Let $\mathrm{\Γ} \= \left\{X_1\, X_2\, ..\. \, X_r\right\}$be a $r$-dimensional Lie algebra of vector fields on a manifold $M$. A vector field, differential form, tensor or connection $S$ is said to be $\mathrm{\Γ}$-invariant if the Lie derivative${\mathrm{\ℒ}}_{X_i}S \= 0$(*) for$i \= 1\,2\, ..\. \, r$.

The procedure InvariantGeometricObjectFields(Gamma, T) calculates the $\mathrm{\Γ}-$invariant geometric object fields which are in the span (over the functions on $M\)$of the geometric object fields given by the second argument T.

The procedure creates the general linear combination $S$of the tensors in T (with coefficients which are functions of the coordinates on $M\)$and then generates the system of first order PDE for the coefficients arising from the invariance conditions (*).The command pdsolve is used to solve these PDE.

If T = [1], then the $\mathrm{\Γ}$-invariant functions on $M$are computed.

If connection = "yes", then $\mathrm{\Γ}$invariant connections are computed.

With output = "list", the program returns a basis for the invariant tensors, over the ring of invariant functions. This option is not available when connection = "yes".

With output = "pde", the pde system defined by the equations (*) is returned.

The exact form for the geometric object fields can be specified by ansatz = t. With this option, the unknown functions in t must be explicitly listed with the unknowns option.

If P = {a1, a2, ... } is a set of parameters appearing in Gamma, then the optional argument parameters = P will invoke the case splitting capabilities of pdsolve. Exceptional parameter values will be determined and a sequence of lists of invariant geometry object fields, one list for each set of parameter values, will be returned.

Other optional arguments for pdsolve may be passed through the command InvariantGeometricObjectFields.

If pdsolve is unable to explicitly solve the pde system defined by LieDerivative(X, t) = 0, then NULL is returned.  

The command InvariantGeometricObjectFields is part of the DifferentialGeometry:-GroupActions package.  It can be used in the form InvariantGeometricObjectFields(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-InvariantGeometricObjectFields(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{GroupActions}\right)\:$$\mathrm{with}\left(\mathrm{JetCalculus}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$$\mathrm{DGsetup}\left(\left[x\,y\right]\,N\right)\:$$\mathrm{DGsetup}\left(\left[x\right]\,\left[u\right]\,J\,2\right)\:$

$\mathrm{ChangeFrame}\left(M\right)$

$J$

$\mathrm{\Γ1}≔\mathrm{evalDG}\left(\left[x\mathrm{D\_y}-y\mathrm{D\_x}\,x\mathrm{D\_z}-z\mathrm{D\_x}\,y\mathrm{D\_z}-z\mathrm{D\_y}\right]\right)$

$\mathrm{\Γ1}:=\left[-\mathrm{D\_x}y\+\mathrm{D\_y}x\,-\mathrm{D\_x}z\+\mathrm{D\_z}x\,-\mathrm{D\_y}z\+\mathrm{D\_z}y\right]$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{\Γ1}\,\left[1\right]\right)$

$\mathrm{\_F1}\left(x^2\+y^2\+z^2\right)$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{\Γ1}\,\left[1\right]\,\mathrm{output}=''list''\right)$

$\left[x^2\+y^2\+z^2\right]$

$T≔\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\right]$

$T:=\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\right]$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{\Γ1}\,T\,\mathrm{output}=''list''\right)$

$\left[\frac{\sqrt{-z^2}x\mathrm{dx}}{z}\+\frac{\sqrt{-z^2}y\mathrm{dy}}{z}\+\sqrt{-z^2}\mathrm{dz}\right]$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{\&Gamma;1}\&comma;T\&comma;\mathrm{output}=''list''\right)\mathbf{assuming}0<z$

$\left[\frac{x\mathrm{dx}}{z}\&plus;\frac{y\mathrm{dy}}{z}\&plus;\mathrm{dz}\right]$

$T≔\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{dx}\&comma;\mathrm{dy}\&comma;\mathrm{dz}\right]\&comma;2\right)$

