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Converted Cartesian coordinates to polar coordinates

  1. Dec 8, 2006 #1
    I don't know where have I gone wrong...
    I converted Cartesian coordinates to polar coordinates:

    [tex] \frac{\partial^2\Psi}{\partial x^2} +\frac{\partial^2\Psi}{\partial y^2}= \frac{1}{2}(\frac{\partial^2}{\partial x^2}+\frac{\partial^2 }{\partial y^2})\Psi^2 - \Psi(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})\Psi
    =\frac{\partial^2\Psi}{\partial r^2}+ \frac{1}{r^2}\frac{\partial^2\Psi}{\partial \Phi^2}[/tex]

    But on the left hand side (the Cartesian components) is just the Laplacian in 2D, but the final answer I got for the polar components is not equivalent to the Laplacian for polar coordinate system.
    I'm missing the term [tex]\frac{1}{r}\frac{\partial\Psi}{\partial r}[/tex]
     
    Last edited: Dec 8, 2006
  2. jcsd
  3. Dec 8, 2006 #2

    AlephZero

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    You are misunderstanding something here.

    [tex] {\frac{\partial^2}{\partial x^2}\Psi^2
    = \frac{\partial}{\partial x}\frac{\partial}{\partial x}\Psi^2
    = \frac{\partial}{\partial x}( 2\Psi \frac{\partial \Psi}{\partial x})
    = 2 ({\frac{\partial \Psi}{\partial x}})^2 + 2\Psi \frac{\partial^2 \Psi}{\partial^2 x} [/tex]

    That doesn't seem consistent with your (wrong) equation.
     
    Last edited: Dec 8, 2006
  4. Dec 8, 2006 #3
    Ooops, I typed the wrong stuffs. I'm sorry.
    It should read:

    [tex] (\frac{\partial\Psi}{\partial x})^2 +(\frac{\partial\Psi}{\partial y})^2= \frac{1}{2}(\frac{\partial^2}{\partial x^2}+\frac{\partial^2 }{\partial y^2})\Psi^2 - \Psi(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})\Psi[/tex]

    Next, I plug in the Laplacian for the polar coordinates, essentially cylindrical coordinate, with z constant, and I end up with:
    [tex](\frac{\partial\Psi}{\partial x})^2 +(\frac{\partial\Psi}{\partial y})^2= (\frac{\partial\Psi}{\partial r})^2+ \frac{1}{r^2}(\frac{\partial\Psi}{\partial \Phi})^2[/tex]

    Next, I am required to get the Euler Lagrange equation for a system. The above is just the potential part. The time derivative kinetic is just [tex]\frac{1}{2}m\dot{\Psi}^2[/tex]
    Taking the Euler Lagrange for the Cartesian, I end up with an expression from the potential part:
    [tex] \frac{\partial^2\Psi}{\partial x^2} +\frac{\partial^2\Psi}{\partial y^2}[/tex]
    And this is just a Laplacian.

    But when taking the Euler Lagrange for the polar coordinates, I end up with an expression [tex]\frac{\partial^2\Psi}{\partial r^2}+\frac{1}{r^2}\frac{\partial^2\Psi}{\partial \Phi^2}[/tex]
    and this is not equal to the Laplacian for the polar coordinates.
    I am missing [tex]\frac{1}{r}\frac{\partial\Psi}{\partial r}[/tex]
     
    Last edited: Dec 8, 2006
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