Let $G$ be a Lie group with multiplication * and $\mathrm{\μ}\: G\to M$a free (left) action of $G$ on a manifold $M$. A right moving frame is a map $\mathrm{rho}\:G \to M$such that $\mathrm{\ρ}\left(\mathrm{\μ}\left(a\, x\right)\right) \= \mathrm{rho}\left(x\right)\*a^{-1}$ for all $a \in G$and $x \in M$.

A cross-section to the action $\mathrm{\μ}\: G\to M$  is a submanifold $K$ of $M\,$ with codim(K) = dim$\left(G\right)$, which is transverse to the orbits of $\mathrm{\μ}$. The cross-section $K$has the property that if $k_1\, k_2 \in K$and $\mathrm{\μ}\left(a\, k_1\right) \= \mathrm{\μ}\left(a\, k_2\right)$then $k_1 \= k_2\.$

The Invariantization command will map any function on $M$ to a $G$invariant function.

The commands RightMovingFrame and Invariantization are part of the DifferentialGeometry:-GroupActions:-MovingFrames package. They can be used in the forms RightMovingFrame(...) and Invariantization(...) only after executing the commands with(DifferentialGeometry), with(GroupActions), and with(MovingFrames), but can always be used by executing DifferentialGeometry:-GroupActions:-MovingFrames:-RightMovingFrame(...) and DifferentialGeometry:-GroupActions:-MovingFrames:-Invariantization(...).

References:

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{with}\left(\mathrm{JetCalculus}\right)\:$$\mathrm{with}\left(\mathrm{GroupActions}\right)\:$

$\mathrm{with}\left(\mathrm{MovingFrames}\right)\:$

$\mathrm{Preferences}\left(''JetNotation''\,''JetNotation2''\right)\:$

$\mathrm{DGsetup}\left(\left[x\right]\,\left[y\right]\,E\,3\,\mathrm{verbose}\right)$

$\mathrm{The\; following\; coordinates\; have\; been\; protected:}$

$\left[x\,y_0\,y_1\,y_2\,y_3\right]$

$\mathrm{The\; following\; vector\; fields\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{D\_x}\,{\mathrm{D\_y}}_0\,{\mathrm{D\_y}}_1\,{\mathrm{D\_y}}_2\,{\mathrm{D\_y}}_3\right]$

$\mathrm{The\; following\; differential\; 1-forms\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{dx}\,{\mathrm{dy}}_0\,{\mathrm{dy}}_1\,{\mathrm{dy}}_2\,{\mathrm{dy}}_3\right]$

$\mathrm{The\; following\; type\; [1,0]\; biforms\; have\; been\; defined\; and\; protected::}$

$\left[\mathrm{Dx}\right]$

$\mathrm{The\; following\; type\; [0,1]\; biforms\; (contact\; 1-forms)\; have\; been\; defined\; and\; protected::}$

$\left[{\mathrm{Cy}}_0\,{\mathrm{Cy}}_1\,{\mathrm{Cy}}_2\,{\mathrm{Cy}}_3\right]$

$\mathrm{frame\; name:\; E}$

$\mathrm{Gamma}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}\,\mathrm{D\_y}\left[0\right]\,x\mathrm{D\_y}\left[0\right]-y\left[0\right]\mathrm{D\_x}\,y\left[0\right]\mathrm{D\_y}\left[0\right]+x\mathrm{D\_x}\right]\right)$

$\mathrm{\Γ}:=\left[\mathrm{D\_x}\,{\mathrm{D\_y}}_0\,-\mathrm{D\_x}{y}_0\+x{\mathrm{D\_y}}_0\,\mathrm{D\_x}x\+{\mathrm{D\_y}}_0{y}_0\right]$

$\mathrm{DGsetup}\left(\left[a\,b\,\mathrm{\theta }\,t\right]\,G\right)$

$\mathrm{frame\; name:\; G}$

$\mathrm{\mu }≔\mathrm{Action}\left(\mathrm{Gamma}\,G\right)$

$\mathrm{\μ}:=\left[x\=a-y_0{\ⅇ}^t\sin \left(\mathrm{\θ}\right)\+x{\ⅇ}^t\cos \left(\mathrm{\θ}\right)\,y_0\=b\+y_0{\ⅇ}^t\cos \left(\mathrm{\θ}\right)\+x{\ⅇ}^t\sin \left(\mathrm{\θ}\right)\right]$

$\mathrm{\μ3}≔\mathrm{simplify}\left(\mathrm{Prolong}\left(\mathrm{\mu }\,3\right)\right)$

