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Homogeneous deformation

  1. Aug 24, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that in the homogeneous deformation, particles which after the deformation lie on the surface of a shere of radius b originally lay on the surface of an ellipsoid.

    2. Relevant equations

    homogeneous deformations are motions of the form:

    xi=ci + AiRXR

    where ci and AiR are constants or functions of time.

    3. The attempt at a solution

    I don't know how to prove this, i think i first need to know the equation of motion of a sphere then relate this to the equation above. Im confused about this.
     
  2. jcsd
  3. Aug 24, 2009 #2

    HallsofIvy

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    So you have <x, y, z> which satisify [itex]x^2+ y^2+ z^2= R^2[/itex] and your deformation if of the form
    [tex]\begin{bmatrix}x' \\ y' \\ z'\end{bmatrix}= \begin{bmatrix}u \\ v\\ w\end{bmatrix}+ \begin{bmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}[/tex]

    Go ahead and do the calculation for x', y', z' in terms of x, y, and z and use the equation for the sphere to show that x', y', z' satisfy the equation for an ellipse.
     
    Last edited: Aug 24, 2009
  4. Aug 24, 2009 #3
    Thanks, but wouldnt the x', y', and z' be in terms of x, y, z, and u, v, w, and all the a's after the matrix multiplication?
     
  5. Aug 24, 2009 #4

    HallsofIvy

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    Well, yes. I didn't mention the components of A since I assumed that was a constant.
     
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