No-Nonsense Classical Mechanics: A Student-Friendly Introduction

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