In this video, we compute velocity in a radial representation $$\mathbf{x} = r \mathbf{\hat{r}}$$.

We use a scalar radial coordinate $$r$$, and leave all the angular dependence implicitly encoded in a radial unit vector $$\mathbf{\hat{r}}$$.

We find the geometric algebra structure of the $$\mathbf{\hat{r}}’$$ in two different ways, to find

$$\mathbf{\hat{r}}’ = \frac{\mathbf{\hat{r}}}{r} \left( \mathbf{\hat{r}} \wedge \mathbf{\hat{x}}’ \right),$$

then derive the conventional triple vector cross product equivalent for reference:

$$\mathbf{\hat{r}}’ = \left( \mathbf{\hat{r}} \times \mathbf{\hat{x}}’ \right) \times \frac{\mathbf{\hat{r}}}{r}.$$

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

$$\frac{m}{2} \mathbf{v}^2 = \frac{1}{2 m} {(m r’)}^2 – \frac{1}{2 m r^2 } L^2.$$

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.

As well as being posted to Google’s censorship-tube, this video can also be found on odysee.