The Invariants method attempts to find the invariants of a LAVF object via integration. If successful, it returns the invariants as a list of expressions.

Let L be a LAVF object and OD be the orbit distribution of L. Then Invariants(L) is equivalent to Integrals(OD). For more detail of the Distribution's methods, see Overview of the Distribution object.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The Invariants command was introduced in Maple 2020.

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