Editing PGE/RotatingFrame (section)

====Transformation Matrix====

Following Chandrasekhar, we let <math>\vec{X}</math> represent the inertial-frame position vector of a fluid element, in which case <math>d\vec{X}/dt</math> is the inertial-frame velocity <math>(\vec{v})</math> of that fluid element, and the acceleration, <math>d\vec{v}/dt</math>, that appears on the LHS of  the [[PGE/Euler#Lagrangian_Representation|Lagrangian representation of the (intertial-frame) Euler equation]] may be rewritten as the second time-derivative of <math>\vec{X}</math>, namely,

<table border="0" align="center" cellpadding="5">
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  <td align="center" colspan="3"><font color="#770000">'''Lagrangian Representation'''</font><br />of the (inertial-frame) Euler Equation</td>
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  <td align="right">
<math>\frac{d^2\vec{X}}{dt^2}</math>
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<math>=</math>
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<math>- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr]_\mathrm{inertial} \, .</math>
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</table>

Chandrasekhar uses the matrix, <math>\mathbf{T}(t)</math>, to represent the (time-dependent) linear transformation that relates <math>\vec{X}</math> to  the corresponding moving-frame position vector, <math>\vec{x}</math>.  Specifically, he sets,

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  <td align="right">
<math>\vec{x}</math>
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<math>=</math>
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<math>\mathbf{T}\vec{X} \, .</math>
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  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (1)</td>
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Applying the same transformation to the inertial-frame velocity, <math>d\vec{X}/dt</math>, gives,
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  <td align="right">
<math>\vec{U}</math>
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<math>=</math>
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<math>\mathbf{T}\frac{d\vec{X}}{dt} \, ,</math>
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  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (14a)</td>
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which Chandrasekhar refers to as the velocity in the <b>inertial frame</b> that has been <font color="darkgreen">"resolved along the instantaneous coordinate axes of the moving frame."</font>  And applying this transformation to the inertial-frame acceleration gives the term, <math>\mathbf{T} [d^2\vec{X}/dt^2]</math>, which Chandrasekhar describes as representing <font color="darkgreen">"&hellip; the acceleration in the <b>inertial frame</b> resolved, however, along the instantaneous directions of the coordinate axes of the moving frame."</font>  Applying the transformation to both sides of the Lagrangian representation of the Euler equation gives,

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  <td align="right">
<math>\mathbf{T} \frac{d^2\vec{X}}{dt^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr]_\mathrm{moving} \, ,</math>
  </td>
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  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, combination of Eqs. (16) &amp; (17)</td>
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where, as Chandrasekhar clarifies, the gradients on the RHS must be <font color="darkgreen">"&hellip; evaluated in the coordinates of the moving frame."</font>