Rotating reference frame: Difference between revisions

Revision as of 22:53, 16 November 2022

Revision as of 02:22, 17 November 2022

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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 matrixtransformation taking basis vectors fromof the inertial- to the rotating frame, with matrix columns equal to the basis vectors of the rotating frame, then the cross product mulitplicationmultiplication by the rotation vector expressed in the rotating frame 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>