Linear Symplectomorphisms and SL(2,R)

Applied Representation Theory | Episode 3
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We describe the subgroup of linear transformations of phase space that preserves the Poisson bracket, and show that it is essentially the special linear group of 2x2 matrices, SL(2,R).

We introduce some gentle proofs to get the physicists in the audience on board with a more Mathematical style.

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📕 Lecture notes: https://pasayten.org/applied-representation-theory-notes

🎓 Recommended Prerequisites: Basics of vector spaces, Lie algebras and multivariable calculus. Also our past two videos.

📚 References: Goldstein is the standard reference on Classical Mechanics, but you can also check out a cool book by Jose & Saletan. The Humphreys text on Lie algebras is a great formal start, although you can get pretty far with Brian C Hall's book or even Georgi's little text on Lie algebras in Particle Physics can help. Peter Woit also has a number of useful write ups on the symplectic groups and sl(2,R) on his professional website, https://www.math.columbia.edu/~woit/

⏭️ Next in series: "The Symplectic Group Sp(2n,R)"

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Timestamps:
0:35 : The general linear group of linear transformations on phase space
1:30 : Quick review of the structure of phase space
2:00 : The problem of linear symplectomorphisms
2:30 : Short notes on the possibility of nonlinear phase space models
2:59 : Linear transformations in two-dimensions
3:23 : A necessary condition for preserving the Poisson Bracket
4:15 : A sufficient condition for preserving the Poisson Bracket
5:54 : Claim: SL(2,R) is a proper subgroup of GL(2,R)
6:36 : Claim: dim SL(2,R) = 3
6:51 : Four examples of SL(2,R) transformation

Видео Linear Symplectomorphisms and SL(2,R) канала Sean Downes