MovingFrames - Maple Help

GroupActions[MovingFrames] - a package for the Fels-Olver method of moving frames

Calling Sequences

RightMovingFrame(mu, G, K)

Invariantization(mu, rho, f)

Parameters

mu        - a free (left) action of a Lie group $G$ on a manifold $M$, given as a transformation from  to $M$

G         - a Maple name or string, the name of the initialized coordinate system for the Lie group $G$

K         - a list of equations defining a cross-section for the action mu

rho       - a right moving frame for the action mu

f         - a Maple expression, defining a function on $M$

Description

 • Let $G$ be a Lie group with multiplication * and a free (left) action of $G$ on a manifold $M$. A right moving frame is a map such that  for all and .
 • A cross-section to the action   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 has the property that if and then
 • 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:

[1] M. Fels and P. Olver, Moving Coframes I. A practical algorithm Acta Appl. Math. 51 (1998)

[2] M. Fels and P. Olver, Moving Coframes II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999) 127-208

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$
 > $\mathrm{with}\left(\mathrm{JetCalculus}\right):$$\mathrm{with}\left(\mathrm{GroupActions}\right):$
 > $\mathrm{with}\left(\mathrm{MovingFrames}\right):$
 > $\mathrm{Preferences}\left("JetNotation","JetNotation2"\right):$

Example 1.

In this example, we shall use the method of moving frames to construct the fundamental differential invariant for the special affine group (translations, rotations, scaling) in the $\mathrm{xy}$ plane.

 > $\mathrm{DGsetup}\left(\left[x\right],\left[y\right],E,3,\mathrm{verbose}\right)$
 ${\mathrm{The following coordinates have been protected:}}$
 $\left[{x}{,}{{y}}_{{0}}{,}{{y}}_{{1}}{,}{{y}}_{{2}}{,}{{y}}_{{3}}\right]$
 ${\mathrm{The following vector fields have been defined and protected:}}$
 $\left[{\mathrm{D_x}}{,}{{\mathrm{D_y}}}_{{0}}{,}{{\mathrm{D_y}}}_{{1}}{,}{{\mathrm{D_y}}}_{{2}}{,}{{\mathrm{D_y}}}_{{3}}\right]$
 ${\mathrm{The following differential 1-forms have been defined and protected:}}$
 $\left[{\mathrm{dx}}{,}{{\mathrm{dy}}}_{{0}}{,}{{\mathrm{dy}}}_{{1}}{,}{{\mathrm{dy}}}_{{2}}{,}{{\mathrm{dy}}}_{{3}}\right]$
 ${\mathrm{The following type \left[1,0\right] biforms have been defined and protected::}}$
 $\left[{\mathrm{Dx}}\right]$
 ${\mathrm{The following type \left[0,1\right] biforms \left(contact 1-forms\right) have been defined and protected::}}$
 $\left[{{\mathrm{Cy}}}_{{0}}{,}{{\mathrm{Cy}}}_{{1}}{,}{{\mathrm{Cy}}}_{{2}}{,}{{\mathrm{Cy}}}_{{3}}\right]$
 ${\mathrm{frame name: E}}$ (2.1)

We start with the infinitesimal generators for the action of the special affine group.

 E > $\mathrm{Gamma}≔\mathrm{evalDG}\left(\left[\mathrm{D_x},\mathrm{D_y}\left[0\right],x\mathrm{D_y}\left[0\right]-y\left[0\right]\mathrm{D_x},y\left[0\right]\mathrm{D_y}\left[0\right]+x\mathrm{D_x}\right]\right)$
 ${\mathrm{Γ}}{:=}\left[{\mathrm{D_x}}{,}{{\mathrm{D_y}}}_{{0}}{,}{-}{\mathrm{D_x}}{}{{y}}_{{0}}{+}{x}{}{{\mathrm{D_y}}}_{{0}}{,}{\mathrm{D_x}}{}{x}{+}{{\mathrm{D_y}}}_{{0}}{}{{y}}_{{0}}\right]$ (2.2)

This is a solvable group so we can use the Action command in the GroupAction package to find the action of the special affine group.

 E > $\mathrm{DGsetup}\left(\left[a,b,\mathrm{\theta },t\right],G\right)$
 ${\mathrm{frame name: G}}$ (2.3)
 G > $\mathrm{\mu }≔\mathrm{Action}\left(\mathrm{Gamma},G\right)$
 ${\mathrm{μ}}{:=}\left[{x}{=}{a}{-}{{y}}_{{0}}{}{{ⅇ}}^{{t}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){+}{x}{}{{ⅇ}}^{{t}}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){,}{{y}}_{{0}}{=}{b}{+}{{y}}_{{0}}{}{{ⅇ}}^{{t}}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){+}{x}{}{{ⅇ}}^{{t}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)\right]$ (2.4)

We use the program Prolong in the JetCalculus package to prolong this action to the 3-jets of E.

