Homework Assignment 4 Canonical Transformations 1 ... Homework Assignment 4 Canonical...

download Homework Assignment 4 Canonical Transformations 1 ... Homework Assignment 4 Canonical Transformations

of 6

  • date post

    26-Jun-2018
  • Category

    Documents

  • view

    270
  • download

    0

Embed Size (px)

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).

  • mioduszewskiRectangle

    mioduszewskiLine

    mioduszewskiOval

    mioduszewskiOval

  • mioduszewskiRectangle

    mioduszewskiOval

    mioduszewskiRectangle

    mioduszewskiLine

    mioduszewskiRectangle

    mioduszewskiRectangle

    mioduszewskiRectangle

    mioduszewskiOval

    mioduszewskiRectangle

    mioduszewskiOval

    mioduszewskiOval

    mioduszewskiOval

    mioduszewskiOval

    mioduszewskiRectangle

    mioduszewskiRectangle

    mioduszewskiLine

    mioduszewskiLine

  • mioduszewskiRectangle

    mioduszewskiOval

    mioduszewskiOval

    mioduszewskiOval

    mioduszewskiOval

  • mioduszewskiRectangle

    mioduszewskiRectangle

    mioduszewskiRectangle

    mioduszewskiRectangle

    mioduszewskiOval

    mioduszewskiOval

  • mioduszewskiOval