Books by Bryan B. Salazar
Lagrangians and Hamiltonians, 2024
A concise but rigorous treatment of variational techniques, focusing primarily on Lagrangian and ... more A concise but rigorous treatment of variational techniques, focusing primarily on Lagrangian and Hamiltonian systems, this book is ideal for physics, engineering and mathematics students. The book begins by applying Lagrange's equations to a number of mechanical systems. It introduces the concepts of generalized coordinates and generalized momentum. Following this, the book turns to the calculus of variations to derive the Euler-Lagrange equations. It introduces Hamilton's principle and uses this throughout the book to derive further results. The Hamiltonian, Hamilton's equations, canonical transformations, Poisson brackets and Hamilton-Jacobi theory are considered next. The book concludes by discussing continuous Lagrangians and Hamiltonians and how they are related to field theory. Written in clear, simple language, and featuring numerous worked examples and exercises to help students master the material, this book is a valuable supplement to courses in mechanics. Contents I LAGRANGIAN MECHANICS 1 Fundamental concepts 1.1 Kinematics 1.2 Generalized coordinates 1.3 Generalized velocity 1.4 Constraints 1.5 Virtual displacements 1.6 Virtual work and generalized force 1.7 Configuration space 1.8 Phase space 1.9 Dynamics 1.9.1 Newton's laws of motion 1.9.2 The equation of motion 1.9.3 Newton and Leibniz 1.10 Obtaining the equation of motion 1.10.1 The equation of motion in Newtonian mechanics 1.10.2 The equation of motion in Lagrangian mechanics 1.11 Conservation laws and symmetry principles 1.11.1 Generalized momentum and cyclic coordinates 1.11.2 The conservation of linear momentum 1.11.3 The conservation of angular momentum 1.11.4 The conservation of energy and the work function 1.12 Problems v vi Contents 2 The calculus of variations 2.1 Introduction 2.2 Derivation of the Euler-Lagrange equation 2.2.1 The difference between δ and d 2.2.2 Alternate forms of the Euler-Lagrange equation 2.3 Generalization to several dependent variables 2.4 Constraints 2.4.1 Holonomic constraints 2.4.2 Non-holonomic constraints 2.5 Problems 3 Lagrangian dynamics 3.1 The principle of d'Alembert. A derivation of Lagrange's equations 3.2 Hamilton's principle 3.3 Derivation of Lagrange's equations 3.4 Generalization to many coordinates 3.5 Constraints and Lagrange's λ-method 3.6 Non-holonomic constraints 3.7 Virtual work 3.7.1 Physical interpretation of the Lagrange multipliers 3.8 The invariance of the Lagrange equations 3.9 Problems II HAMILTONIAN MECHANICS 4 Hamilton's equations 4.1 The Legendre transformation 4.1.1 Application to thermodynamics 4.2 Application to the Lagrangian. The Hamiltonian 4.3 Hamilton's canonical equations 4.4 Derivation of Hamilton's equations from Hamilton's principle 4.5 Phase space and the phase fluid 4.6 Cyclic coordinates and the Routhian procedure 4.7 Symplectic notation 4.8 Problems 5 Canonical transformations; Poisson brackets 5.1 Integrating the equations of motion 5.2 Canonical transformations Contents vii 5.3 Poisson brackets 5.4 The equations of motion in terms of Poisson brackets 5.4.1 Infinitesimal canonical transformations 5.4.2 Canonical invariants 5.4.3 Liouville's theorem 5.4.4 Angular momentum 5.5 Angular momentum in Poisson brackets 5.6 Problems 6 Hamilton-Jacobi theory 6.1 The Hamilton-Jacobi equation 6.2 The harmonic oscillator-an example 6.3 Interpretation of Hamilton's principal function 6.4 Relationship to Schrödinger's equation 6.5 Problems 7 Continuous systems 7.1 A string 7.2 Generalization to three dimensions 7.3 The Hamiltonian density 7.4 Another one-dimensional system 7.4.1 The limit of a continuous rod 7.4.2 The continuous Hamiltonian and the canonical field equations 7.5 The electromagnetic field 7.6 Conclusion 7.7 Problems Further reading Answers to selected problems Index 3 Be aware that the fields of analytical mechanics and the calculus of variations are vast and this book is limited to presenting some fundamental concepts.
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Books by Bryan B. Salazar