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Generalized Galilean transformation

  1. May 18, 2017 #1
    1. The problem statement, all variables and given/known data
    Write the Galilean coordinate transformation equations
    for the case of an arbitrary direction for the relative velocity v of one frame with respect to the other. Assume that the corresponding axes of the two frames remain parallel. (Hint: let v have componentsvx, vy, vz.)

    Write down the equivalent matrix equation.
    2. Relevant equations
    Consider a frame S' moving with uniform velocity v with respect to another inertial frame S.
    Then
    x'=x- vx t, $$
    $$y'=y- vy t, $$
    $$z'=z- vz t, $$
    $$t' = t


    3. The attempt at a solution
    The matrix formulation is
    $$\begin{pmatrix}
    x' \\
    y' \\
    z'\\
    t'
    \end{pmatrix} =\begin{pmatrix}
    x& -v_x \\
    y & -v_y \\
    z & -v_z\\
    0&1\end{pmatrix}
    \begin{pmatrix}
    1 \\
    t\\

    \end{pmatrix}
    $$
    Is this right?
     
  2. jcsd
  3. May 18, 2017 #2

    BvU

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    Gold Member

    I don't think so. Your relevant equations are fine. For a coordinate transformation one expects something of the form $$
    \begin{pmatrix}
    x' \\
    y' \\
    z' \\
    t'
    \end{pmatrix} = M\begin{pmatrix}
    x \\
    y \\
    z \\
    t
    \end{pmatrix}$$
    (compare with the matrix for a simple rotation in 3D, e.g. around the z-axis)
     
  4. May 18, 2017 #3
    Then,
    ##\begin{pmatrix}
    x' \\
    y' \\
    z'\\
    t'
    \end{pmatrix} =\begin{pmatrix}
    1&0&0& -v_x \\
    0&1&0& -v_y \\
    0&0&1 & -v_z\\
    0&0&0&1\end{pmatrix}\begin{pmatrix}
    x \\
    y\\
    z\\
    t

    \end{pmatrix}##
    Is this correct?
    Can you please tell me how to write the matrices side by side?
     
    Last edited: May 18, 2017
  5. May 18, 2017 #4

    BvU

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    Science Advisor
    Homework Helper
    Gold Member

    I think this is what the exercise composer meant, yes.
    And the inverse transformation matrix looks the same, except that the minus signs are now plus signs. Good exercise to check that ##M^{-1}M = MM^{-1} = {\mathbb I}##
    But you did that already in post #1 !

    Generally: right-click on a formula and pick show math as ##\TeX## commands :smile:
     
  6. May 18, 2017 #5
    Thank you.
    I got it. I need to write" ## "before and after the text command for matrices.
     
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