre Transformation as Diffeomorphism from A(Vn+1) to F(Vn+1)
11.4.3 Global Intrinsic Hamiltonian Formulation via the Field Z and the Emancipation of the Lagrangian Formulation
11.5 Symplectic Vector Spaces, the Symplectic Group and the Lie Algebra of Hamiltonian Matrices
11.5.1 Symplectic Vector Spaces
11.5.2 The Symplectic Group
11.5.3 Intermezzo: Matrix Exponential
11.5.4 The Symplectic Group and the Lie Algebra of Hamiltonian Matrices
12 Canonical Hamiltonian Theory, Hamiltonian Symmetries and Hamilton-Jacobi Theory
12.1 Canonical Hamiltonian Theory
12.1.1 Canonical Transformations and Canonical Coordinates
12.1.2 Conservation of Hamilton's Equations
12.2 Liouville's Theorem in Global Form and Poincaré's ``Recurrence'' Theorem
12.2.1 Liouville Theorem and Liouville Equation
12.2.2 Poincaré's ``Recurrence'' Theorem
12.3 [0.72cm][l]ACSymmetries and Conservation Laws in Hamiltonian Mechanics
12.3.1 Hamiltonian Vector Fields and Poisson Bracket
12.3.2 Local One-Parameter Groups of Active Canonical Transformations
12.3.3 Symmetries and Conservation Laws: The Hamiltonian Noether Theorem
12.4 [0.72cm][l]ACPoincaré-Cartan Form and Introduction to Hamilton-Jacobi Theory
12.4.1 Lie's Condition and Canonical Transformations
12.4.2 Generating Functions of Canonical Transformations
12.4.3 Introduction to Hamilton-Jacobi Theory
12.4.4 Local Existence of Complete Integrals: Hamilton's Principal Function
12.4.5 Time-Independent Hamilton-Jacobi Equation
12.4.6 Local Existence of Solutions for Boundary Value Problems of Order Two on Manifolds
13 [1.3cm][l]AC Hamiltonian Symplectic Structures: An Introduction
13.1 The Phase Space F as Symplectic Manifold
13.1.1 The Symplectic Structure of Autonomous Hamiltonian Systems
13.1.2 Symplectic Manifolds and Hamiltonian Mechanics
13.2 Hamiltonian Vector Fields and the Poisson Bracket on Symplectic Manifolds
13.2.1 Hamiltonian Noether Theorem on Symplectic Manifolds
13.2.2 The Action of the Galilean Group on Phase Space
13.2.3 The Action of the Poincaré Group on Phase Space for the Free Particle
13.2.4 The Arnold-Liouville Theorem in a Nutshell
13.3 The Symplectic Structure of F(Vn+1)
13.3.1 F(Vn+1) as Bundle of Symplectic Manifolds
13.3.2 A More General Notion of Phase Spacetime
14 Complement: Elements of the Theory of Ordinary Differential Equations
14.1 Systems of Differential Equations
14.1.1 Reduction to Order One
14.1.2 The Cauchy Problem
14.1.3 First Integrals
14.2 Preparatory Notions and Results for the Existence and Uniqueness Theorems
14.2.1 The Banach Space C0(K; Kn)
14.2.2 Fixed-Point Theorem in Complete Metric Spaces
14.2.3 (Locally) Lipschitz Functions
14.3 Existence and Uniqueness Theorems for the Cauchy Problem
14.3.1 Theorem of Local Existence and Uniqueness for the Cauchy Problem
14.3.2 A Condition for First Integrals
14.3.3 Global Existence and Uniqueness Theorem for the Cauchy Problem
14.3.4 Linear Differential Equations
14.3.5 Structure of the Solution Set of a Linear Equation
14.3.6 Completeness of Maximal Solutions
14.4 Comparison of Solutions and Dependency on Initial Conditions and Parameters
14.4.1 Gronwall Lemma and Consequences
14.4.2 Regularity of the Dependency on Cauchy Data and Related Issues
14.5 Initial Value Problem on Differentiable Manifolds
14.5.1 Cauchy Problem, Global Existence and Uniqueness
14.5.2 Completeness of Maximal Solutions
14.5.3 One-Parameter Groups of Local and Global Diffeomorphisms
14.5.4 Commuting Vector Fields and Their Local Groups
14.5.5 First Integrals and Functionally Independent First Integrals
15 Complement: The Physical Principles at the Foundations of Special Relativity
15.1 The Classical Perspective'S Crisis
15.2 Spacetime and Reference Frames
15.2.1 The Synchronisation Problem
15.3 The Fundamental Physical Postulates of Special Relativity
15.3.1 Constancy of the Speed of Light
15.3.2 Principle of Inertia
15.3.3 Principle of Relativity
15.4 From the Postulates of Special Relativity to the Poincaré Group
15.4.1 RS1 and RS2 Recast in Minkowski Coordinates
15.4.2 Finding the Poincaré Transformations and the Affine Structure of M4
A Elements of Topology, Analysis, Linear Algebra and Geometry
A.1 Review of Elementary Topology
A.2 Integrals of Limits and Derivatives
A.3 Series of Vector-Valued Functions
A.4 Deformation of Curves
A.5 Symmetric Operators on Finite-Dimensional Real Vector Spaces
A.6 Elements of Differential Geometry
A.6.1 Product Manifolds
A.6.2 Differentiable Maps
A.6.3 Embedded Submanifolds and Non-singular Maps
A.6.4 Tangent and Cotangent Spaces
A.6.5 Covariant and Contravariant Vector Fields on Manifolds
A.6.6 Differentials, Curves and Tangent Vectors
A.6.7 Affine and Euclidean Spaces as Differentiable Manifolds
B [1.3cm][l]AC: Advanced Topics in Differential Geometry
B.1 Differentiation on Manifolds and Related Notions
B.1.1 Pushforward and Pullback
B.1.2 Lie Derivative of a Vector Field
B.2 Immersion of Tangent Spaces for Embedded Submanifolds
B.3 Tangent and Cotangent Bundles, Fibre Bundles and Sections
B.4 Theory of Differential Forms and Integration on Differentiable Manifolds
B.4.1 p-Forms and p-Vectors
B.4.2 Differential Forms
B.4.3 Lie Derivative of a p-Form
B.4.4 Integral of Top Forms and Volume Forms on Oriented Manifolds
B.4.5 Integral of Forms on Submanifolds
B.4.6 Manifolds with Boundary and the Stokes-Poincaré Theorem
C Solutions and/or Hints to Suggested Exercises
C.1 Exercises for Chap.1
C.2 Exercises for Chap.2
C.3 Exercises for Chap.3
C.4 Exercises for Chap.4
C.5 Exercises for Chap.5
C.6 Exercises for Chap.6
C.7 Exercises for Chap.7
C.8 Exercises for Chap.10
C.9 Exercises for Chap.11
C.10 Exercises for Chap.12
C.11 Exercises for Complement 14
C.12 Exercises for Appendix A
References
Index