Homework Assignment 4 Canonical Transformations Due Tuesday, Feb. 23, 2010

1. Consider the canonical transformation from {p1, p2, q1, q2} → {P1, P2, Q1, Q2} such that

Q1 = q 2

1 , Q2 =

q2

cos(p2) ,

P1 = p1cos(p2) − 2q2

2q1cos(p2) , P2 = sin(p2) − 2q1.

a) Show the following is true for the Poisson Brackets: [Qi, Pj ]pq = δij , [Qi, Qj ]pq = 0, and [Pi, Pj ]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.

Q1 = q1, P1 = p1 − 2p2

Q2 = p2, P2 = −2q1 − q2

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).

mioduszewski Rectangle

mioduszewski Line

mioduszewski Oval

mioduszewski Oval

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mioduszewski Oval

mioduszewski Rectangle

mioduszewski Line

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mioduszewski Rectangle

mioduszewski Rectangle

mioduszewski Oval

mioduszewski Rectangle

mioduszewski Oval

mioduszewski Oval

mioduszewski Oval

mioduszewski Oval

mioduszewski Rectangle

mioduszewski Rectangle

mioduszewski Line

mioduszewski Line

mioduszewski Rectangle

mioduszewski Oval

mioduszewski Oval

mioduszewski Oval

mioduszewski Oval

mioduszewski Rectangle

mioduszewski Rectangle

mioduszewski Rectangle

mioduszewski Rectangle

mioduszewski Oval

mioduszewski Oval

mioduszewski Oval