Let L be a LAVF object and is Lie algebra (see IsLieAlgebra). Then OrbitDistribution method returns the orbit distribution of L.

The returned orbit distribution of obj is a Distribution object. A Distribution object has access to various methods, see Overview of the Distribution object for more detail.

The orbit distribution is the infinitesimal version of the group orbit.

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

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

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

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

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

$R_x≔-z\left(\frac{\ⅆ}{\ⅆy}\right)+y\left(\frac{\ⅆ}{\ⅆz}\right)$

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

$R_y≔z\left(\frac{\ⅆ}{\ⅆx}\right)-x\left(\frac{\ⅆ}{\ⅆz}\right)$

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

$R_z≔-y\left(\frac{\ⅆ}{\ⅆx}\right)+x\left(\frac{\ⅆ}{\ⅆy}\right)$

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

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

$L≔\mathrm{EliminationLAVF}\left(V\,\left[R\left[x\right]\,R\left[y\right]\,R\left[z\right]\right]\right)$

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

$\mathrm{OD}≔\mathrm{OrbitDistribution}\left(L\right)$

$\mathrm{OD}≔\left\{-\frac{z\left(\frac{\ⅆ}{\ⅆx}\right)}{x}+\frac{\ⅆ}{\ⅆz}\,-\frac{y\left(\frac{\ⅆ}{\ⅆx}\right)}{x}+\frac{\ⅆ}{\ⅆy}\right\}$

$\mathrm{Dimension}\left(\mathrm{OD}\right)$

$2$

$\mathrm{Integrals}\left(\mathrm{OD}\right)$

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

$\mathrm{GetAnnihilator}\left(\mathrm{OD}\right)$

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

The OrbitDistribution command was introduced in Maple 2020.

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