PHY 4222 "Mechanics II"
Class Diary for Spring 2019


Date Notes
Monday 1/7 Introductions, course policies. Syllabus. Textbook: John R. Taylor "Classical Mechanics"
Action item for the students who were not in the Mechanics I class last semester: Look at the diary from Mechanics I and review which sections from the book have been covered so far.
Review of the Newtonian and the Lagrangian formulation of Mechanics. Generalized coordinates.
Wednesday 1/9 Reading material: sections 13.1, 13.2.
Hamiltonian formulation of mechanics. Configuration space, state space and phase space. Generalized momenta. Solving for the generalized velocities in terms of the generalized momenta. Hamiltonian. Hamilton's equations. The procedure for deriving Hamilton's equations of motion (page 532) and illustrative examples: Example 13.1 A bead on a straight wire.
Friday 1/11 Reading material: sections 13.3, 13.4.
More examples:
Example 13.2 Atwood's machine.
Example 13.3 Hamilton's equations for a particle in a central force field. See also Problem 13.19.
Example 13.4 Hamilton's equations for a mass on a cone.
Monday 1/14 Reading material: sections 13.5, 13.6.
Problem 13.10.
Section 13.6. Phase-space orbits. Example 13.5 One-dimensional harmonic oscillator. Phase space orbits. Examples of Liouville's theorem.
Wednesday 1/16 Example 13.6 A falling body (see also Problem 13.2).
Problem 13.12: an example of a Hamiltonian not equal to the energy.
Friday 1/18 Homework 1 due today.
Section 13.4 Ignorable coordinates and conserved quantities. Problem 13.22.
Section 13.5 Lagrange vs Hamilton. Canonical transformations. Problem 13.25.
Begin Chapter 8. Reading material: Section 8.1 and 8.2. Formulation of the problem.
Monday 1/21 MLK Day - no class.
Wednesday 1/23 QUIZ DAY: Chapter 13, sections 1-6.
Chapter 8. Reading material: Sections 8.2 and 8.3. Center of mass and relative coordinates. Separation of variables (see Problem 8.1). Reduced mass (See Problem 8.4). The equations of motion. CM frame.
Friday 1/25 Chapter 8. Reading material: Section 8.4.
Solutions to the quiz. Section 8.4: The equivalent one-dimensional problem. Centrifugal force and centrifugal potential energy. Energy considerations. Effective potential energy. Bounded and unbounded orbits.
Monday 1/28 Chapter 8. Reading material: Section 8.5. Section 8.5: Derivation of the equation of the orbit. Example 8.3: The radial equation for a free particle.
Wednesday 1/30 Chapter 8. Reading material: Sections 8.6-8.8.
The Kepler orbits. Bounded and unbounded orbits. Problem 8.16. Bounded orbits: aphelion, perihelion, properties of an ellipse. Eccentricity. Section 8.7: Unbounded Kepler orbits. Problem 8.30. Properties of parabolas and hyperbolas. Conic sections.
Friday 2/1 Homework 2 accepted today.
The orbital period: Kepler's third law. Relation between energy and eccentricity. Example 8.4: Halley's comet. Section 8.8. Changes of orbit.
Fun reading in preparation for the big game on Sunday:
Projectile motion for quarterbacks: how to throw the perfect pass
Multivariate-data-visualization-based investigation of projectiles in sports
Monday 2/4 Homework 2 due today.
Reading material: Section 12.9. The logistic map (read only up to and including page 506). Maps versus functions. Examples: quadratic function; exponential function or multiplicative map (exponential growth). The logistic map.The carrying capacity of the model. Fixed points. Graphical representation. The fixed points of the logistic map. Test for the stability of a fixed point. Geometrical interpretation.
Wednesday 2/6 QUIZ DAY: Chapter 8 More on chaos. Test for the stability of a fixed point. Geometrical interpretation.
Cool video: Why do colliding blocks compute pi?
