Lagrangian and Hamiltonian Mechanics : A Modern Approach with Core Principles and Underlying Topics

This book serves as a textbook for an analytical mechanics course, a fundamental subject of physics, that pays special attention to important topics that are not discussed in most standard textbooks. Readers are provided with a clear understanding of topics that are usually inaccessible to the undergraduate level and that are critical to learning Lagrangian and Hamiltonian mechanics. Each chapter also includes worked problems and solutions, as well as additional exercises for readers to try. This book begins with the fundamentals of analytical mechanics, concisely introducing readers to the calculus of variations, Hamilton’s Principle, and Lagrange’s equations. While presenting readers with these core topics, the author uses an intuitive approach to delve into essential questions, such as where Galilean invariance lies in Lagrangian mechanics and how Hamilton’s Principle of Least Action encompasses Newton’s three laws, interesting conclusions that often go unnoticed. Infact, Hamilton’s principle is taken throughout as the very origin of classical physical laws, and the choice of appropriate Lagrangians in each case as the real theoretical challenge, meaning that forms of Lagrangian which differ from the standard one are not mere curiosities but, instead, the general rule. This book clarifies common misunderstandings that students face when learning the subject and formally rationalizes concepts that are often difficult to grasp. In addition, the final chapter provides an introduction to a Lagrangian field theory for those interested in learning more advanced topics. Ideal for upper undergraduate and graduate students, this book seeks to teach the intrinsic meaning of the principles and equations taught in an analytical mechanics course and convey their usefulness as powerful theoretical instruments of modern physics. 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