We then compute kinetic energy in this representation, and show how a bivector-valued angular momentum \( L = \mathbf{x} \wedge \mathbf{p} \), falls naturally from that computation, where we have

Prerequisites: calculus (derivatives and chain rule), and geometric algebra basics (vector multiplication, commutation relationships for vectors and bivectors in a plane, wedge and cross product equivalencies, …)

Errata: at around 4:12 I used \( \mathbf{r} \) instead of \( \mathbf{x} \), then kept doing so every time after that when the value for \( L \) was stated.

Months ago, I used Manim to create a outline a geometric algebra treatment of the derivation of the circular velocity and acceleration formulas that you would find in a first year undergrad physics course. I never published it, since overlaying audio and getting the timing of the audio and video right is hard (at least for me.) I’m also faced with the difficulty of not being able to speak properly when attempting to record myself.

Anyways, I finally finished the audio overlays (it was sitting waiting for me to record the final 10s of audio!), and have posted this little 11 minute video, which includes:

A reminder of what circular coordinates are.

A brief outline of what is meant by each of the circular basis vectors.

A derivation of those basis vectors (just basic geometry, and no GA.)

A brief introduction to geometric algebra, and geometric algebra for a plane, including the “imaginary” \( i = \Be_1 \Be_2 \), and it’s use for rotation and polar form.

How to express the circular basis vectors in polar form.

Application of all the ideas above to compute velocity and acceleration.

Circular coordinate examples of velocity and acceleration.

It probably doesn’t actually make sense to try to pack all these ideas into one video, but oh well — that’s what I did.