IsInvariant - Maple Help

IsInvariant

check if a distribution is invariant under Lie group action

 Calling Sequence IsInvariant(dist, L)

Parameters

 dist - a Distribution object. L - a LAVF object

Description

 • 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.

Examples

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

Building a LAVF object

 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left\{\mathrm{\eta }\left(x,y\right),\mathrm{\xi }\left(x,y\right),\mathrm{zeta}\left(z\right)\right\}\right)$
 > $X≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x,y\right)\mathrm{D}\left[y\right]+\mathrm{zeta}\left(z\right)\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${X}{≔}{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{ζ}}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $\mathrm{Sys}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)=\frac{1}{y}\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)=-\frac{x}{y}\mathrm{\xi }\left(x,y\right),\mathrm{diff}\left(\mathrm{zeta}\left(z\right),z,z\right)=0\right],\mathrm{indep}=\left[x,y,z\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta },\mathrm{zeta}\right]\right)$
 ${\mathrm{Sys}}{≔}\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}\frac{{\mathrm{\xi }}}{{y}}{,}{\mathrm{\eta }}{=}{-}\frac{{x}{}{\mathrm{\xi }}}{{y}}{,}\frac{{{ⅆ}}^{{2}}{\mathrm{ζ}}}{{ⅆ}{{z}}^{{2}}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}{,}{z}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}{,}{\mathrm{ζ}}\right]$ (2)
 > $L≔\mathrm{LAVF}\left(X,\mathrm{Sys}\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{ζ}}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[\frac{{{ⅆ}}^{{2}}{\mathrm{ζ}}}{{ⅆ}{{z}}^{{2}}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}\frac{{\mathrm{\eta }}}{{x}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{\mathrm{\xi }}{=}{-}\frac{{\mathrm{\eta }}{}{y}}{{x}}\right]\right\}$ (3)

Building some Distribution objects

 > $T\left[x\right]≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{T}}_{{x}}{≔}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (4)
 > $T\left[y\right]≔\mathrm{VectorField}\left(\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{T}}_{{y}}{≔}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (5)
 > $\mathrm{\Sigma }≔\mathrm{Distribution}\left(T\left[x\right],T\left[y\right]\right)$
 ${\mathrm{\Sigma }}{≔}\left\{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (6)
 > $\mathrm{Gamma}≔\mathrm{Distribution}\left(T\left[x\right]\right)$
 ${\mathrm{Γ}}{≔}\left\{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (7)

Now we test if the following distributions are invariant under L

 > $\mathrm{IsInvariant}\left(\mathrm{\Sigma },L\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{IsInvariant}\left(\mathrm{Gamma},L\right)$
 ${\mathrm{false}}$ (9)

Compatibility

 • The IsInvariant command was introduced in Maple 2020.