| Date | Notes |
| Monday 01/08 |
Introductions, course policies. Syllabus. Textbook: John R. Taylor "Classical Mechanics"
Action item for the students who did not take my Mechanics I class: Look at the diary from Mechanics I and review which sections from the book have been covered so far. |
| Wednesday 01/10 |
Reading material: sections 13.1, 13.2.
Review of the Newtonian and the Lagrangian formulation of Mechanics. Generalized coordinates. Hamiltonian formulation of mechanics. Configuration space, state space and phase space. Generalized momenta. Hamiltonian. Hamilton's equations. |
| Friday 01/12 |
Reading material: sections 13.3, 13.4.
Solving for the generalized velocities in terms of the generalized momenta. The procedure for deriving Hamilton's equations of motion (page 532) and illustrative examples: Example 13.1 A bead on a straight wire. Example 13.2 Atwood's machine. |
| Monday 01/15 | Martin Luther King Jr. Day - no class. |
| wednesday 01/17 |
Reading material: sections 13.4 and 13.5.
Section 13.4 Ignorable coordinates and conserved quantities. Problem 13.22. Section 13.5 Lagrange vs Hamilton. 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. |
| Friday 01/19 |
Reading material: section 13.6. (Skip Section 13.7)
Section 13.6. Phase-space orbits. Example 13.5 One-dimensional harmonic oscillator. Phase space orbits. Canonical transformations. Problem 13.25. Problem 13.10. Hamilton's equations for a two-dimensional system. (Optional extra reading (not on the exam):) Section 13.7. Examples of Liouville's theorem. Harmonic oscillator. Example 13.6 A falling body (see also Problem 13.2). |
| Monday 01/22 |
Homework 1 due today. Problem 13.12: an example of a Hamiltonian not equal to the energy. Begin Chapter 8. Reading material: Sections 8.1 and 8.2 Formulation of the problem. 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. |
| Wednesday 01/24 |
Chapter 8. Reading material: Sections 8.3 and 8.4. Section 8.4: The equivalent one-dimensional problem. Centrifugal force and centrifugal potential energy. |
| Friday 01/26 | Chapter 8. Reading material: Section 8.5. Energy considerations. Effective potential energy. Bounded and unbounded orbits. Section 8.5: Derivation of the equation of the orbit. |
| Monday 01/29 |
Sidebar: Conic sections and their properties. Reading material: Sections 8.6-8.7. Finish the derivation of the equation of the orbit. Example 8.3: The radial equation for a free particle. The Kepler orbits. Bounded and unbounded orbits. Problem 8.16. Bounded orbits: aphelion, perihelion, properties of an ellipse. Eccentricity. |
| Wednesday 01/31 |
Homework 2 due today.
Reading material: Section 8.8. Section 8.7: Unbounded Kepler orbits. Problem 8.30. Properties of parabolas and hyperbolas. Conic sections. The orbital period: Kepler's third law. Relation between energy and eccentricity. Example 8.4: Halley's comet. Section 8.8. Changes of orbit. |
| Friday 02/02 |
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. |
| Monday 02/05 |
More on chaos. Test for the stability of a fixed point. Geometrical interpretation.
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.
Discussion of the last problem from homework 3: Predator-prey model, stability of the fixed points. Cool videos and demos: Chaos in the double pendulum Chaos in the double pendulum illuminated Balls falling on a circle Rabits and foxes Why do colliding blocks compute pi? |
| Wednesday 02/07 |
Homework 3 due today. Review session in preparation for the first exam. The exam covers Chapters 8, 12 and 13. Iterative maps: graphical solution and interpretation of fixed points. Computer exercises: Lotka-Volterra equations, driven damped pendulum. |
| Friday 02/09 |
Periodic testing: first midterm exam.
The exam will cover Chapters 8, 12 and 13.
Fun reading after the exam and before the big game Sunday: |
| Monday 02/12 | 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. Neil deGrasse Tyson's nice explanation of the tides. |
| Wednesday 02/14 |
Review of the solutions to the first exam (if needed). Tidal effect due to the Moon and the Sun. Spring tides and neap tides. Quantitative discussion. Tidal force. Magnitude of the tides. 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). Sidebar on matrices (see pages 618 and 619 in the book): Identity matrix, inverse matrix, orthogonal matrix, determinant of a matrix. Examples of simple rotations (about one of the coordinate axes). |
| Friday 02/16 |
Review of vector operations - see Section 1.2, subsection "Vector Operations" on pages 6-7.
The angular velocity vector as a time derivative of the rotation vector. The useful relation (9.22). Addition of angular velocities. Notation for angular velocities. 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.) |
| Monday 02/19 |
Guest lecture:
Symmetries in physics. Definition of a symmetry. Example: rotations in two dimensions. Properties of 2D rotations. Group theory: formal definition of a group. Orthogonal matrices, the groups O(2) and SO(2). Conserved quantity under rotations. Unitary groups. Unitary matrices. U(1) description of 2D rotations. Conserved quantity. |
| Wednesday 02/21 |
Guest lecture:
Abelian groups. Examples: U(1) and SO(2). Rotations in three dimensions. Lie algebras. Generators. Generators of 2D rotations. Generators of 3D rotations. SU(2) description of rotations in 3 dimensions. Group invariants of SO(3) and SU(2). Summary of the concepts learned during the two guest lectures. A copy of the guest lectures. References for further reading: Using machine learning to derive the generators of the orthogonal and Lorentz and unitary groups. |
| Friday 02/23 |
Section 9.6. The centrifugal force. Order of magnitude comparison of the Coriolis and centrifugal forces. Free-fall acceleration and the plumb line. Section 9.7 The Coriolis Force. Direction of the Coriolis force. Problem 9.8. Problem 9.16. Example 9.2: Simple motion on a turntable. 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. Section 9.9. The Foucault Pendulum. Section 9.10. Coriolis force and Coriolis acceleration. |
| Monday 02/26 |
Homework 4 due today.
