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Just plain dont know what to do or how to even start

  1. Sep 20, 2007 #1
    1. The problem statement, all variables and given/known data
    Well i've been working on these problems for the last 4 hours and still nothing...and out of despairation i turn to you all.

    It has to do with Canonical transformations of the Hamiltonian:
    1) Consider a type 1 generating function (F(q,Q,t)) where the following must be satisfied
    [tex]
    p = \frac{\partial{F}}{\partial{q}} \
    P = -\frac{\partial{F}}{\partial{Q}} \
    [/tex]
    show that for a single degree of freedom the possion bracket [Q,P] = 1 (aka canonical) now ive been able to show it is 0 obviously wrong and yes it makes use of many different forms of differentials any help on this would help

    2) For Two particles interact via a central potential V(r1-r2) the H is
    [tex]
    H= \frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+V(r_1-r_2)
    [/tex]

    [tex]
    Q = \frac{m_1 r_1+m_2 r_2}{M = m_1+m_2}, P = p_1+p_2
    [/tex]

    [tex]
    q= r_1 - r_2, \ p = \frac{m_2 p_1-m_1 p_2}{M}
    [/tex]

    this one im really not sure about, however i think the transformation is type 3 so that may help doing that what does anyone think? any help is always awesome.


    show that the transformation is canonical



    2. Relevant equations
    [tex]
    [Q,P] = \frac{\partial{Q}}{\partial{q}}\frac{\partial{P}}{\partial{p}}-\frac{\partial{P}}{\partial{q}}\frac{\partial{Q}}{\partial{p}}
    [/tex]
     
    Last edited: Sep 20, 2007
  2. jcsd
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