Friday 2/8 Reading material: Sections 12.1, 12.2, 12.3. Definition of chaos. Uncertainty in the initial conditions. Examples of chaotic systems. The necessary conditions for chaotic behavior: non-linearity and complexity. Section 12.1: Linearity and non-linearity. Superposition principle. Section 12.2: The Driven Damped Pendulum. Section 12.3: 1) The linear oscillator; 2) The nearly linear oscillations; 3) Appearance of harmonics of the drive frequency.
Cool videos and demos:
Chaos in the double pendulum
Chaos in the double pendulum illuminated
Rabits and foxes
Fractals from chaos
Monday 2/11 Homework 3 due today.
Review session in preparation for the first exam. The exam covers Chapters 8, 12 and 13.
Illustrations of orbits for central force motion. Phase space orbits. Orbits for inverse 3/2 and inverse 5/2 potentials. Problem 8.25 (Bertrand's theorem).
Overview of Quiz 2. Discussion of the effective potential, comparing different orbits, changes of orbits.
Discussion of the last problem from homework 3: Predator-prey model, stability of the fixed points.
Wednesday 2/13 Periodic testing: first midterm exam The exam will cover Chapters 8, 12 and 13.
Friday 2/15 Review of the solutions to the first exam.
Monday 2/18 Begin Chapter 9. Section 9.1. Acceleration without rotation. Example 9.1: Pendulum in an accelerating car. Effective gravity. Section 9.2. The Tides. Qualitative explanation of the tides. Tidal force. Magnitude of the tides. Tidal effect due to the Moon and the Sun. Spring tides and neap tides.
Wednesday 2/20 Section 9.3 "The Angular Velocity Vector". Arbitrary motion of an extended object: decomposing into a CM motion + rotation with respect to the CM. Euler's rotation theorem. Two ways to represent rotations: a rotation vector or a rotation matrix (example 15.6). The angular velocity vector as a time derivative of the rotation vector. Examples of simple rotations (about one of the coordinate axes). Sidebar on matrices (see pages 618 and 619 in the book): Identity matrix, inverse matrix, orthogonal matrix, determinant of a matrix.
Friday 2/22 Review of vector operations - see Section 1.2, subsection "Vector Operations" on pages 6-7. The useful relation (9.22). Addition of angular velocities. Notation for angular velocities.
Monday 2/25 Section 9.4 Time derivatives in a Rotating Frame. Section 9.5 Newton's second law in a rotating frame. Coriolis force and centrifugal force. (Problem 9.10: azimuthal force.) Section 9.6. The centrifugal force. Order of magnitude comparison of the Coriolis and centrifugal forces. Free-fall acceleration and the plumb line.
Wednesday 2/27 Section 9.7 The Coriolis Force. Direction of the Coriolis force. Problem 9.8. Air circulation in hurricanes and areas of high pressure: in place of Fig. 9.13 in the book, you can find a detailed discussion in my Meteorology notes, see in particular slides 11 and 16. Problem 9.16.
Friday 3/1 Homework 4 accepted today.
Example 9.2: Simple motion on a turntable. Section 9.8: Free fall and the Coriolis force. (Skip Section 9.9 The Foucault pendulum and Section 9.10 Coriolis force and Coriolis acceleration.) Begin Chapter 10. Rotational motion of rigid bodies. Begin Section 10.1 Properties of the center-of-mass (CM).
Monday 3/4 Spring Break - no class.
Wednesday 3/6 Spring Break - no class.
Friday 3/8 Spring Break - no class.
Monday 3/11 Homework 4 due today.
Finish Section 10.1 Properties of the center-of-mass (CM). The total momentum and the CM. The total angular momentum. Kinetic energy. Potential energy of a rigid body. Section 10.2. Rotation about a fixed axis. Moment of inertia. Products of inertia. Section 10.3. Rotation about any axis: the inertia tensor. The system of equations (10.36) relating the compoinents of the angular momentum and the components of the angular velocity. Symmetry property of the inertia tensor.
