Incoming links: Symmetry
Solving differential equations was apparently Lie's original motivation for developing Lie groups. It is therefore likely one of the most understandable ways to approach it.
It appears that Lie's goal was to understand when can a differential equation have an explicitly written solution, much like Galois theory had done for algebraic equations. Both approaches use symmetry as the key tool.
- www.researchgate.net/profile/Michael_Frewer/publication/269465435_Lie-Groups_as_a_Tool_for_Solving_Differential_Equations/links/548cbf250cf214269f20e267/Lie-Groups-as-a-Tool-for-Solving-Differential-Equations.pdf Lie-Groups as a Tool for Solving Differential Equations by Michael Frewer. Slides with good examples.
Strangeness Minus Three (BBC Horizon 1964)
Source. Basically shows Richard Feynman 15 minutes on a blackboard explaining the experimental basis of the eightfold way really well, while at the same time hyperactively moving all over. The word symmetry gets tossed a few times.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! en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras
The fact that there are elements arbitrarily close to the identity, which is only possible due to the group being continuous, is the key factor that simplifies the treatment of Lie groups, and follows the philosophy of continuous problems are simpler than discrete ones.
Bibliography:
- 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.
- www.youtube.com/playlist?list=PLRlVmXqzHjURZO0fviJuyikvKlGS6rXrb "Lie Groups and Lie Algebras" playlist by XylyXylyX (2018). Tutorial with infinitely many hours
- www.staff.science.uu.nl/~hooft101/lectures/lieg07.pdf
- www.physics.drexel.edu/~bob/LieGroups.html
This is a good book. It is rather short, very direct, which is a good thing. At some points it is slightly too direct, but to a large extent it gets it right.
The main goal of the book is to basically to build the Standard Model Lagrangian from only initial symmetry considerations, notably the Poincaré group + internal symmetries.
The book doesn't really show how to extract numbers from that Lagrangian, but perhaps that can be pardoned, do one thing and do it well.
- why the square: physics.stackexchange.com/questions/535/why-does-kinetic-energy-increase-quadratically-not-linearly-with-speed on Physics Stack Exchange. Ron Maimon's answer is great, as it relies only on the following staring points:He also offers a symmetry argument considering the case of potential energy.
- why the half: physics.stackexchange.com/questions/27847/why-is-there-a-frac-1-2-in-frac-1-2-mv2 on Physics Stack Exchange