Interpretation of general rotations in 3D generated by Lie Algebra

Question

I just started learning about Group Theory and Lie Algebras.

I know that an infinitesimal rotation in 3D can be expressed as $R = I + A$ where A is an 3x3 antisymmetric matrix. Now all 3x3 antisymmetric matrices can be written as linear combination of

$J_x = \begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 1 \\ 0 & -1 & 0\end{bmatrix}$ $\qquad$ $J_y = \begin{bmatrix}0 & 0 & -1\\ 0 & 0 & 0 \\ 1 & 0 & 0\end{bmatrix}$ $\qquad$ $J_z = \begin{bmatrix}0 & 1 & 0\\ -1 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$

as $$A = \Theta_x J_x + \Theta_y J_y + \Theta_z J_z$$ I therefore obtain a general rotation by exponentiating: $$R(\Theta_x, \Theta_y, \Theta_z) = e^{\Theta_x J_x + \Theta_y J_y + \Theta_z J_z}$$ This rotation is paramatrized by the three real numbers $\Theta_x$, $\Theta_y$ and $\Theta_z$.

My question is: Is there an easy and intuitive way to see how I can interpret the general rotation the combined parameters describe? For example, is it possible to make a connection between them and axis-angle representation of a rotation?

Answer 1

Yes, there is! The infinitesimal rotation $J_x$ correspond to the rotation which fixes the $x$-axis and rotate the other two. The same holds for the other three.

The "blind-eye" proof in which you don't know the answer can be done by actually computing $e^{tJ_x}$ for a small parameter t (the "infinitesimality" of the rotation). But there is a trick to see "why" this should happen. Note that for any matrix $A$

$$ \partial_t ( e^{tA}) |_{t=0} = \partial_t \sum_{n=0}^{\infty} t^n \frac{A^n}{n! } |_{t=0} = \sum_{n=1}^{\infty} t^{n-1} \frac{A^n}{n! } |_{t=0} = A$$

and $ e^{tA} |_{t=0} = Id$. These two conditions intuitively determines $e^{tA}$ for small $t$. Now note that the matrix

$$ B_x(t) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & cos(t) & sin(t) \\ 0 & -sin(t) & cos(t) \\ \end{pmatrix} $$ is the identity in $t=0$, and it's derivative in zero is $J_x$. So it is very likely to be $e^{tJ_x}$!!

Edit. In general, searching online, I found a paper on rotations which introduces many parametrizations of $SO(3) $; two of them are euler angles and exponentiation from infitesimal rotation. It seems there is no nice change from one to another. If you have explicit coefficients you can indeed find the other forms as usual. For example, here I illustrate how to find the simplest parametrizations, i.e. the axis of the rotation and the angle, with a simple algorithm:

1. Note that the $M = \exp (a J_x + b J_y + c J_z)$, which have an explicit formula in terms of sines and cosines, always have two complex eigenvalues ( which are $w, \bar{w}$) and one real eigenvalue (of norm 1). Since the determinant is 1 and the two complex numbers have product $w \bar{w} = | w| =1$, the real eigenvalue must be 1. Solving

$$ Mv = v$$

You can find the axis of the rotation.

2. By computing the characteristic polynomial and dividing by $\lambda-1$, you get a quadratic equation which yields $w, \bar{w}$. Writing $w = e^{i\theta}$ , $\theta$ is the angle you are looking for.