Wednesday 3/13 QUIZ DAY
Calculating moments of inertia and products of inertia. Examples 10.1, 10.2 and 10.3. Problem 10.26.
Friday 3/15 Section 10.4: Principal axes of inertia. Existence of Principal Axes theorem. Section 10.5: Finding the Principal Axes; Eigenvalue Equations. Eigenvalues and eigenvectors. Example 10.4: Principal axes of a cube about a corner. Diagonalization of the inertia tensor.
Monday 3/18 Homework 5 due today.
Review session in preparation for the second exam. Section 10.6: Precession of a top due to a weak torque. Precession of the equinoxes. Problem 10.10. Compound pendulum: Problems 10.13 and 10.18. "Sweet spot", i.e., center of percussion. Problem 10.16.
Additional practice problems: 10.3, 10.5, 10.7, 10.8.
Wednesday 3/20 Periodic testing: second exam The exam covers Chapters 9 and 10.
Friday 3/22 Begin Chapter 11. Coupled Oscillators and Normal Modes. Section 11.5. The general case.
Monday 3/25 Section 11.1. Two masses and three springs. Section 11.2. Identical springs and equal masses. Normal coordinates (see problem 11.9).
Wednesday 3/27 Section 11.3. Two weakly coupled oscillators. Section 11.4. Lagrangian approach: the double pendulum.
Friday 3/29 Section 11.6. Three coupled pendulums. Problem 11.27.
Monday 4/1 Homework 6 due today.
Begin Chapter 14. Scattering experiments in high energy physics. The need for probabilistic description. Section 14.1. The scattering angle and impact parameter. Section 14.2 The collision cross-section.
Cool video: Why do colliding blocks compute pi?
Wednesday 4/3 QUIZ DAY Continue with Chapter 14. Example 14.1 Shooting crows in a tree. Section 14.3. Generalizations of the cross-section. Elastic versus inelastic cross-sections. Various types of inelastic processes: ionization, capture, fission. Begin Section 14.4. The Differential scattering cross-section.
Friday 4/5 Continue with Chapter 14. Solid angle. The differential cross section. Example 14.4. Section 14.5 Calculating the differential cross-section. Hard sphere scattering.
Monday 4/8 Homework 7 due today.
Finish Chapter 14. Section 14.6 Rutherford scattering. Problems 14.19 and 14.21.
Wednesday 4/10 QUIZ DAY Begin Chapter 16 (wave equations). Continuum mechanics versus discrete mechanics. The continuum hypothesis. Section 16.1. Transverse motion of a taut string. Derivation of the wave equation in 1 dimension. Section 16.2. The wave equation. General solutions to the wave equation.
Friday 4/12 Example 16.1. Evolution of a triangular pulse. Section 16.3: Boundary conditions; waves on a finite string. Normal modes. Fundamental frequency. Harmonics. Skip the subsection "The general solution" starting on page 691.
Monday 4/15 Section 16.4: The three-dimensional wave equation. The Laplacian. Plane waves and spherical waves. Section 16.5: Volume and surface forces. Pressure, tension, shear. (Skip the subsection "When is pressure isotropic" starting on page 699.) Section 16.6: Stress and strain: the elastic moduli.
Wednesday 4/17 Section 16.7: The stress tensor. (Skip the derivation of eq. (16.66) and the subsection "The stress tensor is symmetric".) Example 16.3. The stress tensor in a static fluid. Example 16.4. Numerical example of stress. Section 16.8. The strain tensor for a solid. Example 16.5. Dilatation. Decomposition of the general strain tensor. Relation between stress and strain - Hooke's law.
Friday 4/19 Homework 8 due today.
Review session in preparation for the third exam.
Monday 4/22 Periodic testing: third exam
Wednesday 4/24 Review of the solutions to the third exam.