PH 611,2,3,4: Theoretical Mechanics and Statistical Mechanics
AY 2013/14 (D. Belitz)
Chapter 1: The Mathematical Principles of Mechanics
Chapter 2: Mechanics of Point Masses
$1: Point masses in potentials
1.1 A single free particle
1.2 Galileo's principle of relativity
1.3 Potentials
1.4 Equations of motion
1.5 Systems of point masses
$2: Simple examples for the motion of point masses
2.1 Galileo's law of falling bodies
2.2 A particle on an inclined plane
2.3 A particle on a rotating pole
2.4 The harmonic oscillator
$3: One-dimensional conservative systems
3.1 Definition of the model
3.2 Solution of the equations of motion
3.3 Unbounded motinon
3.4 Bounded motion
3.5 Equilibrium points
$4: Motion in a central field
4.1 Reduction to a one-dimensional problem
4.2 Kepler's second law
4.3 The radial motion
4.4 The equation of the orbit
4.5 Classification of orbits
4.6 Kepler's problem
4.7 Kepler's laws of planetary motion
$5: Introduction to perturbation theory
5.1 The general concept
5.2 Digression: Fourier series
5.3 Oscillations and Fourier series
5.4 The anharmonic oscillator
5.5 Perturbation theory for the central field problem
5.6 Perturbations of Kepler's problem
5.7 Digression: Introduction to potential theory
5.8 Mercury's perihelion advance due to an oblate sun
$6: Scattering theory
6.1 Scattering experiments
6.2 Classical theory for the scattering cross section
6.3 Scattering by a central potential
6.4 Rutherford scattering
6.5 Scattering by a hard sphere
$7: N-particle systems
7.1 Closed N-particle systems
7.2 The two-body problem
Chapter 3: The Rigid Body
$1: Mathematical preliminaries: Vector spaces and tensor spaces
1.1 Metric vector spaces
1.2 Coordinate transformations
1.3 Proper coordinate systems
1.4 Proper coordinate transformations
1.5 Tensor spaces
$2: Orthogonal transformations
2.1 3-d Euclidian space
2.2 Special orthogonal transformations: The group SO(3)
2.3 Rotations about a fixed axis
2.4 Infinitesimal rotations: The generators of SO(3)
2.5 Euler angles
$3: The rigid body model
3.1 Parameterization of a rigid body
3.2 Angular velocity
3.3 The inertia tensor
3.4 Classification of rigid bodies
3.5 Angular momentum of a rigid body
3.6 Continuum model of a rigid body
3.7 Euler equations
$4: Simple examples of rigid-body motion
4.1 The physical pendulum
4.2 A cylinder on an inclined plane
4.3 The force-free symmetric top
4.4 The force-free asymmetric top
Chapter 4: Principles of Statistical Mechanics
$1: Hamilton's equations ( pp. 127 - 136 , p. 137 )
1.1 Hamilton's equations, and phase space
1.2 Phase flow
1.3 Liouville's theorem
1.4 Poicare's theorem
1.5 Necessity of a statistical description of many-particle systems
$2: Elements of probability theory
2.1 The definition of probability
2.2 Discrete random variables
2.3 Continuous random variables
2.4 Averages and fluctuations
2.5 The binomial distribution
2.6 The Gaussian or normal distribution
2.7 The central limit theorem
$3: Review of thermodynamics
3.1 Statistical description of large systems
3.2 The equilibrium state
3.3 Interacting systems
3.4 Reversible and irreversible processes
3.5 Energy, temperature, and entropy
3.6 The laws of thermodynamics
3.7 Thermodynamic potentials
$4: Statistical ensembles
4.1 Gibbsian ensembles
4.2 The microcanonical ensemble
4.3 The canonical ensemble
4.4 The grand canonical ensemble
4.5 The thermodynamic limit, and the Duhem-Gibbs relation
4.6 Systems in magnetic fields
$5: Quantum statistical mechanics
5.1 The postulates of quantum statistical mechanics
5.2 The statistical operator for various ensembles
5.3 Fermions and bosons
5.4 The Fermi-Dirac distribution
5.5 The Bose-Einstein distribution
Chapter 5: Selected Applications
$1: Classical systems
1.1 The classical, monatomic ideal gas
1.2 The Gibbs paradox
1.3 The equipartition theorem
1.4 Maxwell's velocity distribution I: Canonical ensemble
1.5 Maxwell's velocity distribution II: Microcanonical ensemble
$2: The ideal Fermi gas
2.1 Distribution functions, and the equation of state (for both fermions and bosons)
2.2 The degenerate electron gas
2.3 The specific heat of the degenerate electron gas (Sommerfeld expansion)
2.4 Pauli paramagnetism
2.5 Landau diamagnetism
$3: The ideal Bose gas
3.1 Bose-Einstein condensation
3.2 Chemical potential and particle number for T < T_{0}
3.3 The specific heat of an ideal Bose gas
$4: Blackbody radiation
4.1 Planck's law
4.2 Discussion of Planck's law
$1: The Boltzmann equation
1.1 The fundamental problem of kinetic theory
1.2 The collision operator I: The `scattering-out' contribution
1.3 The collision operator II: The `scattering-in' contribution
1.4 The Boltzmann collision operator
$4: Fundamental properties, and simple solutions, of the Boltzmann equation
2.1 Collision invariants, conservation laws, and continuity equations
2.2 Boltzmann's H-theorem
2.3 Equilibrium solution of the Boltzmann equation