I have a question regarding the logarithmic map $log: G\mapsto \mathfrak{g}$ and exponential map $exp: \mathfrak{g}\mapsto G$ between the Galilei group and its Lie algebra. The Galilei group of two dimensions can be represented by the matrix Lie group $$Gal(2)=\begin{bmatrix}R&v&a\\0&1&s\\0&0&1\end{bmatrix}$$ and its Lie algebra by $$\mathfrak{gal(2)}=\begin{bmatrix}X&w&b\\0&0&t\\0&0&0\end{bmatrix}.$$ Because the Galilei group $Gal(2)\cong \mathbb{R}^3\rtimes SE(2)\cong \mathbb{R}^3\rtimes (\mathbb{R}^2\rtimes SO(2))$ and there is a closed-form for the exponential and logarithmic maps for the special Euclidean group $SE(2)$ I wonder, if such a closed form also exists for the Galilei group $Gal(2)$? If yes, would that also apply to $Gal(3)$?
Lino
Yes, of course. You may intuit the name of the game from the simplest ever Affine Lie Group, and verify the 2×2 matrix expressions. I'll get you started on Gal(2), but of course this also works for Gal(3), etc.
In your case, you are working with upper triangular 4×4 matrices. I'll use vectors and 2×2 matrices for the upper two rows, which, somewhat confusingly, you conflated with scalar symbols.
The generic group element is $$G=\begin{bmatrix}{\mathbf R}(\theta')&\vec v&\vec a\\0&1&s\\0&0&1\end{bmatrix}$$
and the generic Lie algebra element is $$ g=\begin{bmatrix}{\mathbf X}&\vec w&\vec b\\0&0&t\\0&0&0\end{bmatrix}.$$
First, recall the SO(2) rotation is generated by $$ {\mathbf X}=i\theta \sigma_2= \theta \begin{bmatrix} 0&1\\-1&0\end{bmatrix} \qquad \implies \\ \exp {\mathbf X}= \begin{bmatrix} \cos\theta &\sin\theta \\-\sin\theta &\cos\theta \end{bmatrix} = {\mathbf R}(\theta), $$ with ${\mathbf R}( 0)={\mathbf 1} $.
So go to g which is a linear combination of 6 generators/parameters, $\theta; t; \vec w; \vec b$. Evaluate the group elements, exponentials of each of them separately: $$ \exp \begin{bmatrix} {\mathbf X} & 0&0\\0&0&0\\0&0&0\end{bmatrix} = \begin{bmatrix} {\mathbf R}(\theta) & 0& 0\\0&1&0\\0&0&1\end{bmatrix} ; $$ $$ \exp \begin{bmatrix} 0& 0&0\\0&0&t\\0&0&0\end{bmatrix} = \begin{bmatrix}{\mathbf 1} & 0& 0\\0&1&t\\0&0&1\end{bmatrix} ; $$ $$ \exp \begin{bmatrix} 0& \vec w&0\\0&0&0\\0&0&0\end{bmatrix} = \begin{bmatrix}{\mathbf 1} & \vec w&0\\0&1&0\\0&0&1\end{bmatrix} ; $$$$ \exp \begin{bmatrix} 0& 0&\vec b\\0&0&0\\0&0&0\end{bmatrix} = \begin{bmatrix}{\mathbf 1}& 0&\vec b\\0&1&0\\0&0&1\end{bmatrix} . $$
The product of all 6 (4) of them, in any order, will be an upper triangular matrix of the generic G type above; for instance , with the first one on the right, you get $$\begin{bmatrix}{\mathbf R}(\theta)&\vec w&\vec b + \vec w t\\0&1&t\\0&0&1\end{bmatrix}.$$
But also, by adroit orderings and CBH compositions, for example the trivial $$ \exp \begin{bmatrix} 0& \vec w&0\\0&0&0\\0&0&0\end{bmatrix} \exp \begin{bmatrix} 0& 0&\vec b\\0&0&0\\0&0&0\end{bmatrix}=\exp \begin{bmatrix} 0 & \vec w&\vec b\\0&0&0\\0&0&0\end{bmatrix} , $$ it will be the single exponential of an expression like g, $$\begin{bmatrix}\tilde{\mathbf X}&\tilde{\vec{w}}&\tilde{ \vec{b}}\\0&0&\tilde t\\0&0&0\end{bmatrix}.$$
but with its parameters being functions of the 6 parameters of your input. You compare the two expressions, and you get a relatively simple expression of group parameters as a function of algebra parameters. The exercise is not as trivial as the 2×2 matrix affine group expression I linked at the beginning, but it is likely to be tractable. (I wouldn't be shocked if you found it in some book, but I have not seen it.)