Homework Assignment 4 Canonical Transformations Due Tuesday, Feb. 23, 2010 1. Consider the canonical transformation from {p 1 ,p 2 ,q 1 ,q 2 }→{P 1 ,P 2 ,Q 1 ,Q 2 } such that Q 1 = q 2 1 , Q 2 = q 2 cos(p 2 ) , P 1 = p 1 cos(p 2 ) - 2q 2 2q 1 cos(p 2 ) , P 2 = sin(p 2 ) - 2q 1 . a) Show the following is true for the Poisson Brackets: [Q i ,P j ] pq = δ ij ,[Q i ,Q j ] pq = 0, and [P i ,P j ] pq = 0. b) Find a generating function for this transformation. 2. Show that the following transformation is canonical by showing that the symplectic condition is satisfied. Q = ln 1 q sin(p) , P = q cot(p). 3. Find a generating function for the following transformation. Q 1 = q 1 , P 1 = p 1 - 2p 2 Q 2 = p 2 , P 2 = -2q 1 - q 2 Hint: A combination of two types of generating functions will work. Note that the transformation is a combi- nation of the identity transformation and the exchange transformation (which we went over in class).