Exponential map#
Warning
Draft
2D Euclidean space#
Infinitesimal rotations in 2D Euclidean space are expressed as:
\[R^{♭♭} = θ \: dx ∧ dy\]
We take the exponential map:
\[e^{R^{♭♭}} = \mathbb{1} + R^{♭♭} + \frac{1}{2!} \left( R^{♭♭} \right)^2 + \frac{1}{3!} \left( R^{♭♭} \right)^3 + \cdots\]
We have:
\[\begin{split}dx \⌟ \mathbb{1} &= dx \\
dy \⌟ \mathbb{1} &= dy \\\end{split}\]
\[\begin{split}dx \⌟ R^{♭♭} = + θ dy \\
dy \⌟ R^{♭♭} = - θ dx \\\end{split}\]
Calculations
dx
\[\begin{split}dx \⌟ R^{♭♭} &= dx \⌟ θ dx ∧ dy \\
&= θ dy\end{split}\]
dy
\[\begin{split}dy \⌟ R^{♭♭} &= dy \⌟ θ dx ∧ dy \\
&= - dy \⌟ θ dy ∧ dx \\
&= - θ dx\end{split}\]
\[\begin{split}dx \⌟ \left(R^{♭♭}\right)^2 &= - θ^2 dx \\
dy \⌟ \left(R^{♭♭}\right)^2 &= - θ^2 dy \\\end{split}\]
Calculations
dx
\[\begin{split}dx \⌟ \left(R^{♭♭} \right)^2 & = \left(dx \⌟ R^{♭♭} \right) \⌟ R^{♭♭} \\
& = θ dy \⌟ R^{♭♭} \\
& = θ dy \⌟ \left(θ dx ∧ dy\right) \\
& = θ^2 dy \⌟ \left( - dy ∧ dx \right) \\
& = - θ^2 dx \\\end{split}\]
dy
\[\begin{split}dy \⌟ \left(R^{♭♭} \right)^2 & = \left(dy \⌟ R^{♭♭} \right) \⌟ R^{♭♭} \\
& = - θ dx \⌟ R^{♭♭} \\
& = - θ dx \⌟ \left(θ dx ∧ dy\right) \\
& = - θ^2 dx \⌟ \left( dx ∧ dy \right) \\
& = - θ^2 dy \\\end{split}\]
\[\begin{split}dx \⌟ \left(R^{♭♭}\right)^3 &= - θ^3 dy \\
dy \⌟ \left(R^{♭♭}\right)^3 &= + θ^3 dx \\\end{split}\]
Calculations
dx
\[\begin{split}dx \⌟ \left(R^{♭♭} \right)^3 & = \left(dx \⌟ \left(R^{♭♭}\right)^2 \right) \⌟ R^{♭♭} \\
& = \left( - θ^2 dx \right) \⌟ R^{♭♭} \\
& = - θ^2 dx \⌟ \left(θ dx ∧ dy\right) \\
& = - θ^3 dy\end{split}\]
dy
\[\begin{split}dy \⌟ \left(R^{♭♭} \right)^3 & = \left(dy \⌟ \left(R^{♭♭}\right)^2 \right) \⌟ R^{♭♭} \\
& = \left( - θ^2 dy \right) \⌟ R^{♭♭} \\
& = - θ^2 dy \⌟ \left(θ dx ∧ dy\right) \\
& = - θ^3 dy \⌟ \left(- θ dy ∧ dx\right) \\
& = + θ^3 dx\end{split}\]
We then get:
\[\begin{split}dx \⌟ e^{θ dx ∧ dy}
= \begin{bmatrix}
\left( + 1 - \frac{1}{2!} θ^2 + \cdots \right) dx \\
\left( + θ - \frac{1}{3!} θ^3 + \cdots \right) dy \\
\end{bmatrix}
= \begin{bmatrix}
+ cos(θ) dx \\
+ sin(θ) dy \\
\end{bmatrix}\end{split}\]
\[\begin{split}dy \⌟ e^{θ dx ∧ dy}
= \begin{bmatrix}
\left( - θ + \frac{1}{3!} θ^3 + \cdots) \right) dx \\
\left( + 1 - \frac{1}{2!} θ^2 + \cdots) \right) dy \\
\end{bmatrix}
= \begin{bmatrix}
- sin(θ) dx \\
+ cos(θ) dy \\
\end{bmatrix}\end{split}\]
3D Euclidean space#
Commutation relations#
\[\begin{split}L_x &= dy ∧ dz \\
L_y &= dz ∧ dx \\
L_z &= dx ∧ dy \\\end{split}\]
\[\begin{split}\left[ L_x, L_y \right] &= L_x L_y - L_y L_x = L_z \\
&= dy ∧ dz \⌟ dz ∧ dx - dz ∧ dx \⌟ dy ∧ dz = dx ∧ dy\end{split}\]
\[\begin{split}\small
V^{♭} ⌟ \left[ dy ∧ dz, dz ∧ dx \right]
&= \begin{bmatrix} a dx \\ b dy \\ c dz \end{bmatrix} \⌟ dy ∧ dz \⌟ dz ∧ dx
- \begin{bmatrix} a dx \\ b dy \\ c dz \end{bmatrix} \⌟ dz ∧ dx \⌟ dy ∧ dz
\\ &= \begin{bmatrix} + b dz \\ - c dy \end{bmatrix} \⌟ dz ∧ dx
- \begin{bmatrix} - a dz \\ + c dx \end{bmatrix} \⌟ dy ∧ dz
\\ &= + b dx - a dy
\\ &= \begin{bmatrix}
+ b dx \\
- a dy \\
\end{bmatrix}\end{split}\]
\[\begin{split}V^{♭} ⌟ \left[ dx ∧ dy \right]
&= \begin{bmatrix}
a \: dx \\
b \: dy \\
c \: dz \\
\end{bmatrix}
\⌟
dx ∧ dy \\
&= \begin{bmatrix}
- b \: dx \\
+ a \: dy \\
\end{bmatrix}\end{split}\]