Homework Assignment 4 Canonical Transformations 1. Consider … · 2019. 5. 15. · Homework...

6
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).

Transcript of Homework Assignment 4 Canonical Transformations 1. Consider … · 2019. 5. 15. · Homework...

Page 1: Homework Assignment 4 Canonical Transformations 1. Consider … · 2019. 5. 15. · Homework Assignment 4 Canonical Transformations Due Tuesday, Feb. 23, 2010 1. Consider the 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 = 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).

Page 2: Homework Assignment 4 Canonical Transformations 1. Consider … · 2019. 5. 15. · Homework Assignment 4 Canonical Transformations Due Tuesday, Feb. 23, 2010 1. Consider the canonical
mioduszewski
Rectangle
mioduszewski
Line
mioduszewski
Oval
mioduszewski
Oval
Page 3: Homework Assignment 4 Canonical Transformations 1. Consider … · 2019. 5. 15. · Homework Assignment 4 Canonical Transformations Due Tuesday, Feb. 23, 2010 1. Consider the canonical
mioduszewski
Rectangle
mioduszewski
Oval
mioduszewski
Rectangle
mioduszewski
Line
mioduszewski
Rectangle
mioduszewski
Rectangle
mioduszewski
Rectangle
mioduszewski
Oval
mioduszewski
Rectangle
mioduszewski
Oval
mioduszewski
Oval
mioduszewski
Oval
mioduszewski
Oval
mioduszewski
Rectangle
mioduszewski
Rectangle
mioduszewski
Line
mioduszewski
Line
Page 4: Homework Assignment 4 Canonical Transformations 1. Consider … · 2019. 5. 15. · Homework Assignment 4 Canonical Transformations Due Tuesday, Feb. 23, 2010 1. Consider the canonical
mioduszewski
Rectangle
mioduszewski
Oval
mioduszewski
Oval
mioduszewski
Oval
mioduszewski
Oval
Page 5: Homework Assignment 4 Canonical Transformations 1. Consider … · 2019. 5. 15. · Homework Assignment 4 Canonical Transformations Due Tuesday, Feb. 23, 2010 1. Consider the canonical
mioduszewski
Rectangle
mioduszewski
Rectangle
mioduszewski
Rectangle
mioduszewski
Rectangle
mioduszewski
Oval
mioduszewski
Oval
Page 6: Homework Assignment 4 Canonical Transformations 1. Consider … · 2019. 5. 15. · Homework Assignment 4 Canonical Transformations Due Tuesday, Feb. 23, 2010 1. Consider the canonical
mioduszewski
Oval