The IsInvariant method decides whether Distribution object dist is invariant under the action of the Lie transformation group generated by the vector fields from a LAVF object L. It returns the values true or false.

For this method to make sense, Distribution dist must be in involution (see IsInvolutive), and L must specify a Lie algebra (see IsLieAlgebra).  An involutive distribution $\mathrm{\Sigma }$ is invariant under a Lie group action if the foliation induced by $\mathrm{\Sigma }$ maps to itself (i.e. leaves map to leaves).

This method is associated with the Distribution object. For more detail see Overview of the Distribution object.

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

$\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left\{\mathrm{\eta }\left(x\,y\right)\,\mathrm{\xi }\left(x\,y\right)\,\mathrm{\zeta }\left(z\right)\right\}\right)$

$X≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x\,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x\,y\right)\mathrm{D}\left[y\right]+\mathrm{\zeta }\left(z\right)\mathrm{D}\left[z\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$X≔\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)+\mathrm{\zeta }\left(\frac{\ⅆ}{\ⅆz}\right)$

$\mathrm{Sys}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)=\frac{1}{y}\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)=-\frac{x}{y}\mathrm{\xi }\left(x\,y\right)\,\mathrm{diff}\left(\mathrm{\zeta }\left(z\right)\,z\,z\right)=0\right]\,\mathrm{indep}=\left[x\,y\,z\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\zeta }\right]\right)$

$\mathrm{Sys}≔\left[{\mathrm{\xi }}_x=0\,{\mathrm{\xi }}_y=\frac{\mathrm{\xi }}{y}\,\mathrm{\eta }=-\frac{x\mathrm{\xi }}{y}\,\frac{{\ⅆ}^2\mathrm{\zeta }}{\ⅆz^2}=0\right],\mathrm{indep}=\left[x\,y\,z\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\zeta }\right]$

$L≔\mathrm{LAVF}\left(X\,\mathrm{Sys}\right)$

$L≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)+\mathrm{\zeta }\left(\frac{\ⅆ}{\ⅆz}\right)\right]\&where\left\{\left[\frac{{\ⅆ}^2\mathrm{\zeta }}{\ⅆz^2}=0\,{\mathrm{\eta }}_x=\frac{\mathrm{\eta }}{x}\,{\mathrm{\eta }}_y=0\,\mathrm{\xi }=-\frac{\mathrm{\eta }y}{x}\right]\right\}$

$T\left[x\right]≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$T_x≔\frac{\ⅆ}{\ⅆx}$

$T\left[y\right]≔\mathrm{VectorField}\left(\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$T_y≔\frac{\ⅆ}{\ⅆy}$

$\mathrm{\Sigma }≔\mathrm{Distribution}\left(T\left[x\right]\,T\left[y\right]\right)$

$\mathrm{\Sigma }≔\left\{\frac{\ⅆ}{\ⅆx}\,\frac{\ⅆ}{\ⅆy}\right\}$

$\mathrm{Gamma}≔\mathrm{Distribution}\left(T\left[x\right]\right)$

$\mathrm{\Γ}≔\left\{\frac{\ⅆ}{\ⅆx}\right\}$

$\mathrm{IsInvariant}\left(\mathrm{\Sigma }\,L\right)$

$\mathrm{true}$

$\mathrm{IsInvariant}\left(\mathrm{Gamma}\,L\right)$

$\mathrm{false}$

The IsInvariant command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.