This page contains syllabi, problem sets, past exams, and other resources for the courses that I am currently teaching and have taught in the past.

Planet 9 and Kuiper Belt objects (Caltech/R. Hurt/WorldWide Telescope)

Since ancient times, humans have observed the night sky. One of the most striking feature easily observable with the naked eye are planets, the wandering stars. For centuries astronomers have recorded and predicted their motion. This course introduces graduate students to three topics in the wide field of Planetary Dynamics. Note that students can opt to take only one or two out of the three mini-courses being offered. But note that each mini-course builds on the knowledge developed during the previous mini-course(s).

You can download a tentative syllabus. If you would like to take this course, please get in touch via e-mail.

Simulation of the formation of large scale structure and galaxies in the early universe (Max Planck Institute for Astrophysics).

*From the calendar:* Scientific computing is a rapidly growing field because computers can solve previously intractable problems and simulate natural processes governed by equations that do not have analytic solutions. During the first part of this course, students will learn numerical algorithms for various standard tasks such as root finding, integration, data fitting, interpolation and visualization. In the second part, students will learn how to model real-world systems from various branches of science. At the end of the course, students will be expected to write small programs by themselves. Assignments will regularly include programming exercises.

Please see Quercus for updates regarding this course.

Launch of a rocket built by students in PHYB54 on the roof of the Science Wing.

- What is the Reynolds number?
- What is the terminal speed in a linear or quadratic medium?
- Guesstimate the characteristic drag timescale of an every day object as it moves through air!

- Where is the centre of mass of the Earth Moon system (approximately)?
- Which body contains most of the angular momentum in the solar system? Quantify!
- Look up the specifications for an A8-5 rocket motor. Write down the thrust duration, the maximum thrust and the total impulse in SI units.

- What it the kinetic energy of the Earth going around the Sun (in kWh)?
- What it the potential energy in the Earth-Sun system (in kWh)? Compare to the kinetic energy!
- Come up with one explicit example of a position dependent force which is not conservative!

- Write down the general solution for the simple harmonic oscillator in four different ways: as a linar combination of two trigonometric functions, one trigonometric function with a phase, a linear combination of two exponentials, and as the real part of one exponential.
- While your on a bus or in a car, try to estimate the frequency when you drive over a bump in the road. Also try to estimate if you are in a regime of over or underdamping.

- What is the differential equation for the driven, damped harmonic oscillator!
- At what driving frequency does one get the maximum amplitude in a driven, damped harmonic oscillator? Write down the condition in terms of the natural frequency and the damping parameter.

- How is the Lagrangian defined?
- Show that Newton's second law implies the Lagrange equations! (two line proof)
- What are two major advantages of using Lagrange equations to solve problems in dynamics compared to just using Newton's laws?

- Who was Emmy Noether?
- What is Noether's theorem?

- How many orbital parameters do you need to describe the orbit of an asteroid orbiting the Sun? Make a list with symbols and names!
- Sketch the orbit of Earth, Mars and the NASA InSight spacecraft! What is this type of orbit called?

- Sketch the normal modes of a system consisting of two carts (Fig 11.1) where the masses of the two carts are equal and all spring constants are equal!
- Sketch again the normal modes, but this time for a system where the oscillators are weakly coupled!

- Write the equation of motion for the DDP in your journal!
- Describe where the Feigenbaum number appears in the discussion of the DDP!
- What is the Lyapunov exponent?

- What are the differences between a bifurcation diagram, a state-space orbit, and a Poincare section?
- What is the recursion equation for the logistic map?

Simulation of the formation of large scale structure and galaxies in the early universe (Max Planck Institute for Astrophysics).

*From the calendar:* Scientific computing is a rapidly growing field because computers can solve previously intractable problems and simulate natural processes governed by equations that do not have analytic solutions. During the first part of this course, students will learn numerical algorithms for various standard tasks such as root finding, integration, data fitting, interpolation and visualization. In the second part, students will learn how to model real-world systems from various branches of science. At the end of the course, students will be expected to write small programs by themselves. Assignments will regularly include programming exercises.

Launch of a rocket built by students in PHYB54 on the roof of the Science Wing.

Final exam from 2017. Download as PDF.

Puri Gagarin the Cosmocat visiting the UTSC Observatory.

Simulation of the formation of large scale structure and galaxies in the early universe (Max Planck Institute for Astrophysics).

Launch of a rocket built by students in PHYB54 on the roof of the Science Wing.

Image of M51 taken with the UTSC Observatory.