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The Theoretical Minimum: What You Need to Know to Start Doing Physics by George Hrabovsky, Leonard Susskind

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Lecture 10: Poisson Brackets, Angular Momentum, and Symmetries

Lenny asked, “Hey George, can we hang fish on a Poisson Bracket?”

George smiled. “Only if they’re theoretical.”

An Axiomatic Formulation of Mechanics

Let’s abstract a set of rules that enable one to manipulate Poisson Brackets (from now on I’ll use the abbreviation PB) without all the effort of explicitly calculating them. You can check (consider it homework) that the rules really do follow from the definition of PB’s. Let A, B, and C be functions of the p’s and q’s. In the last lecture, I defined the PB:

 

(1)

•The first property is antisymmetry: If you interchange the two functions ...

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