Angular Momentum
Table of Contents
Time derivatives in a rotating frame
Let ,
and
be unit basis vectors in the rotating frame, with a rotational speed of
about the axis
.
Letting
be the unit vector in this frame, we have:
![\begin{equation*}
\frac{d}{dt} \hat{\mathbf{u}} = \boldsymbol{\Omega} \times \hat{\mathbf{u}}
\end{equation*}](../../assets/latex/angular_momentum_cab9b56ab738135011f2e0153e534f1394152ce2.png)
Then if we have a function , taking the derivative in the rotational frame :
![\begin{equation*}
\begin{split}
\frac{d \mathbf{f}}{dt} &= \Bigg( \frac{d \mathbf{f}_x}{dt} \hat{\mathbf{i}} + \mathbf{f}_x \frac{d \hat{\mathbf{i}}}{dt} \Bigg)
+ \Bigg( \frac{d \mathbf{f}_y}{dt} \hat{\mathbf{j}} + \mathbf{f}_y \frac{d \hat{\mathbf{j}}}{dt} \Bigg)
+ \Bigg( \frac{d \mathbf{f}_z}{dt} \hat{\mathbf{k}} + \mathbf{f}_z \frac{d \hat{\mathbf{k}}}{dt} \Bigg) \\
&= \Bigg( \frac{d \mathbf{f}_x}{dt} \hat{\mathbf{i}} + \frac{d \mathbf{f}_y}{dt} \hat{\mathbf{j}} + \frac{d \mathbf{f}_z}{dt} \hat{\mathbf{k}} \Bigg)
+ \Bigg( \mathbf{f}_x \frac{d \hat{\mathbf{i}}}{dt} + \mathbf{f}_y \frac{d \hat{\mathbf{j}}}{dt} + \mathbf{f}_z \frac{\hat{\mathbf{k}}}{dt} \Bigg) \\
&= \Bigg( \frac{d \mathbf{f}}{dt} \Bigg)_\text{rot} + \Omega \times \mathbf{f}(t)
\end{split}
\end{equation*}](../../assets/latex/angular_momentum_82e25516674533f587f5c4cffcdd7b7148125b3e.png)
Which describes the change of wrt.
using the rotational frame.
Euler's Equations (rigid body dynamics)
From Time derivatives in a rotating frame we have:
![\begin{equation*}
\frac{d L_\text{in}}{dt} \overset{\text{def}}{=} \frac{d}{dt} \big( I_\text{in} \cdot M \big) = M_\text{in}
\end{equation*}](../../assets/latex/angular_momentum_5012cbf015be9cb2cb3001cb3a1a4afae827d36c.png)
where is the moment of inertia tensor calculated in the intertial frame .
It's often more useful to change to coordinates to the rotating frame .
With rotating frame we mean that the coordinate axes are rotating about a fixed axis.
Why is this useful? In this frame the moment of inertia tensor is constant (and diagonal)!
Moment of inertia
Covariance and contravariance
- Describe how the quantitive description of a certain geometric or physical entities change with a change of basis
Covariance
- Components of a covector change in the same way as changes in scale of the reference axes → the covector and the reference axes are covariant
Contravariance
- Components of a contr