$T:=\left[\mathrm{dx}\mathrm{dx}\&comma;\frac{1}{2}\mathrm{dx}\mathrm{dy}\&plus;\frac{1}{2}\mathrm{dy}\mathrm{dx}\&comma;\frac{1}{2}\mathrm{dx}\mathrm{dz}\&plus;\frac{1}{2}\mathrm{dz}\mathrm{dx}\&comma;\mathrm{dy}\mathrm{dy}\&comma;\frac{1}{2}\mathrm{dy}\mathrm{dz}\&plus;\frac{1}{2}\mathrm{dz}\mathrm{dy}\&comma;\mathrm{dz}\mathrm{dz}\right]$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{\&Gamma;1}\&comma;T\&comma;\mathrm{output}=''list''\right)$

$\left[\mathrm{dx}\mathrm{dx}\&plus;\mathrm{dy}\mathrm{dy}\&plus;\mathrm{dz}\mathrm{dz}\&comma;xz\mathrm{dx}\mathrm{dz}\&plus;x^2\mathrm{dx}\mathrm{dx}\&plus;xy\mathrm{dx}\mathrm{dy}\&plus;xy\mathrm{dy}\mathrm{dx}\&plus;y^2\mathrm{dy}\mathrm{dy}\&plus;zy\mathrm{dy}\mathrm{dz}\&plus;xz\mathrm{dz}\mathrm{dx}\&plus;zy\mathrm{dz}\mathrm{dy}\&plus;z^2\mathrm{dz}\mathrm{dz}\right]$

$T≔\mathrm{GenerateTensors}\left(\left[\left[\mathrm{dx}\&comma;\mathrm{dy}\&comma;\mathrm{dz}\right]\&comma;\left[\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]\right]\right)$

$T:=\left[\mathrm{dx}\mathrm{D\_x}\&comma;\mathrm{dx}\mathrm{D\_y}\&comma;\mathrm{dx}\mathrm{D\_z}\&comma;\mathrm{dy}\mathrm{D\_x}\&comma;\mathrm{dy}\mathrm{D\_y}\&comma;\mathrm{dy}\mathrm{D\_z}\&comma;\mathrm{dz}\mathrm{D\_x}\&comma;\mathrm{dz}\mathrm{D\_y}\&comma;\mathrm{dz}\mathrm{D\_z}\right]$

$\mathrm{\_EnvExplicit}≔\mathrm{true}$

$\mathrm{\_EnvExplicit}:=\mathrm{true}$

$\mathrm{Inv}≔\mathrm{InvariantGeometricObjectFields}\left(\mathrm{\&Gamma;1}\&comma;T\&comma;\mathrm{output}=''list''\right)\mathbf{assuming}0<z$

$\mathrm{Inv}:=\left[\mathrm{D\_x}\mathrm{dx}\&plus;\mathrm{D\_y}\mathrm{dy}\&plus;\mathrm{D\_z}\mathrm{dz}\&comma;-\mathrm{D\_x}\mathrm{dx}z-\mathrm{D\_y}\mathrm{dy}z\&plus;\mathrm{D\_z}\mathrm{dx}x\&plus;\mathrm{D\_z}\mathrm{dy}y\&comma;-\mathrm{D\_x}\mathrm{dy}z\&plus;\mathrm{D\_y}\mathrm{dx}z-\mathrm{D\_z}\mathrm{dx}y\&plus;\mathrm{D\_z}\mathrm{dy}x\right]$

$\mathrm{ChangeFrame}\left(M\right)\&colon;$

$\mathrm{\&Gamma;2}≔\left[\mathrm{D\_x}\&comma;\exp \left(-y\right)\mathrm{D\_z}\&comma;x\mathrm{D\_x}+\mathrm{D\_y}\right]$

$\mathrm{\&Gamma;2}:=\left[\mathrm{D\_x}\&comma;{\&ExponentialE;}^{-y}\mathrm{D\_z}\&comma;\mathrm{D\_x}x\&plus;\mathrm{D\_y}\right]$

$T≔\left[\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]$

$T:=\left[\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]$

$Z≔\mathrm{InvariantGeometricObjectFields}\left(\mathrm{\&Gamma;2}\&comma;T\&comma;\mathrm{output}=''list''\right)$

