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**Author(s):** José Rachid Mohallem

**Publisher:** Springer

**Year:** 2024

**Language:** English

Preface

Contents

1 Concepts and Principles

1.1 Fundamentals

1.1.1 Dynamical Variables, Explicit and ImplicitDependences

1.2 Calculus of Variations

1.3 Hamilton's Principle

1.4 Constructing Lagrangians from Symmetries

1.5 ``Magic'' Lagrangians (One Particle)

1.6 Adapting Notation

1.7 Generalized Potentials

1.8 Equivalent Lagrangians

1.9 Newton's Laws from Hamilton's Principle

1.9.1 Newton's First Law: An Equivalent Lagrangian

1.9.2 Newton's Third Law: Space Homogeneity

1.10 Connection to Quantum Mechanics: The Classical Limit

1.11 Final Considerations on General Lagrangian Mechanics

2 Lagrangian Mechanics of Systems with L=T-V

2.1 Constraints and Generalized Coordinates

2.1.1 Holonomic Constraints

2.1.2 Generalized Coordinates an Velocities

2.1.3 Configuration Space

2.1.4 Transformation Equations

2.2 The Lagrangian

2.2.1 Virtual Displacements

2.3 Hamilton's Principle in Generalized Coordinates

2.3.1 Invariance of Lagrange's Equations Under Point Transformations

2.4 Newton's Second Law from Lagrange's Equations

2.5 Lagrange's Equations from Newton's Second Law

2.5.1 Absence of Constraints

2.5.2 Presence of Constraints: D'Alembert's Principle

2.5.3 The Lagrangian for Planar Motion of Rigid Bodies

2.6 Moving Constraints and Reference Frames

2.7 Symmetries of the Lagrangian and Conservation Theorems

2.7.1 Constants of Motion

2.7.2 Symmetries and Cyclical Coordinates

2.8 Nöther's Theorem

2.8.1 Back to Conservation of the Angular Momentum

2.9 Energy—Jacobi's h Integral

2.9.1 Energy Conservation

2.9.2 Energy and Galilean Relativity

2.9.3 Rheonomic Systems and Constants of Motion

2.10 The General Motion of a Rigid Body

2.10.1 Combined Translation and Rotation

2.11 Final Considerations About Lagrangian Mechanics

3 Hamiltonian Mechanics

3.1 Canonical Variables and the Hamiltonian Function

3.2 Hamilton's Equations

3.2.1 The Phase Space

3.3 What Is Really a Canonical Pair?

3.4 Hamilton's Principle in Phase Space

3.4.1 Symmetries and Cyclic Coordinates

3.4.2 Some Examples

3.5 Canonical Transformations

3.5.1 General Transformations

3.5.2 Canonical Transformation with K=H (Direct Substitution)

3.5.3 Canonical Transformation with K≠H

3.5.4 Poisson's Brackets

3.6 Infinitesimal Canonical Transformations and TemporalEvolution

3.6.1 Temporal Evolution as an Active Canonical Transformation

3.6.2 Infinitesimal Variation of a Dynamical Quantity

3.6.3 Poisson Brackets and Constants of Motion

3.7 Hamilton-Jacobi's Theory

3.7.1 Case K=0

3.7.2 Separation of Cyclic Variables

3.7.3 Case K(=H) Constant, Separation of Time

3.8 Hamilton-Jacobi's Perturbation Theory (HJ-PT)

3.8.1 Action-Angle Variables

3.9 Special Topic: A Classical Version of a Feshbach Resonance

3.10 Adiabatic Invariants

3.11 A Transition to Quantum Mechanics: Canonical Quantization

3.12 Final Considerations on Hamiltonian Mechanics

4 Lagrangian Theory of Classical Fields

4.1 Some Considerations Concerning Invariance Under Change of Inertial Frames

4.2 Classical Fields

4.3 Equations of Motion for Fields

4.4 Searching for Field Lagrangians

4.4.1 A Static Field

4.4.2 A Relativistic Field

4.4.3 Particle-Field Interactions

4.5 Final Considerations on Field Theory

Bibliography

Index

Contents

1 Concepts and Principles

1.1 Fundamentals

1.1.1 Dynamical Variables, Explicit and ImplicitDependences

1.2 Calculus of Variations

1.3 Hamilton's Principle

1.4 Constructing Lagrangians from Symmetries

1.5 ``Magic'' Lagrangians (One Particle)

