It is me again. I ask you to say if I infringing any SO rule, like those related to good questions and so.
My supervisor helps me learn about symmetry in geometry. His question for me is clear: we can find the symmetries of a vector field $a$ by finding a connection $\nabla$ for such the Lie derivative $L_a \nabla$ to be 0. He suspects the result may be related to the controllability of a dynamical system in a topological space. Since finding a symmetry connection is not trivial, he suggested pursuing the following direction: we can find the 1-form $\omega$ to satisfy $L_a \omega = 0$.
Great! I like reading broad theoretic textbooks, but I still need an example. Thus, he gave me this example: Let us take the Brockett integrator $(u^1, u^2, x^2 \, u^1 - x^1 u^2)$ as vector field $a \in T_x\mathbb{R}^3$. We define an extended vector field $\bar{a}$ as extension $(a, 0, 0)$. My first solution guess was to use Cartan's formula for the Lie derivative of a $n$-form $L_a \omega$, given by expression $\iota_a(d\omega)+d(\iota_a \, \omega)$ and expanded below, in Einstein's notation.
$$[(\partial_j a^i) \alpha_i + a^i (\partial_i \alpha_j)] dx^j$$
I was able to only find the 2 1-forms $d u^1$ and $d u^2$. Since the Brockett integrator is a transformation of unicycle $(v \cos(\theta), v \sin(\theta), \omega)$, such that input variables $v$ and $\omega$ are respectively linear and angular speeds, I think I can use the translation and rotation symmetries somehow, but as I said, I am not an expert, just a curious student. Both state and input maps from Brockett integrator to unicycle are below:
$$\begin{bmatrix} x^1 \\ x^2 \\ x^3 \end{bmatrix} = \begin{bmatrix} \theta \\ x \cos{(\theta)} + y \sin{(\theta)} \\ x \sin{(\theta)} - y \cos{(\theta)} \end{bmatrix}$$
$$\begin{bmatrix} u^1 \\ u^2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & x \sin{(\theta)} - y \cos{(\theta)} \end{bmatrix} \begin{bmatrix} v \\ \omega_{\theta} \end{bmatrix}$$
$\mathbf{Question}$: how can I use translation and rotation symmetry with new coordinates $(x, y, \theta, v, \omega)$ for calculating 1-form $\bar{\omega}$ on partial differential equation $L_\bar{a} \bar{\omega} = 0$, for vector field $\bar{a}=(v \cos{\theta}, v \sin{\theta}, \omega_{\theta}, 0, 0)$? The next step is coordinate transformation $(x^1, x^2, x^3, u^1, u^2) = \psi(x, y, \theta, v, \omega_\theta)$ and its inverse $\psi^{-1}$.Additionally, how can I find the symmetry connection coefficients for connection $\nabla$ given 1-form $\omega$?