$Z:=\left[\mathrm{D\_z}z-\mathrm{D\_y}\&comma;\mathrm{D\_z}\&comma;{\&ExponentialE;}^y\mathrm{D\_x}\right]$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{\&Gamma;2}\&comma;T\&comma;\mathrm{unknowns}=\left[A\left(x\&comma;y\&comma;z\right)\&comma;B\left(x\&comma;y\&comma;z\right)\&comma;C\left(x\&comma;y\&comma;z\right)\right]\&comma;\mathrm{output}=''pde''\right)$

$\left[\frac{\partial }{\partial x}A\left(x\&comma;y\&comma;z\right)\&comma;\frac{\partial }{\partial x}B\left(x\&comma;y\&comma;z\right)\&comma;\frac{\partial }{\partial x}C\left(x\&comma;y\&comma;z\right)\&comma;{\&ExponentialE;}^{-y}\left(\frac{\partial }{\partial z}A\left(x\&comma;y\&comma;z\right)\right)\&comma;{\&ExponentialE;}^{-y}\left(\frac{\partial }{\partial z}B\left(x\&comma;y\&comma;z\right)\right)\&comma;{\&ExponentialE;}^{-y}\left(B\left(x\&comma;y\&comma;z\right)\&plus;\frac{\partial }{\partial z}C\left(x\&comma;y\&comma;z\right)\right)\&comma;-A\left(x\&comma;y\&comma;z\right)\&plus;x\left(\frac{\partial }{\partial x}A\left(x\&comma;y\&comma;z\right)\right)\&plus;\frac{\partial }{\partial y}A\left(x\&comma;y\&comma;z\right)\&comma;x\left(\frac{\partial }{\partial x}B\left(x\&comma;y\&comma;z\right)\right)\&plus;\frac{\partial }{\partial y}B\left(x\&comma;y\&comma;z\right)\&comma;x\left(\frac{\partial }{\partial x}C\left(x\&comma;y\&comma;z\right)\right)\&plus;\frac{\partial }{\partial y}C\left(x\&comma;y\&comma;z\right)\&comma;0\right]\&comma;\left[A\left(x\&comma;y\&comma;z\right)\&comma;B\left(x\&comma;y\&comma;z\right)\&comma;C\left(x\&comma;y\&comma;z\right)\right]$

$Z≔\mathrm{evalDG}\left(a\left(x\right)\mathrm{D\_x}+b\left(x\&comma;y\right)\mathrm{D\_y}+c\left(x\&comma;y\&comma;z\right)\mathrm{D\_z}\right)$

$Z:=a\left(x\right)\mathrm{D\_x}\&plus;b\left(x\&comma;y\right)\mathrm{D\_y}\&plus;c\left(x\&comma;y\&comma;z\right)\mathrm{D\_z}$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{\&Gamma;2}\&comma;Z\&comma;\mathrm{unknowns}=\left[a\left(x\right)\&comma;b\left(x\&comma;y\right)\&comma;c\left(x\&comma;y\&comma;z\right)\right]\&comma;\mathrm{output}=''list''\right)$

$\left[\mathrm{D\_z}z-\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]$

$\mathrm{ChangeFrame}\left(J\right)\&colon;$

$\mathrm{\&Gamma;3}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}\&comma;\mathrm{D\_u}\left[\right]\&comma;u\left[\right]\mathrm{D\_x}-x\mathrm{D\_u}\left[\right]\right]\right)$

$\mathrm{\&Gamma;3}:=\left[\mathrm{D\_x}\&comma;{\mathrm{D\_u}}_{\&lsqb;\&rsqb;}\&comma;\mathrm{D\_x}{u}_{\&lsqb;\&rsqb;}-x{\mathrm{D\_u}}_{\&lsqb;\&rsqb;}\right]$

$\mathrm{Gamma3a}≔\mathrm{map}\left(\mathrm{Prolong}\&comma;\mathrm{\&Gamma;3}\&comma;2\right)$

