kinetic energy

New video: Velocity and angular momentum with geometric algebra

September 7, 2023 math and physics play , , , , ,


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.

Video: Spherical basis vectors in geometric algebra

August 8, 2023 Geometric Algebra for Electrical Engineers , ,

I’ve made a new manim-based video with a geometric algebra application.

In the video, the geometric algebra form for the spherical unit vectors are derived, then unpacked to find the conventional vector algebra form. We will then use our new tools to find the expression for the kinetic energy of a particle in spherical coordinates.

Prerequisites: calculus (derivatives and chain rule), complex numbers (exponential polar form), and geometric algebra basics (single sided rotations, vector multiplication, vector commutation sign changes, …)

You can find the video on Google’s censorship-tube, or on odysee.

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