Editing Rotating reference frame (section)

=== Time derivatives in the two frames ===
Introduce unit vectors <math>\hat{\boldsymbol{\imath}},\ \hat{\boldsymbol{\jmath}},\ \hat{\boldsymbol{k}}</math>, now representing standard unit basis vectors in the general rotating frame. As they rotate they will remain normalized and perpendicular to each other. If they rotate at the speed of <math>\Omega(t)</math> about an axis along the rotation vector <math>\boldsymbol {\Omega}(t)</math> then each unit vector <math>\hat{\boldsymbol{u}}</math> of the rotating coordinate system (such as <math>\hat{\boldsymbol{\imath}},\ \hat{\boldsymbol{\jmath}},</math> or <math>\hat{\boldsymbol{k}}</math>) abides by the following equation:
<math display=block>\frac{\mathrm{d}}{\mathrm{d}t}\hat{\boldsymbol{u}} = \boldsymbol{\Omega} \times \boldsymbol{\hat{u}} \ .</math>
So if <math>R(t)</math> denotes the transformation taking basis vectors of the inertial- to the rotating frame, with matrix columns equal to the basis vectors of the rotating frame, then the cross product multiplication by the rotation vector is given by <math>\boldsymbol{\Omega}\times = R'(t)\cdot R(t)^T</math>.

If <math>\boldsymbol{f}</math> is a vector function that is written as<ref group=note>So <math>f_1, f_2, f_3</math> are <math>\boldsymbol{f}</math>'s coordinates with respect to the rotating basis vector <math>\hat{\boldsymbol{\imath}},\ \hat{\boldsymbol{\jmath}},\ \hat{\boldsymbol{k}}</math> (<math>\boldsymbol{f}</math>'s coordinates with respect to the inertial frame are not used). Consequently, at any given instant, the rate of change of <math>\boldsymbol{f}</math> with respect to these rotating coordinates is <math>\frac{\mathrm{d}f_1}{\mathrm{d}t}\hat{\boldsymbol{\imath}} + \frac{\mathrm{d}f_2}{\mathrm{d}t}\hat{\boldsymbol{\jmath}} + \frac{\mathrm{d}f_3}{\mathrm{d}t}\hat{\boldsymbol{k}}.</math> So for example, if <math>f_1 \equiv 1</math> and <math>f_2 = f_3 \equiv 0</math> are constants, then <math>\boldsymbol{f} \equiv \hat{\boldsymbol{\imath}}</math> is just one of the rotating basis vectors and (as expected) its time rate of change with respect to these rotating coordinates is identically <math>\boldsymbol{0}</math> (so the formula for <math>\frac{\mathrm{d}}{\mathrm{d}t} \boldsymbol{f}</math> given below implies that the derivative at time <math>t</math> of this rotating basis vector <math>\boldsymbol{f} \equiv \hat{\boldsymbol{\imath}}</math> is <math>\frac{\mathrm{d}}{\mathrm{d}t} \boldsymbol{i} = \boldsymbol{\Omega}(t) \times \boldsymbol{i}(t)</math>); however, its rate of change with respect to the non-rotating inertial frame will not be constantly <math>\boldsymbol{0}</math> except (of course) in the case where <math>\hat{\boldsymbol{\imath}}</math> is not moving in the inertial frame (this happens, for instance, when the axis of rotation is fixed as the <math>z</math>-axis (assuming standard coordinates) in the inertial frame and also <math>\hat{\boldsymbol{\imath}} \equiv (0, 0, 1)</math> or <math>\hat{\boldsymbol{\imath}} \equiv (0, 0, -1)</math>).</ref>
<math display=block>\boldsymbol{f}(t)=f_1(t) \hat{\boldsymbol{\imath}}+f_2(t) \hat{\boldsymbol{\jmath}}+f_3(t) \hat{\boldsymbol{k}}\ ,</math>
and we want to examine its first derivative then (using the [[product rule]] of differentiation):<ref name=Lanczos>{{cite book |url=https://books.google.com/books?num=10&btnG=Google+Search |title=The Variational Principles of Mechanics |author=Cornelius Lanczos |date=1986 |isbn=0-486-65067-7 |publisher=[[Dover Publications]] |edition=Reprint of Fourth Edition of 1970 |no-pp=true |pages=Chapter 4, §5}}</ref><ref name=Taylor>{{cite book |title=Classical Mechanics |author=John R Taylor |page= 342 |publisher=University Science Books |isbn=1-891389-22-X |date=2005 |url=https://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1}}</ref>
<math display=block>\begin{align}
  \frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{f}
    &= \frac{\mathrm{d}f_1}{\mathrm{d}t}\hat{\boldsymbol{\imath}} + \frac{\mathrm{d}\hat{\boldsymbol{\imath}}}{\mathrm{d}t}f_1 + \frac{\mathrm{d}f_2}{\mathrm{d}t}\hat{\boldsymbol{\jmath}} + \frac{\mathrm{d}\hat{\boldsymbol{\jmath}}}{\mathrm{d}t}f_2 + \frac{\mathrm{d}f_3}{\mathrm{d}t}\hat{\boldsymbol{k}} + \frac{\mathrm{d}\hat{\boldsymbol{k}}}{\mathrm{d}t}f_3 \\
    &= \frac{\mathrm{d}f_1}{\mathrm{d}t}\hat{\boldsymbol{\imath}} + \frac{\mathrm{d}f_2}{\mathrm{d}t}\hat{\boldsymbol{\jmath}} + \frac{\mathrm{d}f_3}{\mathrm{d}t}\hat{\boldsymbol{k}} + \left[\boldsymbol{\Omega} \times \left(f_1 \hat{\boldsymbol{\imath}} + f_2 \hat{\boldsymbol{\jmath}} + f_3 \hat{\boldsymbol{k}}\right)\right] \\
    &= \left( \frac{\mathrm{d}\boldsymbol{f}}{\mathrm{d}t}\right)_{\mathrm{r}} + \boldsymbol{\Omega} \times \boldsymbol{f}
\end{align}</math>
where <math>\left( \frac{\mathrm{d}\boldsymbol{f}}{\mathrm{d}t}\right)_{\mathrm{r}}</math> denotes the rate of change of <math>\boldsymbol{f}</math> as observed in the rotating coordinate system. As a shorthand the differentiation is expressed as:
<math display=block>\frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{f} = \left[ \left(\frac{\mathrm{d}}{\mathrm{d}t}\right)_{\mathrm{r}} + \boldsymbol{\Omega} \times \right] \boldsymbol{f} \ .</math>

This result is also known as the [[transport theorem]] in analytical dynamics and is also sometimes referred to as the ''basic kinematic equation''.<ref>{{cite web|last=Corless|first=Martin|title=Kinematics|url=https://engineering.purdue.edu/AAE/Academics/Courses/aae203/2003/fall/aae203F03supp.pdf|archive-url=https://web.archive.org/web/20121024121222/https://engineering.purdue.edu/AAE/Academics/Courses/aae203/2003/fall/aae203F03supp.pdf|url-status=dead|archive-date=24 October 2012|work=Aeromechanics I Course Notes|publisher=[[Purdue University]]|access-date=18 July 2011|page=213}}</ref>