Variational Principles in Physics

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
6.2.2 Hamiltonian Formalism
6.3 Scalar Field
6.4 Electromagnetic Field
6.5 Equations of First Order in Time
6.5.1 Diffusion Equation
6.5.2 Schrödinger Equation
6.6 Problem
7 Motion in a Curved Space
7.1 The Equivalence Principle
7.2 Curved Spaces
7.2.1 Generalities
7.2.2 The Light Rays, Geodesics of Our Space
7.2.3 Metric Tensor
7.2.4 Examples
7.3 Free Motion in a Curved Space
7.3.1 Lagrangian
7.3.2 Equations of Motion
7.3.3 Simple Examples
7.4 Geodesic Lines
7.4.1 Definition
7.4.2 Equation of the Geodesics
7.4.3 Examples
7.4.4 Maupertuis Principle and Geodesics
7.5 Gravitation and the Curvature of Space-Time
7.5.1 Newtonian Gravitation and Relativity
7.5.2 The Schwarzschild Metric
7.5.3 Gravitation and Time Flow
7.5.4 Precession of Mercury's Perihelion
7.5.5 Gravitational Deflection of Light Rays
7.6 Gravitational Optics and Mirages
7.6.1 Gravitational Lensing
7.6.2 Gravitational Mirages
7.6.3 Observation of a Double Quasar
7.6.4 Baryonic Dark Matter
7.7 Exercises
7.8 Problem. Motion on the Sphere S3
8 Gravitational Waves
8.1 Evolution of Space-Time
8.2 Detection of Gravitational Waves
8.3 Gravitational Waves
8.3.1 Quadrupole Waves; Orders of Magnitude
8.3.2 Formation and Propagation of the Gravitational Waves
8.3.3 Linearization of Einstein's Equations
8.3.4 Polarization: Transverse Traceless Waves
8.3.5 Detection of the Waves
8.3.6 Generation of Gravitational Waves
8.3.7 Radiated Power
8.4 Binary System
8.4.1 Motion of Two Stars
8.4.2 Transmitted Gravitational Waves
8.4.3 Binary System Energy Loss
8.5 Double Pulsar Discovery PSR B1913+16
9 Feynman's Path Integrals in Quantum Mechanics
9.1 The Initial Click
9.2 Feynman's Principle
9.3 The Path Integral
9.3.1 Recollections of Analytical Mechanics
9.3.2 Quantum Amplitudes
9.3.3 Superposition Principle and Feynman's Principle
9.3.4 Path Integral
9.3.5 Amplitude of Successive Events
9.4 Free Particle
9.4.1 Propagator of a Free Particle
9.4.2 Evolution Equation of the Free Propagator
9.4.3 Normalization and Interpretation of the Propagator
9.4.4 Fourier and Schrödinger Equations
9.4.5 Energy and Momentum of a Free Particle
9.4.6 Interference and Diffraction
9.5 Wave Function and the Schrödinger Equation
9.5.1 Free Particle
9.5.2 Particle in a Potential
9.5.3 Hamiltonian Operator and Consequences
9.5.4 Conservation of Probability
9.5.5 Stationary States
9.6 The Momentum
9.6.1 Momentum Measurement
9.6.2 Probability Amplitude
9.6.3 Fourier Transformation
9.7 Concluding Remarks
9.7.1 Classical Limit
9.7.2 The Difficulty of Spin 1/2
9.7.3 Optics and Analytical Mechanics
9.7.4 The Role of the Phase
9.8 Exercises
Appendix Solutions of the Problems and Exercises
Appendix References
Index