Let L1, L2 be LAVF objects on the same space. Then IsInvariant(L1,L2) checks if L1 is invariant under action of L2 (i.e. if $\left[\mathrm{L1}\,\mathrm{L2}\right]\subseteq \mathrm{L1}$).

The call IsInvariant(L1, L2) is equivalent to the call AreCommuting(L1,L2,L1). See AreCommuting for more detail.

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

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

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

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

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

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

$\mathrm{T2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{T2}≔\left[{\mathrm{\xi }}_x=0\,{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_x=0\,{\mathrm{\eta }}_y=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{E2}≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{LE2}≔\mathrm{LAVF}\left(V\,\mathrm{E2}\right)$

$\mathrm{LE2}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{LT2}≔\mathrm{LAVF}\left(V\,\mathrm{T2}\right)$

$\mathrm{LT2}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=0\,{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{IsLieAlgebra}\left(\mathrm{LE2}\right)$

$\mathrm{true}$

$\mathrm{IsLieAlgebra}\left(\mathrm{LT2}\right)$

$\mathrm{true}$

$\mathrm{IsInvariant}\left(\mathrm{LT2}\,\mathrm{LE2}\right)$

$\mathrm{true}$

The IsInvariant command was introduced in Maple 2020.

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