BCH 公式 | Baker Campbell Hausdorff formula

1 BCH 公式

BCH 公式 (Baker Campbell Hausdorff formula) 如下：
\(\ln (\exp (\boldsymbol{A}) \exp (\boldsymbol{B}))=\boldsymbol{A}+\boldsymbol{B}+\frac{1}{2}[\boldsymbol{A}, \boldsymbol{B}]+\frac{1}{12}[\boldsymbol{A},[\boldsymbol{A}, \boldsymbol{B}]]-\frac{1}{12}[\boldsymbol{B},[\boldsymbol{A}, \boldsymbol{B}]]+\cdots\tag{1}\)

2 线性近似

在李群和李代数中，我们用到了 BCH 公式的近似线性表示：
\(\ln \left(\exp \left(\phi_{1}^{\wedge}\right) \exp \left(\phi_{2}^{\wedge}\right)\right)^{\vee} \approx\left\{\begin{array}{ll}J_{l}\left(\phi_{2}\right)^{-1} \phi_{1}+\phi_{2} & \text { if } \phi_{1} \text { is small } \\J_{r}\left(\phi_{1}\right)^{-1} \phi_{2}+\phi_{1} & \text { if } \phi_{2} \text { is small }\end{array}\right.\tag{2}\)

其中：
\(\boldsymbol{J}_{l}=\boldsymbol{J}=\frac{\sin \theta}{\theta} \boldsymbol{I}+\left(1-\frac{\sin \theta}{\theta}\right) \boldsymbol{a} \boldsymbol{a}^{T}+\frac{1-\cos \theta}{\theta} \boldsymbol{a}^{\wedge}\tag{3}\)

\(\boldsymbol{J}_{l}^{-1}=\frac{\theta}{2} \cot \frac{\theta}{2} \boldsymbol{I}+\left(1-\frac{\theta}{2} \cot \frac{\theta}{2}\right) \boldsymbol{a} \boldsymbol{a}^{T}-\frac{\theta}{2} \boldsymbol{a}^{\wedge}\tag{4}\)

以及：
\(J_{r}(\phi)=J_{l}(-\phi)\tag{5}\)

这样就解决了李群乘法和李代数加法的问题。

参考文献

[1] https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula
[2] https://dyhgo.xyz/%E6%9D%8E%E7%BE%A4%E4%B8%8E%E6%9D%8E%E4%BB%A3%E6%95%B0/