1.6 Adapting Notation

1.7 Generalized Potentials

1.8 Equivalent Lagrangians

1.9 Newton's Laws from Hamilton's Principle

1.9.1 Newton's First Law: An Equivalent Lagrangian

1.9.2 Newton's Third Law: Space Homogeneity

1.10 Connection to Quantum Mechanics: The Classical Limit

1.11 Final Considerations on General Lagrangian Mechanics

2 Lagrangian Mechanics of Systems with L=T-V

2.1 Constraints and Generalized Coordinates

2.1.1 Holonomic Constraints

2.1.2 Generalized Coordinates an Velocities

2.1.3 Configuration Space

2.1.4 Transformation Equations

2.2 The Lagrangian

2.2.1 Virtual Displacements

2.3 Hamilton's Principle in Generalized Coordinates

2.3.1 Invariance of Lagrange's Equations Under Point Transformations

2.4 Newton's Second Law from Lagrange's Equations

2.5 Lagrange's Equations from Newton's Second Law

2.5.1 Absence of Constraints

2.5.2 Presence of Constraints: D'Alembert's Principle

2.5.3 The Lagrangian for Planar Motion of Rigid Bodies

2.6 Moving Constraints and Reference Frames

2.7 Symmetries of the Lagrangian and Conservation Theorems

2.7.1 Constants of Motion

2.7.2 Symmetries and Cyclical Coordinates

2.8 Nöther's Theorem

2.8.1 Back to Conservation of the Angular Momentum

2.9 Energy—Jacobi's h Integral

2.9.1 Energy Conservation

2.9.2 Energy and Galilean Relativity

2.9.3 Rheonomic Systems and Constants of Motion

2.10 The General Motion of a Rigid Body

2.10.1 Combined Translation and Rotation

2.11 Final Considerations About Lagrangian Mechanics

3 Hamiltonian Mechanics

3.1 Canonical Variables and the Hamiltonian Function

3.2 Hamilton's Equations

3.2.1 The Phase Space

3.3 What Is Really a Canonical Pair?

3.4 Hamilton's Principle in Phase Space

3.4.1 Symmetries and Cyclic Coordinates

3.4.2 Some Examples

3.5 Canonical Transformations

3.5.1 General Transformations

3.5.2 Canonical Transformation with K=H (Direct Substitution)

3.5.3 Canonical Transformation with K≠H

3.5.4 Poisson's Brackets

3.6 Infinitesimal Canonical Transformations and TemporalEvolution

3.6.1 Temporal Evolution as an Active Canonical Transformation

3.6.2 Infinitesimal Variation of a Dynamical Quantity

3.6.3 Poisson Brackets and Constants of Motion

3.7 Hamilton-Jacobi's Theory

3.7.1 Case K=0

3.7.2 Separation of Cyclic Variables

3.7.3 Case K(=H) Constant, Separation of Time

3.8 Hamilton-Jacobi's Perturbation Theory (HJ-PT)

3.8.1 Action-Angle Variables

3.9 Special Topic: A Classical Version of a Feshbach Resonance

3.10 Adiabatic Invariants

3.11 A Transition to Quantum Mechanics: Canonical Quantization

3.12 Final Considerations on Hamiltonian Mechanics

4 Lagrangian Theory of Classical Fields

4.1 Some Considerations Concerning Invariance Under Change of Inertial Frames

4.2 Classical Fields

4.3 Equations of Motion for Fields

4.4 Searching for Field Lagrangians

4.4.1 A Static Field

4.4.2 A Relativistic Field

4.4.3 Particle-Field Interactions

4.5 Final Considerations on Field Theory

Bibliography

Index