[NOTES/ME-02002]-The SO(3) Group

Let $K'$ and $K''$ be two systems of coordinate axes obtained by application of a rotation $(\hat{n}_1,\theta_1)$ followed by $(\hat{n}_2,\theta_2)$ \begin{equation} K\stackrel{(\hat{n}_1,\theta_1)}{\longrightarrow} K' \stackrel{(\hat{n}_2,\theta_2)}{\longrightarrow} K'' \end{equation} Let $x,x',x''$ denote components of position vector of a point $x$ with respect to the three sets of coordinate systems.Thus \begin{equation} x'=R_{\hat{n}_1}(\theta_1) x, \end{equation} and \begin{equation} x''=R_{\hat{n}_2}(\theta_2)x' =R_{\hat{n}_2}(\theta_2)R_{\hat{n}_1}(\theta_1) x . \end{equation} Thus \begin{equation} x''=R_{\hat{n}_3}(\theta_3) x \end{equation} where \begin{equation} R_{\hat{n}_3}=R_{\hat{n}_2}(\theta_2)R_{\hat{n}_1}(\theta_1). \end{equation} is an orthogonal matrix with unit determinant and hence it corresponds to a rotation about an axis $\hat{n_3}$ by some angle $\theta$. \begin{equation} K\longrightarrow K' \longrightarrow K''. \end{equation} It is now straight forward to check that the set of all rotations form a group. This is most easily checked by noting that the set \begin{equation} SO_{3}=\{R|R=3 \times 3 \text{ orthogonal matrix with } \det R=1\} \end{equation} is a group under matrix multiplication. This is the group of all (proper) rotations in the three dimension. $SO_{3}$ is called special orthogonal group and $S$ refers to the special property of determinant being unity. The set of all orthogonal $N \times N$ matrices (with determinant $\pm1$) is called orthogonal group $O(N)$. The orthogonal group may also be defined as the set of all transformations $R$ such that $x^{T} y=x'^T y'$ where $x\longrightarrow x'=R x, y'=Ry$. This set includes all proper $(\det R=+1)$ as well as improper rotations, $(\det R=-1)$ . The group $SO(3)$ is a non-Abelian group since the order of combining two rotations is important when the axes are different: \begin{equation} R_{\hat{n}_1}(\theta_1)R_{\hat{n}_2}(\theta_2)\ne R_{\hat{n}_2}(\theta_2)R_{\hat{n}_1}(\theta_1) . \end{equation}

Rotations about a fixed axis: The set of all rotations about a {\it fixed axis} form a subgroup.

1. Closure property: The rotation by angle $\theta$ followed by a rotation by angle $\phi$ about the same axis is a rotation by angle $\theta+\phi$. \begin{equation} R_{\hat{n}}(\theta)R_{\hat{n}}(\phi)=R_{\hat{n}}(\theta+\phi). \end{equation}
2. Associative property is obeyed for matrices. It can also be seen to hold because of \begin{equation} \Big(R_{\hat{n}}(\theta_1)R_{\hat{n}}(\theta_2)\Big)R_{\hat{n}}(\theta_3)=R_{ \hat{n}}(\theta_1+\theta_2+\theta_3) =R_{\hat{n}}\Big(R_{\hat{n}}(\theta_2)R_{\hat{n}}(\theta_3)\Big). \end{equation}
3. Existence of identity : The rotation by $\theta=0$ is identity transfor mation and is represented by the identity matrix $I$.
4. Inverse of a rotation $(\hat{n},\theta)$ is the rotation $(\hat{n},-\theta)$.

The group of rotations about a fixed axis is a commutative (Abelian) group because both $R_{\hat{n}}(\theta)R_{\hat{n}}(\phi)$ and $ R_{\hat{n}}(\phi)R_{\hat{n}}(\theta)$ are equal to $R_{n}(\theta+\phi)$.