 E > $\mathrm{μ3}≔\mathrm{simplify}\left(\mathrm{Prolong}\left(\mathrm{\mu },3\right)\right)$
 ${\mathrm{μ3}}{:=}\left[{x}{=}{a}{-}{{y}}_{{0}}{}{{ⅇ}}^{{t}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){+}{x}{}{{ⅇ}}^{{t}}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){,}{{y}}_{{0}}{=}{b}{+}{{y}}_{{0}}{}{{ⅇ}}^{{t}}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){+}{x}{}{{ⅇ}}^{{t}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){,}{{y}}_{{1}}{=}\frac{{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}{+}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}{{-}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}{+}{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}{,}{{y}}_{{2}}{=}{-}\frac{{{y}}_{{2}}{}{{ⅇ}}^{{-}{t}}}{{-}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}^{{3}}{+}{3}{}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{3}}{}{{y}}_{{1}}^{{2}}{+}{3}{}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}{+}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}^{{3}}{-}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{3}}{-}{3}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}^{{2}}}{,}{{y}}_{{3}}{=}\frac{\left({-}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}{}{{y}}_{{3}}{+}{3}{}{{y}}_{{2}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){+}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{{y}}_{{3}}\right){}{{ⅇ}}^{{-}{2}{}{t}}}{{-}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{4}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}^{{5}}{+}{5}{}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{5}}{}{{y}}_{{1}}^{{4}}{+}{10}{}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{4}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}^{{3}}{+}{2}{}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}^{{5}}{-}{10}{}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{5}}{}{{y}}_{{1}}^{{2}}{-}{10}{}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{3}}{}{{y}}_{{1}}^{{4}}{-}{5}{}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{4}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}{-}{10}{}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}^{{3}}{-}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}^{{5}}{+}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{5}}{+}{10}{}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{3}}{}{{y}}_{{1}}^{{2}}{+}{5}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{{y}}_{{1}}^{{4}}}\right]$ (2.5)
 E > $\mathrm{_EnvExplicit}≔\mathrm{true}$
 ${\mathrm{_EnvExplicit}}{:=}{\mathrm{true}}$ (2.6)

We calculate a moving frame for this prolonged action.

 E > $\mathrm{\rho }≔\mathrm{RightMovingFrame}\left(\mathrm{μ3},G,\left[x=0,y\left[0\right]=0,y\left[1\right]=0,y\left[2\right]=1\right]\right)$
 ${\mathrm{ρ}}{:=}\left[{a}{=}{-}\frac{{{y}}_{{2}}{}\left({{y}}_{{0}}{}{{y}}_{{1}}{+}{x}\right)}{{\left({{y}}_{{1}}^{{2}}{+}{1}\right)}^{{2}}}{,}{b}{=}\frac{{{y}}_{{2}}{}\left({x}{}{{y}}_{{1}}{-}{{y}}_{{0}}\right)}{{\left({{y}}_{{1}}^{{2}}{+}{1}\right)}^{{2}}}{,}{\mathrm{θ}}{=}{\mathrm{arctan}}{}\left({-}\frac{{{y}}_{{1}}}{\sqrt{{{y}}_{{1}}^{{2}}{+}{1}}}{,}\frac{{1}}{\sqrt{{{y}}_{{1}}^{{2}}{+}{1}}}\right){,}{t}{=}{\mathrm{ln}}{}\left(\frac{{{y}}_{{2}}}{{\left({{y}}_{{1}}^{{2}}{+}{1}\right)}^{{3}{/}{2}}}\right)\right]{,}\left[{a}{=}{-}\frac{{{y}}_{{2}}{}\left({{y}}_{{0}}{}{{y}}_{{1}}{+}{x}\right)}{{\left({{y}}_{{1}}^{{2}}{+}{1}\right)}^{{2}}}{,}{b}{=}\frac{{{y}}_{{2}}{}\left({x}{}{{y}}_{{1}}{-}{{y}}_{{0}}\right)}{{\left({{y}}_{{1}}^{{2}}{+}{1}\right)}^{{2}}}{,}{\mathrm{θ}}{=}{\mathrm{arctan}}{}\left(\frac{{{y}}_{{1}}}{\sqrt{{{y}}_{{1}}^{{2}}{+}{1}}}{,}{-}\frac{{1}}{\sqrt{{{y}}_{{1}}^{{2}}{+}{1}}}\right){,}{t}{=}{\mathrm{ln}}{}\left({-}\frac{{{y}}_{{2}}}{{\left({{y}}_{{1}}^{{2}}{+}{1}\right)}^{{3}{/}{2}}}\right)\right]$ (2.7)

We use this moving frame to find the fundamental differential invariant on the 3-jet.

 E > $\mathrm{\kappa }≔\mathrm{expand}\left(\mathrm{simplify}\left(\mathrm{Invariantization}\left(\mathrm{μ3},\mathrm{\rho }\left[1\right],y\left[3\right]\right)\right)\right)$
 ${\mathrm{κ}}{:=}\frac{{{y}}_{{1}}^{{2}}{}{{y}}_{{3}}}{{{y}}_{{2}}^{{2}}}{-}{3}{}{{y}}_{{1}}{+}\frac{{{y}}_{{3}}}{{{y}}_{{2}}^{{2}}}$ (2.8)