The key and central motivation for studying Lie groups and their Lie algebras appears to be to characterize symmetry in Lagrangian mechanics through Noether's theorem, just start from there. Notably local symmetries appear to map to forces, and local means "around the identity", notably: local symmetries of the Lagrangian imply conserved currents.
More precisely: local symmetries of the Lagrangian imply conserved currents.
TODO Ciro Santilli really wants to understand what all the fuss is about:
Oh, there is a low dimensional classification! Ciro is a sucker for classification theorems! https://en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras
- https://youtu.be/kpeP3ioiHcw?t=2655 "Particle Physics Topic 6: Lie Groups and Lie Algebras" by Alex Flournoy (2016). Good SO(3) explicit exponential expansion example. Then next lecture shows why SU(2) is the representation of SO(3). Next ones appear to eventually get to the physical usefulness of the thing, but I lost patience. Not too far out though.
- https://www.youtube.com/playlist?list=PLRlVmXqzHjURZO0fviJuyikvKlGS6rXrb "Lie Groups and Lie Algebras" playlist by XylyXylyX (2018). Tutorial with infinitely many hours
- Table of contents | 217, 1k, 22
- 2. Lie algebra | 21, 98, 2
- 3. Continuous symmetry | 18, 308, 2
- 3.1. Local symmetry | 152, 290, 1
- 4. Important Lie groups | 0, 693, 13
- Applications of Lie groups to differential equations | 102
- Continuous symmetry | 18, 308, 2
- Derivation of the Schrodinger equation | 91, 136, 1
- Group of Lie type | 35
- Lagrangian mechanics | 896, 2k, 18
- Noether's theorem | 243, 253, 1
- Physics | 276, 33k, 581
- Representation theory | 61
- Symmetry | 29
- What does it mean that photons are force carriers for electromagnetism? | 375