Section 9.8: Free fall and the Coriolis force. Problem 9.27. Begin Chapter 10. Rotational motion of rigid bodies. 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. |
| Wednesday 02/28 | Section 10.3. Rotation about any axis: the inertia tensor. The system of equations (10.36) relating the components of the angular momentum and the components of the angular velocity. Symmetry property of the inertia tensor. Calculating moments of inertia and products of inertia. Eigenvalues of the inertia tensor. Diagonalization of the inertia tensor. Section 10.4: Principal axes of inertia. Existence of Principal Axes theorem. |
| Friday 03/01 |
Example 10.1. Calculating simple moments and products of inertia. Example 10.2. Inertia tensor for a solid cube. Example 10.3. Inertia tensor for a solid cone. Problem 10.26. A copy of the lecture can be found here. |
| Monday 03/04 |
Homework 5 due today.
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. Review session in preparation for the second exam. The following moments of inertia formula sheet will be available on the exam. The tennis racket theorem (the Dzhanibekov effect). Additional practice problems: 10.3, 10.5, 10.7, 10.8. Problem 10.10. Problem 10.13: compound pendulum. Note: we are skipping sections 10.6-10.10. For those who need a refresher on linear algebra, see the series series by 3blue1brown, in particular eigenvectors and eigenvalues (chapter14). |
| Wednesday 03/06 | Periodic testing: second exam The exam covers Chapters 9 and 10 EXCEPT sections 10.6, 10.7, 10.8, 10.9 and 10.10. |
| Friday 03/08 |
Guest lecture:
Diagonalizing real symmetric matrices: finding eigenvalues and eigenvectors.
A copy of the guest lectures. |
| Monday 03/11 | Spring break - no class. |
| Wednesday 03/13 | Spring break - no class. |
| Friday 03/15 | Spring break - no class. |
| Monday 03/18 | Begin Chapter 11. Coupled Oscillators and Normal Modes. Section 11.5. The general case. |
| Wednesday 03/20 | Section 11.1. Two masses and three springs. Section 11.2. Identical springs and equal masses. Normal coordinates (see problem 11.9). Problem 11.27. |
| Friday 03/22 | Section 11.3. Two weakly coupled oscillators. |
| Monday 03/25 |
Section 11.4. Lagrangian approach: the double pendulum.
Section 11.6. Three coupled pendulums. Extra practice: Problems 11.14, 11.28, 11.20 and 11.29. |
| Wednesday 03/27 |
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. 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. |
| Friday 03/29 |
Discussion of the different types of cross-sections at the Large Hadron Collider. Section 14.4. The Differential scattering cross-section. Solid angle. Example 14.4. Section 14.5 Calculating the differential cross-section. Hard sphere scattering. Cool video: Why do colliding blocks compute pi? |
| Monday 04/01 |
April Fool's Day. Section 14.6 Rutherford scattering. Problems 14.19 and 14.21. |
| Wednesday 04/03 |
Homework 7 due today. 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 04/05 | Example 16.1. Evolution of a triangular pulse. Standing wave (end of Sec. 16.2). 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 04/08 | Section 16.4: The three-dimensional wave equation. The Laplacian. Plane waves and spherical waves. |
| Wednesday 04/10 |
Homework 8 due today. Chapter 15. Special relativity. Section 15.8. Review of rotations in ordinary 3 dimensions. Invariance of the Euclidean metric under rotations. Four-dimensional space-time. Four-vectors. Minkowski metric. Lorentz transformations: ordinary rotations. |
| Friday 04/12 |
Review the previous notes from our class on February 19 and 21, and especially rotations in 3 dimensions.
Guest lecture: Continue Section 15.8. Lorentz transformations = rotations in spacetime. The Lorentz group SO(3,1). Explicit form of a Lorentz transformation. Lorentz boost along the x axis. Relation between the beta and gamma parameters. Section 15.9. The invariant scalar product in Minkowski space (+++-). The notes of today's guest lecture. |
| Monday 04/15 |
Guest lecture:
Section 15.2: Galielean transformations. Section 15.6: Physical interpretation of the Lorentz boost. Section 15.9. The invariant scalar product. Section 15.13. Energy, the fourth component of 4-momentum. The invariant mass. Problem 15.60: Decay of a massive particle to two identical particles. The notes of today's guest lecture. |
| Wednesday 04/17 |
Homework 9 due today. Wrap up Chapter 15 and review for the test. Section 15.18. Electrodynamics and relativity. Electromagnetic field tensor. The electromagnetic 4-potential. Section 15.10. The light cone. Speed of light. Forward light cone. Backward light cone. Time-like and space-like intervals between events in spacetime. Causality. Time-ordering of events. Events as seen by different inertial observers. The notes of today's lecture. |
| Friday 04/19 |
Periodic testing: third exam.
The exam covers the material from Chapters 11, 14, 15 and 16. The formula sheet for the exam. |
| Monday 04/22 | Guest lecture: Section 16.5: Volume and surface forces. Pressure, tension, shear. Section 16.6: Stress and strain: the elastic moduli. Section 16.7: The stress tensor. Example 16.3. The stress tensor in a static fluid. Example 16.4. Numerical example of stress. |
| Wednesday 04/24 | Guest lecture: 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. |