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### Transcript of Homework Assignment 4 Canonical Transformations 1 ... Homework Assignment 4 Canonical...

• Homework Assignment 4Canonical TransformationsDue Tuesday, Feb. 23, 2010

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

Q1 = q2

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 = ij , [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

qsin(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).

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