$\mathrm{Gamma3a}:=\left[\mathrm{D\_x}\&comma;{\mathrm{D\_u}}_{\&lsqb;\&rsqb;}\&comma;u_{\&lsqb;\&rsqb;}\mathrm{D\_x}-x{\mathrm{D\_u}}_{\&lsqb;\&rsqb;}-\left(u_1^2\&plus;1\right){\mathrm{D\_u}}_1-3u_1{u}_{1\&comma;1}{\mathrm{D\_u}}_{1\&comma;1}\right]$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Gamma3a}\&comma;\left[1\right]\&comma;\mathrm{coefficientvariables}=\left[x\&comma;u\left[\right]\&comma;u\left[1\right]\&comma;u\left[1,1\right]\right]\right)$

$\mathrm{\_F1}\left(\frac{u_{1\&comma;1}}{{\left(u_1^2\&plus;1\right)}^{3\&sol;2}}\right)$

$\mathrm{Gamma2b}≔\mathrm{map}\left(\mathrm{Prolong}\&comma;\mathrm{Gamma3a}\&comma;3\right)$

$\mathrm{Gamma2b}:=\left[\mathrm{D\_x}\&comma;{\mathrm{D\_u}}_{\&lsqb;\&rsqb;}\&comma;u_{\&lsqb;\&rsqb;}\mathrm{D\_x}-x{\mathrm{D\_u}}_{\&lsqb;\&rsqb;}-\left(u_1^2\&plus;1\right){\mathrm{D\_u}}_1-3u_1{u}_{1\&comma;1}{\mathrm{D\_u}}_{1\&comma;1}-\left(4u_1{u}_{1\&comma;1\&comma;1}\&plus;3u_{1\&comma;1}^2\right){\mathrm{D\_u}}_{1\&comma;1\&comma;1}\right]$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Gamma2b}\&comma;\left[1\right]\right)$

$\mathrm{\_F1}\left(\frac{u_{1\&comma;1}}{{\left(u_1^2\&plus;1\right)}^{3\&sol;2}}\&comma;\frac{u_1^2{u}_{1\&comma;1\&comma;1}-3u_1{u}_{1\&comma;1}^2\&plus;u_{1\&comma;1\&comma;1}}{{\left(u_1^2\&plus;1\right)}^3}\right)$

$S≔\mathrm{Dx}$

$S:=\mathrm{Dx}$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Gamma2b}\&comma;\left[S\right]\&comma;\mathrm{coefficientvariables}=\left[x\&comma;u\left[\right]\&comma;u\left[1\right]\right]\right)$

$\mathrm{\_C1}\sqrt{u_1^2\&plus;1}\mathrm{Dx}$

$S≔\mathrm{Dx}\&wedge\mathrm{Cu}\left[\right]$

$S:=\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\&lsqb;\&rsqb;}$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Gamma2b}\&comma;\left[S\right]\&comma;\mathrm{coefficientvariables}=\left[x\&comma;u\left[\right]\&comma;u\left[1\right]\&comma;u\left[1,1\right]\right]\right)$

$\mathrm{\_F1}\left(\frac{u_{1\&comma;1}}{{\left(u_1^2\&plus;1\right)}^{3\&sol;2}}\right)\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\&lsqb;\&rsqb;}$

$\mathrm{ChangeFrame}\left(N\right)$

$J$

$\mathrm{\&Gamma;4}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}\&comma;x\mathrm{D\_x}+\mathrm{\alpha }y\mathrm{D\_y}\right]\right)$

$\mathrm{\&Gamma;4}:=\left[\mathrm{D\_x}\&comma;\mathrm{D\_y}y\mathrm{\&alpha;}\&plus;\mathrm{D\_x}x\right]$

$F≔\left[\mathrm{dx}\&comma;\mathrm{dy}\right]$

$F:=\left[\mathrm{dx}\&comma;\mathrm{dy}\right]$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{\&Gamma;4}\&comma;F\&comma;\mathrm{parameters}=\left\{\mathrm{\alpha }\right\}\right)$

$\mathrm{\_F1}\left(y\right)\mathrm{dy}\&comma;\mathrm{\_C2}{y}^{-\frac{1}{\mathrm{\&alpha;}}}\mathrm{dx}\&plus;\frac{\mathrm{\_C1}\mathrm{dy}}{y}\&comma;\left[\left\{\mathrm{\&alpha;}\&equals;0\right\}\&comma;\left\{\mathrm{\&alpha;}\&equals;\mathrm{\&alpha;}\right\}\right]$

