In this short (hehe) video, I set up and solve the Geodesic Problem on a Plane. A geodesic is a special curve that represents the shortest distance between two points on a particular surface. Here, because the geodesic problem is being solved on a plane, I show that the geodesic on a plane is a straight line. To do this, I set up a length functional, and then using the Euler-Lagrange equation, I solve for the equation of the geodesic path. The computation here isn't that difficult, but I said I'd cover geodesics when I started this series so I figured I'd do this for the sake of completeness.
Now, I didn't mention this in the video because I thought it was a bit tangential to the topic, but geodesics become quite important in General Relativity. General Relativity essentially has two important (sets of) equations. The first are the Einstein Field Equations, which describe how spacetime (which can be thought of as a 4-D surface) curves under the influence of mass and energy. The second set are the geodesic equations, which describe how light and matter travel in this 'curved' spacetime. In other words, light travels along geodesics in spacetime (that's why light 'bends' in gravity).
The geodesic equation in General Relativity can, in fact, be derived using Euler-Lagrange. In this case, the dS isn't just sqrt(dx^2 + dy^2) but is a lot more complicated, and involves the metric tensor. I'll cover this in more depth once I start General Relativity, but I figured I'd briefly discuss an application for the interested folks out there.
ERRATA: At 3:29, I meant to say y' instead of y.
Pre-reqs: The first two videos of this playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9CD8igcUr9Fmn5WXLpE8ZE_
Euler-Lagrange Video: https://www.youtube.com/watch?v=sFqp2lCEvwM
Lecture Notes: https://drive.google.com/open?id=1bjirQAVyzE39-3GHnnVLTVSuOATD-1o5
Special thanks to my Patrons:
- Jennifer Helfman
- Justin Hill
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard