Rotating reference frame: Difference between revisions

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Introduce the unit vectors <math>\hat{\boldsymbol{\imath}},\ \hat{\boldsymbol{\jmath}},\ \hat{\boldsymbol{k}}</math> representing standard unit basis vectors in the rotating frame. As they rotate they will remain normalized. If we let them rotate at the speed of <math>\Omega(t)</math> about an axis <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}(t) \times \boldsymbol{\hat{u}} \ .</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> 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>