The OrbitDimension method calculates the dimension of the orbit distribution of a LAVF object.

The InvariantCount method calculates the count of scalar invariants of a LAVF object. By default (type="all"), all invariants are counted.

If type="essential" is specified, then only essential invariants are counted. An invariant is essential, roughly speaking, if the group action cannot be expressed without it.

Let L be a LAVF object. Then IsTransitive(L) returns true if and only if the action of L is transitive, that is, InvariantCount(L) = 0.

Let L be a LAVF object and let OD be the orbit distribution of L. Then OrbitDimension(L) equals to Dimension(OD) and InvariantCount(L) equals to Codimension(OD). See Overview of the Distribution object for more detail.

These methods are associated with the LAVF object. For more detail, see Overview of the LAVF object.

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

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

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

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

$R_x≔-z\frac{\partial }{\partial y}+y\frac{\partial }{\partial z}$

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

$R_y≔z\frac{\partial }{\partial x}-x\frac{\partial }{\partial z}$

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

$R_z≔-y\frac{\partial }{\partial x}+x\frac{\partial }{\partial y}$

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

$V≔\mathrm{\xi }\frac{\partial }{\partial x}+\mathrm{\eta }\frac{\partial }{\partial y}+\mathrm{\ζ}\frac{\partial }{\partial z}$

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

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

$\mathrm{OrbitDimension}\left(L\right)$

$2$

$\mathrm{InvariantCount}\left(L\right)$

$1$

$\mathrm{IsTransitive}\left(L\right)$

$\mathrm{false}$

$\mathrm{Invariants}\left(L\right)$

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

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

$Y≔\left(ay+bz\right)\frac{\partial }{\partial x}$

$\mathrm{L2}≔\mathrm{EliminationLAVF}\left(V\,\left[Y\right]\,\mathrm{coefficients}=\left[a\,b\right]\right)$

$\mathrm{L2}≔\left[\mathrm{\xi }\frac{\partial }{\partial x}+\mathrm{\eta }\frac{\partial }{\partial y}+\mathrm{\ζ}\frac{\partial }{\partial z}\right]\&where\left\{\left[{\mathrm{\xi }}_{z,z}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\xi }}_y=\frac{-\left({\mathrm{\xi }}_z\right)z+\mathrm{\xi }}{y}\,\mathrm{\eta }=0\,\mathrm{\ζ}=0\right]\right\}$

$\mathrm{OrbitDimension}\left(\mathrm{L2}\right)$

$1$

$\mathrm{IsTransitive}\left(\mathrm{L2}\right)$

$\mathrm{false}$

$\mathrm{InvariantCount}\left(\mathrm{L2}\right)$

$2$

$\mathrm{InvariantCount}\left(\mathrm{L2}\,''essential''\right)$

$1$

$\mathrm{InvariantCount}\left(\mathrm{L2}\,''inessential''\right)$

$1$

$\mathrm{Invariants}\left(\mathrm{L2}\right)$

$\left[y\,z\right]$

$\mathrm{Invariants}\left(\mathrm{L2}\,''essential''\right)$

$\left[\frac{z}{y}\right]$

The OrbitDimension, InvariantCount and IsTransitive commands were introduced in Maple 2020.

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