No-Nonsense Classical Mechanics: A Student-Friendly Introduction

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Bookmarks are added by using Master PDF. Preface Contents Bird’s-Eye View of Classical Mechanics Part I What Everybody Ought to Know About Classical Mechanics 2 Fundamental Concepts 2.1 Basic Quantities 2.1.1 Mass 2.1.2 Momentum and Angular Momentum 2.1.3 Energy 2.2 Equations of Motion 2.3 Mathematical Arenas 2.3.1 Configuration Space 2.3.2 Phase Space 3 Newtonian Mechanics 4 Lagrangian Mechanics 4.1 Action and the Lagrangian 4.2 Variational Calculus 4.3 The Euler-Lagrange Equation 4.3.1 Meaning of the Euler-Lagrange Equation 5 Hamiltonian Mechanics 5.1 Hamilton’s Equations 5.2 Hamilton’s General Equation 6 Summary Part II Essential Systems and Tools 7 Algorithms and Tools 7.1 The Newtonian Algorithm 7.2 The Lagrangian Algorithm 7.2.1 Constraints 7.2.2 Point Transformations and Generalized Coordinates 7.2.3 Gauge Transformations 7.3 The Hamiltonian Algorithm 7.3.1 Canonical Transformations and Canonical Coordinates 7.3.2 Canonical Point and Gauge Transformations 7.3.3 Infinitesimal Canonical Transformation 7.3.4 Generating Functions 8 The Harmonic Oscillator 8.1 Newtonian Description 8.2 Lagrangian Description 8.3 Hamiltonian Description 8.4 Solving the Equation of Motion 8.5 Solving the Harmonic Oscillator Using a Canonical Transformation 9 The Pendulum 9.1 Newtonian Description 9.2 Lagrangian Description 9.3 Hamiltonian Description 9.4 Solving the Equation of Motion Part III Get an Understanding of Classical Mechanics You Can Be Proud Of 10 Noether’s Theorem 10.1 Symmetries 10.2 Noether’s Theorem Intuitively 10.3 Noether’s Theorem in the Hamiltonian Formalism 10.3.1 Noether’s Extended Theorem 10.3.2 Noether’s Converse Theorem 10.4 Noether’s Theorem in the LagrangianFormalism 10.4.1 Noether’s Extended Theorem 10.5 Summary 11 Additional Formulations of Classical Mechanics 11.1 Hamilton-Jacobi Mechanics 11.1.1 Meaning of Hamilton’s Principal Function 11.1.2 Harmonic Oscillator 11.2 Statistical Mechanics 11.2.1 Probability Density 11.2.2 Conservation of Phase Space Paths 11.2.3 Liouville’s Equation 11.2.4 Liouville’s Theorem 11.3 Koopman-von Neumann Mechanics 11.3.1 Hilbert Space 11.3.2 Koopman-von Neumann Equation 12 The Origins of Classical Mechanics 12.1 The Cube of Physics 12.2 The Origin of the Least Action Principle 12.2.1 The Origin of the Classical Path 12.2.2 The Origin of the Hamilton-Jacobi Equation 12.3 The Origin of the Classical Lagrangian 12.3.1 Special Relativity 12.3.2 Time Dilation 12.3.3 The Lagrangian of Special Relativity 12.3.4 The Free Classical Lagrangian 12.3.5 Understanding the Minus Sign 12.3.6 General Relativity 12.4 Lagrangians in Modern Physics 13 Further Reading Recommendations One Last Thing Part IV Appendices A Calculus A.1 Product Rule A.2 Integration by Parts A.3 Total and Partial Derivatives B The Legendre Transform C Lagrange Multipliers D Invariance, Covariance and Functional Form E Active vs. Passive Transformations and Symmetries vs. Redundancies F Taylor Expansion G Vector Calculus Bibliography Index