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Table of Contents
Intro
Preface
Contents
1 Structure of Physical Theories
2 Variational Principles
2.1 Fermat's Least Time Principle
2.2 Variational Calculus of Euler and Lagrange
2.2.1 First Integrals, Cyclic Variables
2.2.2 Mirages and Curved Rays
2.3 Maupertuis, Principle of Least Action
2.3.1 Electrostatic Potential
2.4 Thermodynamic Equilibrium: Maximal Disorder
2.4.1 Principle of Equal Probability of States
2.4.2 Most Probable Distribution and Equilibrium
2.4.3 Lagrange Multipliers
2.4.4 Boltzmann Factor
2.4.5 Equalization of Temperatures
2.4.6 The Ideal Gas
2.4.7 Boltzmann's Entropy
2.4.8 Heat and Work
2.5 Exercises
2.6 Problem. Win a Downhill
3 The Analytical Mechanics of Lagrange
3.1 Lagrangian Formalism and Least Action
3.1.1 Least Action Principle
3.1.2 Lagrange-Euler Equations
3.1.3 Operation of the Optimization Principle
3.2 Invariances and Conservation Laws
3.2.1 Conjugate Momenta and Generalized Momenta
3.2.2 Cyclic Variables
3.2.3 Energy and Translations in Time
3.2.4 Noether Theorem: Symmetries and Conservation Laws
3.2.5 Momentum and Translations in Space
3.2.6 Angular Momentum and Rotations
3.2.7 Dynamical Symmetries
3.3 Velocity-Dependent Forces
3.3.1 Dissipative Systems
3.3.2 Lorentz Force
3.3.3 Gauge Invariance
3.3.4 Momentum
3.4 Lagrangian of a Relativistic Particle
3.4.1 Lorentz Transformation
3.4.2 Free Particle
3.4.3 Energy and Momentum
3.4.4 Interaction with an Electromagnetic Field
3.5 Exercises
3.6 Problem. Strategy of a Regatta
4 Hamilton's Canonical Formalism
4.1 Hamilton's Canonical Formalism
4.1.1 Canonical Equations
4.2 Poisson Brackets, Phase Space
4.2.1 Time Evolution, Constants of the Motion
4.2.2 Relation Between Analytical and Quantum Mechanics
4.3 Canonical Transformations in Phase Space
4.4 Evolution in Phase Space: Liouville's Theorem
4.5 Charged Particle in an Electromagnetic Field
4.5.1 Hamiltonian
4.5.2 Gauge Invariance
4.6 Dynamical Systems
4.6.1 The Contribution of Henri Poincaré
4.6.2 Poincaré and Chaos in the Solar System
4.6.3 Poincaré's Recurrence Theorem
4.6.4 The Butterfly Effect
the Lorenz Attractor
4.7 Exercises
4.8 Problem. Closed Chain of Coupled Oscillators
5 Action, Optics, Hamilton-Jacobi Equation
5.1 Geometrical Optics, Characteristic Function of Hamilton
5.2 Action and the Hamilton-Jacobi Equation
5.2.1 The Action as a Function of Coordinates and Time
5.2.2 Least Action Principle
5.2.3 Hamilton-Jacobi Equation
5.2.4 Conservative Systems, Reduced Action, Maupertuis Principle
5.3 Semi-Classical Approximation in Quantum Mechanics
5.4 Hamilton-Jacobi Formalism
5.5 Exercises
6 Lagrangian Field Theory
6.1 Vibrating String
6.2 Field Equations
6.2.1 Generalized Lagrange-Euler Equations
Preface
Contents
1 Structure of Physical Theories
2 Variational Principles
2.1 Fermat's Least Time Principle
2.2 Variational Calculus of Euler and Lagrange
2.2.1 First Integrals, Cyclic Variables
2.2.2 Mirages and Curved Rays
2.3 Maupertuis, Principle of Least Action
2.3.1 Electrostatic Potential
2.4 Thermodynamic Equilibrium: Maximal Disorder
2.4.1 Principle of Equal Probability of States
2.4.2 Most Probable Distribution and Equilibrium
2.4.3 Lagrange Multipliers
2.4.4 Boltzmann Factor
2.4.5 Equalization of Temperatures
2.4.6 The Ideal Gas
2.4.7 Boltzmann's Entropy
2.4.8 Heat and Work
2.5 Exercises
2.6 Problem. Win a Downhill
3 The Analytical Mechanics of Lagrange
3.1 Lagrangian Formalism and Least Action
3.1.1 Least Action Principle
3.1.2 Lagrange-Euler Equations
3.1.3 Operation of the Optimization Principle
3.2 Invariances and Conservation Laws
3.2.1 Conjugate Momenta and Generalized Momenta
3.2.2 Cyclic Variables
3.2.3 Energy and Translations in Time
3.2.4 Noether Theorem: Symmetries and Conservation Laws
3.2.5 Momentum and Translations in Space
3.2.6 Angular Momentum and Rotations
3.2.7 Dynamical Symmetries
3.3 Velocity-Dependent Forces
3.3.1 Dissipative Systems
3.3.2 Lorentz Force
3.3.3 Gauge Invariance
3.3.4 Momentum
3.4 Lagrangian of a Relativistic Particle
3.4.1 Lorentz Transformation
3.4.2 Free Particle
3.4.3 Energy and Momentum
3.4.4 Interaction with an Electromagnetic Field
3.5 Exercises
3.6 Problem. Strategy of a Regatta
4 Hamilton's Canonical Formalism
4.1 Hamilton's Canonical Formalism
4.1.1 Canonical Equations
4.2 Poisson Brackets, Phase Space
4.2.1 Time Evolution, Constants of the Motion
4.2.2 Relation Between Analytical and Quantum Mechanics
4.3 Canonical Transformations in Phase Space
4.4 Evolution in Phase Space: Liouville's Theorem
4.5 Charged Particle in an Electromagnetic Field
4.5.1 Hamiltonian
4.5.2 Gauge Invariance
4.6 Dynamical Systems
4.6.1 The Contribution of Henri Poincaré
4.6.2 Poincaré and Chaos in the Solar System
4.6.3 Poincaré's Recurrence Theorem
4.6.4 The Butterfly Effect
the Lorenz Attractor
4.7 Exercises
4.8 Problem. Closed Chain of Coupled Oscillators
5 Action, Optics, Hamilton-Jacobi Equation
5.1 Geometrical Optics, Characteristic Function of Hamilton
5.2 Action and the Hamilton-Jacobi Equation
5.2.1 The Action as a Function of Coordinates and Time
5.2.2 Least Action Principle
5.2.3 Hamilton-Jacobi Equation
5.2.4 Conservative Systems, Reduced Action, Maupertuis Principle
5.3 Semi-Classical Approximation in Quantum Mechanics
5.4 Hamilton-Jacobi Formalism
5.5 Exercises
6 Lagrangian Field Theory
6.1 Vibrating String
6.2 Field Equations
6.2.1 Generalized Lagrange-Euler Equations