$\mathrm{\μ3}:=\left[x\=a-y_0{\ⅇ}^t\sin \left(\mathrm{\θ}\right)\+x{\ⅇ}^t\cos \left(\mathrm{\θ}\right)\,y_0\=b\+y_0{\ⅇ}^t\cos \left(\mathrm{\θ}\right)\+x{\ⅇ}^t\sin \left(\mathrm{\θ}\right)\,y_1\=\frac{\cos \left(\mathrm{\θ}\right)y_1\+\sin \left(\mathrm{\θ}\right)}{-\sin \left(\mathrm{\θ}\right)y_1\+\cos \left(\mathrm{\θ}\right)}\,y_2\=-\frac{y_2{\ⅇ}^{-t}}{-{\cos \left(\mathrm{\θ}\right)}^2\sin \left(\mathrm{\θ}\right)y_1^3\+3{\cos \left(\mathrm{\θ}\right)}^3{y}_1^2\+3{\cos \left(\mathrm{\θ}\right)}^2\sin \left(\mathrm{\θ}\right)y_1\+\sin \left(\mathrm{\θ}\right)y_1^3-{\cos \left(\mathrm{\θ}\right)}^3-3\cos \left(\mathrm{\θ}\right)y_1^2}\,y_3\=\frac{\left(-\sin \left(\mathrm{\θ}\right)y_1{y}_3\+3y_2^2\sin \left(\mathrm{\θ}\right)\+\cos \left(\mathrm{\θ}\right)y_3\right){\ⅇ}^{-2t}}{-{\cos \left(\mathrm{\θ}\right)}^4\sin \left(\mathrm{\θ}\right)y_1^5\+5{\cos \left(\mathrm{\θ}\right)}^5{y}_1^4\+10{\cos \left(\mathrm{\θ}\right)}^4\sin \left(\mathrm{\θ}\right)y_1^3\+2{\cos \left(\mathrm{\θ}\right)}^2\sin \left(\mathrm{\θ}\right)y_1^5-10{\cos \left(\mathrm{\θ}\right)}^5{y}_1^2-10{\cos \left(\mathrm{\θ}\right)}^3{y}_1^4-5{\cos \left(\mathrm{\θ}\right)}^4\sin \left(\mathrm{\θ}\right)y_1-10{\cos \left(\mathrm{\θ}\right)}^2\sin \left(\mathrm{\θ}\right)y_1^3-\sin \left(\mathrm{\θ}\right)y_1^5\+{\cos \left(\mathrm{\θ}\right)}^5\+10{\cos \left(\mathrm{\θ}\right)}^3{y}_1^2\+5\cos \left(\mathrm{\θ}\right)y_1^4}\right]$

$\mathrm{\_EnvExplicit}≔\mathrm{true}$

$\mathrm{\_EnvExplicit}:=\mathrm{true}$

$\mathrm{\rho }≔\mathrm{RightMovingFrame}\left(\mathrm{\μ3}\,G\,\left[x=0\,y\left[0\right]=0\,y\left[1\right]=0\,y\left[2\right]=1\right]\right)$

Warning, multiple moving frames

$\mathrm{\ρ}:=\left[a\=-\frac{y_2\left(y_0{y}_1\+x\right)}{{\left(y_1^2\+1\right)}^2}\,b\=\frac{y_2\left(x{y}_1-y_0\right)}{{\left(y_1^2\+1\right)}^2}\,\mathrm{\θ}\=\arctan \left(-\frac{y_1}{\sqrt{y_1^2\+1}}\,\frac{1}{\sqrt{y_1^2\+1}}\right)\,t\=\ln \left(\frac{y_2}{{\left(y_1^2\+1\right)}^{3\/2}}\right)\right]\,\left[a\=-\frac{y_2\left(y_0{y}_1\+x\right)}{{\left(y_1^2\+1\right)}^2}\,b\=\frac{y_2\left(x{y}_1-y_0\right)}{{\left(y_1^2\+1\right)}^2}\,\mathrm{\θ}\=\arctan \left(\frac{y_1}{\sqrt{y_1^2\+1}}\,-\frac{1}{\sqrt{y_1^2\+1}}\right)\,t\=\ln \left(-\frac{y_2}{{\left(y_1^2\+1\right)}^{3\/2}}\right)\right]$

$\mathrm{\kappa }≔\mathrm{expand}\left(\mathrm{simplify}\left(\mathrm{Invariantization}\left(\mathrm{\μ3}\,\mathrm{\rho }\left[1\right]\,y\left[3\right]\right)\right)\right)$

$\mathrm{\κ}:=\frac{y_1^2{y}_3}{y_2^2}-3y_1\+\frac{y_3}{y_2^2}$