$\mathrm{InvariantGeometricObjectFields}\left(\mathrm{\&Gamma;4}\&comma;F\&comma;\mathrm{output}=''list''\&comma;\mathrm{parameters}=\left\{\mathrm{\alpha }\right\}\right)$

$\left[\mathrm{dy}\right]\&comma;\left[\frac{\mathrm{dy}}{y}\&comma;y^{-\frac{1}{\mathrm{\&alpha;}}}\mathrm{dx}\right]\&comma;\left[\left\{\mathrm{\&alpha;}\&equals;0\right\}\&comma;\left\{\mathrm{\&alpha;}\&equals;\mathrm{\&alpha;}\right\}\right]$

$\mathrm{LD}≔\mathrm{Library}:-\mathrm{Retrieve}\left(''Winternitz''\&comma;1\&comma;\left[4\&comma;10\right]\&comma;\mathrm{alg1}\right)$

$\mathrm{LD}:=\left[\left[\mathrm{e2}\&comma;\mathrm{e3}\right]\&equals;\mathrm{e1}\&comma;\left[\mathrm{e2}\&comma;\mathrm{e4}\right]\&equals;-\mathrm{e3}\&comma;\left[\mathrm{e3}\&comma;\mathrm{e4}\right]\&equals;\mathrm{e2}\right]$

$\mathrm{DGsetup}\left(\mathrm{LD}\right)$

$\mathrm{Lie\; algebra:\; alg1}$

$S≔\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{\&theta;1}\&comma;\mathrm{\&theta;2}\&comma;\mathrm{\&theta;3}\&comma;\mathrm{\&theta;4}\right]\&comma;2\right)$

$S:=\left[\mathrm{\&theta;1}\mathrm{\&theta;1}\&comma;\frac{1}{2}\mathrm{\&theta;1}\mathrm{\&theta;2}\&plus;\frac{1}{2}\mathrm{\&theta;2}\mathrm{\&theta;1}\&comma;\frac{1}{2}\mathrm{\&theta;1}\mathrm{\&theta;3}\&plus;\frac{1}{2}\mathrm{\&theta;3}\mathrm{\&theta;1}\&comma;\frac{1}{2}\mathrm{\&theta;1}\mathrm{\&theta;4}\&plus;\frac{1}{2}\mathrm{\&theta;4}\mathrm{\&theta;1}\&comma;\mathrm{\&theta;2}\mathrm{\&theta;2}\&comma;\frac{1}{2}\mathrm{\&theta;2}\mathrm{\&theta;3}\&plus;\frac{1}{2}\mathrm{\&theta;3}\mathrm{\&theta;2}\&comma;\frac{1}{2}\mathrm{\&theta;2}\mathrm{\&theta;4}\&plus;\frac{1}{2}\mathrm{\&theta;4}\mathrm{\&theta;2}\&comma;\mathrm{\&theta;3}\mathrm{\&theta;3}\&comma;\frac{1}{2}\mathrm{\&theta;3}\mathrm{\&theta;4}\&plus;\frac{1}{2}\mathrm{\&theta;4}\mathrm{\&theta;3}\&comma;\mathrm{\&theta;4}\mathrm{\&theta;4}\right]$

$\mathrm{InvariantGeometricObjectFields}\left(\left[\mathrm{e1}\&comma;\mathrm{e2}\right]\&comma;S\&comma;\mathrm{output}=''list''\right)$

$\left[\frac{1}{2}\mathrm{\&theta;1}\mathrm{\&theta;4}\&plus;\frac{1}{2}\mathrm{\&theta;3}\mathrm{\&theta;3}\&plus;\frac{1}{2}\mathrm{\&theta;4}\mathrm{\&theta;1}\&comma;\mathrm{\&theta;2}\mathrm{\&theta;2}\&comma;\frac{1}{2}\mathrm{\&theta;2}\mathrm{\&theta;4}\&plus;\frac{1}{2}\mathrm{\&theta;4}\mathrm{\&theta;2}\&comma;\mathrm{\&theta;4}\mathrm{\&theta;4}\right]$