$SU(2)$ Lie group

Question

I have been studying Lie groups for a bit of fun for a while now and think they are fascinating. I have recently been told that $SU(2)$ can be used in some way to keep track of navigational systems in bodies as they complete whole turns.

Does anyone know anything on this subject that can explain it to me clearly and maybe have any idea where I can read up on it?

Answer 1

This is connected to the relationship of SU(2) to the unit quaternions. Computationally and practically, there are some perks to using quaternions for rotations.

You can read about that at Wikipedia, and I can also recommend Quaternions and Rotation Sequences by Jack B. Kuipers. (Its subtitle is "A primer with Applications to Orbits, Aerospace, and Virtual Reality".) It sounds like you might get something out of it.

Answer 2

Let $\sigma_i$ be the Pauli matrices. A typical representation of $\mathfrak{su}(2)$ is $i\sigma_i$.

Any 3-vector $x$ can be represented as $$X = \sigma\cdot x,$$ where $\sigma\cdot x = \sum_{i=1}^3 \sigma_i x_i$. ($i X$ is a pure imaginary quaternion.) The components of $x$ can be found from $X$, using the fact that $\mathrm{Tr}\, \sigma_i \sigma_j = 2\delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta. We have $$x_i = \frac{1}{2} \mathrm{Tr}\, \sigma_i X.$$

Rotate the vector $x$ about the axis $n$ by the angle $\theta$. The rotated vector $x'$ is represented by $$\begin{equation*} X' = R^{-1} X R \tag{1} \end{equation*}$$ where $$R = e^{i\theta n\cdot \sigma/2} = \mathbb{I} \cos\frac{\theta}{2} + i n\cdot\sigma \sin \frac{\theta}{2}.$$ (In the language of quaternions, $R$ is a versor.) Note that $R^{-1}(\theta) = R(-\theta)$. Equation (1) makes the fact that $\mathrm{SU}(2)$ is the double cover of $\mathrm{SO}(3)$ explicit---in $\mathrm{SU}(2)$ $R(\theta+2\pi) = -R(\theta)$, but in $\mathrm{SO}(3)$ these rotations are indistinguishable. The fact that (1) represents the appropriate rotation in 3-space can be proved by showing, for example, that it gives Rodrigues' rotation formula, relating $x'$ to $x$ in the appropriate way. Roughly, $X$ transforms like a vector, that is, like the $(1/2,1/2)$ representation of $\mathrm{SU}(2)$---its left index transforms with $R^{-1}$ and its right index with $R$.

It is a good exercise to show using this formalism that the vector $x$ rotated about the $z$-axis by the angle $\theta$ is $$x' = \left(\begin{array}{c} x_1 \cos\theta - x_2\sin\theta \\ x_1 \sin\theta + x_2\cos\theta \\ x_3 \end{array}\right) = \left(\begin{array}{ccc} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{array}\right) \left(\begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array}\right)$$ as required.

Figure 1. Gimbal lock.

As mentioned in the interesting link provided by @JyrkiLahtonen in the comments, gimbal lock can affect gimbals as well as mathematical representations of rotations. Rotations by Euler angles suffer this problem. Let $R(\alpha,\beta,\gamma)$ be the usual representation of a rotation in 3-space by Euler angles. If $\beta = 0$, then $R(\alpha,0,\gamma) = R(\alpha+\gamma,0,0) = R(0,0,\alpha+\gamma)$. We have lost a degree of freedom---we can only rotate about the $z$-axis. $\mathrm{SU}(2)$ representations of rotations in 3-space don't have this problem. For example, no matter how we choose $\theta$, we will not lose any degrees of freedom in $n$ due to that choice. If you are familiar with topology, these statements are a consequence of the fact that the three torus is not a covering space of $\mathrm{SO}(3)$, but that $\mathrm{Spin}(3) = \mathrm{SU}(2)$ is its universal cover.

I recommend using Wikipedia and Google to learn more about this interesting subject.