**Solutions to problems in classical dynamics:**

- One should probably start with the Forced Simple Harmonic Oscillator, which is the way we can discuss resonance.
- Discussion of the solution to the two body gravitational, also called Kepler Problem.
- We all know that a particle under the influence of a constant gravitational field executes a parabolic trajectory. The effect of drag makes the trajectory much more interesting and allows us to talk about Taylor series, as well as non-dimensionalization and scale analysis.

**Some fun problems from dynamical systems:
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**Derivations from Linear and Nonlinear PDE**

- Derivation of D’Alembert’s Solution for a forced wave equation.
- Derivation of the wave equation from the limit of coupled (non-linear) springs.
- Ray theory and the eikonal equation
- Convergence of Fourier Series
- A derivation of the Korteweg-deVries Equation for surface water waves.

**Problems in 2D Euler and Quasi-Geostrophy**

- A comparison of the velocity field of circular patches in 2D Euler versus Shallow water Quasi-geostrophy and Surface Quasi-geostrophy. I also include a discussion of how circular patches are affected by the Beta effect.
- This is a derivation of the 3D quasigeostrophic equations. It begins with the incompressible hydrostatic primitive equations and is intended to be a jumping point for thinking about connecting QG to the tropical non-linear theories.

**Classical fluid instability**

- A discussion of Rayleigh and Fjortoft’s criterion for shear flow instability, along with a presentation of Howard’s semicircle theorem.
- My take on the Eady problem – it’s really a 2 layer SQG problem – and my rant about how the “boundary condition” that everyone uses is actually a red herring. You don’t need a boundary condition if you realize that it is a 2-layer SQG problem.

**Some kinematic results from vector calculus**

- A conversation I had with my colleague, Mohammed Hafez, regarding the computation of the vector potential.