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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).

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

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 = 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

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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