What kinds of motion obeys a higher order form of angular motion? Angular velocity $\vec{\omega}$ can be defined in terms of velocity $\vec{v}$ and position $\vec{s}$ as:
$$ \vec{\omega} = \frac{\vec{s} \times \vec{v}}{\left\lvert s\right\rvert^2} $$
Constant angular motion therefore obeys the equation:
$$ \frac{\vec{s} \times \vec{v}}{\left\lvert s\right\rvert^2} = \vec{c} $$
What kind of motion obeys a higher order version of the equation?
$$ \vec{\theta} = \int \vec{\omega} \, \mathrm{d} t $$
$$ \vec{c} = \frac{\vec{\theta} \times \vec{\omega}}{\left\lvert \theta \right\rvert^2} $$
I know how to solve differential equations but not vector differential equations.
 A: Constant angular velocity traces out motion like:
$$ \vec{s} = \cos \left(\left\lvert \omega \right\rvert t\right) \hat{a} + \sin \left(\left\lvert \omega \right\rvert t\right) \left(\hat{\omega} \times \hat{a} \right) + b \hat{\omega}$$
A higher order version of this would trace out motion like:
$$ \vec{\theta} = \cos \left(\left\lvert c\right\rvert t\right) \hat{a} + \sin \left(\left\lvert c \right\rvert t\right) \left(\hat{c} \times \hat{a}\right) + b \hat{c}$$
for some constants $\hat{a}$, $\vec{c}$ and b.
Such motion could be visualized as tracing out a circle on a sphere.
Note the third component in the equations. The thing moving in a circular manner can be offset along the axis of rotation and still have a constant angular velocity.
A: When studying three phase electric motors, people learn about the PARK TRANSFORM.
Very clever math routine to analyse the starting of this kind of motors.
The math behind this procedure is quite complicated.
If interested, google expressions like "ANGULAR VELOCITY STARTING ELECTRIC MOTORS" or PARK TRANSFORM STARTING THREE PHASE ELECTRIC MOTORS".
See the graphs of angular velocity against time.
Articles of interest:
1) Simulation of three-phase induction motor in Scilab, by Madejski Rafał
2) Modelling and Analysis of Squirrel Cage Induction Motor with Leading Reactive Power Injection, by Adisa A. Jimoh, Pierre-Jac Venter and Edward K. Appiah
