Appendix/Mathematics/EulerAngles - Revision history

Jet53man: /* Using Matrix Notation */

2024-06-14T22:19:20Z

Using Matrix Notation

← Older revision		Revision as of 18:19, 14 June 2024
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	===Using Matrix Notation===		===Using Matrix Notation===

			<table border="1" align="right" cellpadding="1">
			<tr><td align="center">Astrophysics Example</td></tr>
			<tr><td align="center">[[File:EulerAnglesExample02.jpeg\|250px\|Euler Angles Example02]]</td></tr>
			</table>
	With more finesse, we can write the general rotation matrix that links the ''body frame'', <math>(x_1, x_2, x_3)</math>, to the ''inertial/laboratory'' frame, <math>(X, Y, Z)</math>, as the product of the three rotations about the corresponding axes:		With more finesse, we can write the general rotation matrix that links the ''body frame'', <math>(x_1, x_2, x_3)</math>, to the ''inertial/laboratory'' frame, <math>(X, Y, Z)</math>, as the product of the three rotations about the corresponding axes:
	<table border="0" align="center" cellpadding="3">		<table border="0" align="center" cellpadding="3">

2022-01-07T04:47:30Z

Lagrangian Particle Motion

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     - \frac{\zeta_2}{\zeta_3}\biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]\frac{c^2}{b^2}
 = - 0.3448 ~~~ \Rightarrow ~~~ \theta = -0.3320
 \, .
 </math>
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 </tr>
 </table>
 <table border="0" cellpadding="5" align="center">
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 = 0.8399
 \, .</math>
    </td>
 </tr>

2022-01-07T03:44:57Z

Lagrangian Particle Motion

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 \, .
 </math>
    </td>
 </tr>

2022-01-07T03:41:38Z

Lagrangian Particle Motion

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 </table>
 Let's try the numerical values from the [[ThreeDimensionalConfigurations/RiemannTypeI#Example_b1.25c0.470|example Type I ellipsoid used in an accompanying discussion]], namely, <math>b/a = 1.25, c/a = 0.4703, \zeta_2 = -2.2794, \zeta_3 = -1.9637</math>.
 =See Also=

2022-01-07T00:57:48Z

Lagrangian Particle Motion

← Older revision		Revision as of 20:57, 6 January 2022
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	</tr>		</tr>
	</table>		</table>
			Let's try the numerical values from the [[ThreeDimensionalConfigurations/RiemannTypeI#Example_b1.25c0.470\|example Type I ellipsoid used in an accompanying discussion]], namely, <math>b/a = 1.25, c/a = 0.4703, \zeta_2 = -2.2794, \zeta_3 = -1.9637</math>.

	=See Also=		=See Also=

2022-01-07T00:49:19Z

Riemann Type I Ellipsoids

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    </td>
 </tr>
 </table>

2022-01-06T20:05:33Z

Riemann Type I Ellipsoids

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 </math>
    </td>
 </tr>
 </table>

2022-01-06T19:44:37Z

Rewrite Velocity-Component Expressions

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 \biggl[ \frac{z_0 \zeta_2}{a^2 + c^2} \biggr]
 - ( 1 + \Lambda^2 )y'
 \, .
 </math>

2022-01-06T19:35:52Z

Riemann Type I Ellipsoids

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 =Riemann Type I Ellipsoids=
 In an accompanying discussion of Riemann Type I ellipsoids, we have presented a [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again|methodical derivation of orbital parameters]] in a "tipped" reference frame.  In the context of our present discussion of Euler angles, it seems clear that this separate coordinate-frame transformation involved only one nonzero Euler angle; specifically, <math>(\phi, \theta, \psi) = (0, \theta, 0)</math>.  In terms of Euler angles, then, the transformation between the ellipsoid's unprimed frame and the primed "tipped" frame is given by the [[#FormMatrix|above expressions]] &hellip;
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 </table>
 Now, according to EFE &#8212; see eq. 154 of Chapter 7, &sect;51 (p. 156)  &#8212;
 <table border="0" cellpadding="5" align="center">
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 + 1
 \biggr\}y'
 \, .
 </math>

2022-01-06T02:04:15Z

Riemann Type I Ellipsoids

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 <tr>
    <td align="right"><math>u_1^'</math></td>
    <td align="center"><math>=</math></td>
    <td align="left">
 <math>
-\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \biggl[z_0 + y'\sin\theta + z'\cos\theta\biggr]
+\biggl[ \frac{a^2}{a^2 + c^2} \biggr] z_0 \zeta_2
-+ \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \biggl[ z' \sin\theta - y'\cos\theta \biggr]
+- \zeta_3 \cos\theta \biggl\{
-\, ,
+\biggl[ \frac{a^2}{a^2 + c^2} \biggr]
 </math>
    </td>