[Math] Rotating Frame and Angular Velocity

linear algebramatricesmatrix equationsquaternionsrotations

We have an equation $ \frac{dr}{dt}=\Omega \times \bf r \tag 1$

SPECIFICATIONS

  1. $\times$ means cross product,$\Omega$ constant angular velocity,${\bf r}$ is the postion vector of an object
  2. Given object has a position vector ${\bf r}$ in some non-rotating inertial reference frame
  3. This object is in a non-inertial reference frame which rotates with constant angular velocity $\mbox{ $\Omega$}$ about an axis passing through the origin of the inertial frame.
  4. Our object appears stationary in the rotating reference frame.
    In the non-rotating frame, the object's position vector ${\bf r}$ will appear to precess about the origin with angular velocity $\mbox{$\Omega$}$

Question

  1. What will be the case when $\Omega $ is not constant? Means varying with time.Will that be the case as follows? $ \frac{dr}{dt}=\Omega(t) \times \bf r \tag 2$
  2. In some other way imagine if I am happened to know $ \frac{dr}{dt}$,$\bf r$ at each s and able to find a vector $f(t)$ such that $ \frac{dr}{dt}=f(t) \times \bf r \tag 3$. Then can I say r is rotating with a varying angular velocity $f(t)=\Omega(t)$ related to the non moving frame?

Best Answer

If $\Omega$ is not constant, the equation still holds. But except in special cases, you will have to solve it by numerical integration.

If you have a certain rotation motion, and you found that $f(t)$ fits the equation $dr/dt=f(t)\times r$, it is not necessarily your angular velocity, but it will be true that $\Omega(t)=f(t)+\lambda(t) r(t)$, with $\lambda$ real. If you prescribe a rotation by giving $f(t)$, it surely defines a rotation motion, and can be recovered by numerical integration.

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