- #1

sergiokapone

- 302

- 17

Is any way to get Rodrigues' rotation formula from matrix exponential

\begin{equation}

e^{i\phi (\star\vec{n}) } = e^{i\phi (\vec{n}\cdot\hat{\vec{S}}) } = \hat{I} + (\star\vec{n})\sin\phi + (\star\vec{n})^2( 1 - \cos\phi ).

\end{equation}

using SO(3) groups comutators properties ONLY like

\begin{equation}\hat S_k\hat S_j-\hat S_j\hat S_k=i\varepsilon_{kjl}\hat S_l,\qquad \hat S^2=2\hat I ?\end{equation}

where ##\vec{n} = (n_x,n_y,n_z)^{\top}##, ##\vec{n}^2 = 1##, and

\begin{equation}\label{}

(\vec{n}\cdot\hat{\vec{S}}) = \star\vec{n} =

\begin{pmatrix}

0 & -n_z & n_y \\

n_z & 0 & -n_x \\

-n_y & n_x & 0 \\

\end{pmatrix},

\end{equation}

\begin{multline}\label{}

\hat{\vec{S}} =

\begin{pmatrix}

0 & 0 & 0 \\

0 & 0 & i \\

0 & -i & 0 \\

\end{pmatrix}

\vec{e}_x

+

\begin{pmatrix}

0 & 0 & -i \\

0 & 0 & 0 \\

i & 0 & 0 \\

\end{pmatrix}

\vec{e}_y

+

\begin{pmatrix}

0 & i & 0 \\

-i & 0 & 0 \\

0 & 0 & 0 \\

\end{pmatrix}

\vec{e}_z =

\hat{S}_x \vec{e}_x +

\hat{S}_y \vec{e}_y +

\hat{S}_z \vec{e}_z

.

\end{multline}

References: https://en.wikipedia.org/wiki/Axis–angle_representation#Exponential_map_from_so(3)_to_SO(3)

\begin{equation}

e^{i\phi (\star\vec{n}) } = e^{i\phi (\vec{n}\cdot\hat{\vec{S}}) } = \hat{I} + (\star\vec{n})\sin\phi + (\star\vec{n})^2( 1 - \cos\phi ).

\end{equation}

using SO(3) groups comutators properties ONLY like

\begin{equation}\hat S_k\hat S_j-\hat S_j\hat S_k=i\varepsilon_{kjl}\hat S_l,\qquad \hat S^2=2\hat I ?\end{equation}

where ##\vec{n} = (n_x,n_y,n_z)^{\top}##, ##\vec{n}^2 = 1##, and

\begin{equation}\label{}

(\vec{n}\cdot\hat{\vec{S}}) = \star\vec{n} =

\begin{pmatrix}

0 & -n_z & n_y \\

n_z & 0 & -n_x \\

-n_y & n_x & 0 \\

\end{pmatrix},

\end{equation}

\begin{multline}\label{}

\hat{\vec{S}} =

\begin{pmatrix}

0 & 0 & 0 \\

0 & 0 & i \\

0 & -i & 0 \\

\end{pmatrix}

\vec{e}_x

+

\begin{pmatrix}

0 & 0 & -i \\

0 & 0 & 0 \\

i & 0 & 0 \\

\end{pmatrix}

\vec{e}_y

+

\begin{pmatrix}

0 & i & 0 \\

-i & 0 & 0 \\

0 & 0 & 0 \\

\end{pmatrix}

\vec{e}_z =

\hat{S}_x \vec{e}_x +

\hat{S}_y \vec{e}_y +

\hat{S}_z \vec{e}_z

.

\end{multline}

References: https://en.wikipedia.org/wiki/Axis–angle_representation#Exponential_map_from_so(3)_to_SO(3)

Last edited: