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.

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

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

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

$X≔\mathrm{VectorField}\left(\mathrm{\ξ}\left(x\,y\right){\mathrm{D}}_x\+\mathrm{\η}\left(x\,y\right){\mathrm{D}}_y\+\mathrm{zeta}\left(z\right){\mathrm{D}}_z\,\mathrm{space}\=\left[x\,y\,z\right]\right)$

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

$\mathrm{Sys}≔\mathrm{LHPDE}\left(\left[\frac{\partial }{\partial x}\mathrm{\ξ}\left(x\,y\right)\=0\,\frac{\partial }{\partial y}\mathrm{\ξ}\left(x\,y\right)\=\frac{1\mathrm{\ξ}\left(x\,y\right)}{y}\,\mathrm{\η}\left(x\,y\right)\=-\frac{x\mathrm{\ξ}\left(x\,y\right)}{y}\,\frac{{\ⅆ}^2}{\ⅆz^2}\mathrm{zeta}\left(z\right)\=0\right]\,\mathrm{indep}\=\left[x\,y\,z\right]\,\mathrm{dep}\=\left[\mathrm{\ξ}\,\mathrm{\η}\,\mathrm{zeta}\right]\right)$

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

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

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

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

$T_x≔\frac{\ⅆ}{\ⅆx}$

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

$T_y≔\frac{\ⅆ}{\ⅆy}$

$\mathrm{\Σ}≔\mathrm{Distribution}\left(T_x\,T_y\right)$

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

$\mathrm{Gamma}≔\mathrm{Distribution}\left(T_x\right)$

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

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

$\mathrm{true}$

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

$\mathrm{false}$

The IsInvariant command was introduced in Maple 2020.

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