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Coordinate conjugate to momentum.

  1. Aug 26, 2010 #1
    Let's take a system, for simplicity with only one degree of freedom, described by a certain lagrangian

    [tex]L[x,\dot x][/tex]

    I define the momentum

    [tex]p=\frac{\partial L}{\partial\dot x}[/tex]

    Now, if I make a change of coordinates

    [tex]x\longmapsto y\qquad\qquad\qquad(1)[/tex]

    I obtain a second lagrangian

    [tex]M[y,\dot y]=L[x(y(t)),\partial_t x(y(t))][/tex]

    and I can define a second momentum

    [tex]q=\frac{\partial M}{\partial\dot y}[/tex]

    My question is, if instead of the transformation (1) I want to consider the transformation of the momenta

    [tex]p\longmapsto q\qquad\qquad\qquad(2)[/tex]

    How can I find the corresponding transformation (1)? In other words, given that I know how to do (1)-->(2), how can I do (2)-->(1)?
     
    Last edited: Aug 26, 2010
  2. jcsd
  3. Aug 26, 2010 #2
    I think you need to read some material about canonical transformations.
    Maybe wiki is a good start:

    http://en.wikipedia.org/wiki/Canonical_transformation

    Once you are on the starting block, read about generating functions.
    The last section "Modern mathematical description" is like a summary.
     
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