EliminationLAVF - Maple Help

LieAlgebrasOfVectorFields

 EliminationLAVF
 construct a LAVF object from the symmetry vector fields

 Calling Sequence EliminationLAVF(V, vfList)

Parameters

 V - a VectorField object vfList - a list of VectorField objects representing solution vector fields

Description

 • The command EliminationLAVF(...) constructs a LAVF object based on the solution vector fields vf. A valid LAVF object then has access to at least 60 methods methods which allow it to be manipulated and its contents queried. For more detail, see Overview of the LAVF object.
 • It first finds the Lie determining system of all linear combination of vector fields in vfList, and expresses the system DQ in terms of the variable names from V. It then constructs a LAVF object using V and DQ.
 • The command ultimately uses LieAlgebrasOfVectorFields[EliminationSystem] to construct the determining system as a LHPDE object.
 • This command is part of the LieAlgebrasOfVectorFields package. For more detail, see Overview of the LieAlgebrasOfVectorFields package.
 • This command can be used in the form EliminationLAVF(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used in the form LieAlgebrasOfVectorFields:-EliminationLAVF(...).

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left\{\mathrm{\eta },\mathrm{\xi },\mathrm{zeta}\right\}\left(x,y,z\right)\right):$

We consider three 3-D rotation vector fields that generating standard SO(3) action on R3(x,y,z). First we construct these vector fields.

 > $\mathrm{R1}≔\mathrm{VectorField}\left(x\mathrm{D}\left[y\right]-y\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${\mathrm{R1}}{≔}{-}{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $\mathrm{R2}≔\mathrm{VectorField}\left(x\mathrm{D}\left[z\right]-z\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${\mathrm{R2}}{≔}{-}{z}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)
 > $\mathrm{R3}≔\mathrm{VectorField}\left(y\mathrm{D}\left[z\right]-z\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${\mathrm{R3}}{≔}{-}{z}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{y}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (3)
 > $X≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,y,z\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x,y,z\right)\mathrm{D}\left[y\right]+\mathrm{zeta}\left(x,y,z\right)\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${X}{≔}{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{ζ}}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (4)

Then we use the command to construct a LAVF object as the vector fields system for SO(3)

 > $\mathrm{SO3}≔\mathrm{EliminationLAVF}\left(X,\left[\mathrm{R1},\mathrm{R2},\mathrm{R3}\right]\right)$
 ${\mathrm{SO3}}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{ζ}}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{=}\frac{{-}{\mathrm{\eta }}{}{y}{-}{\mathrm{ζ}}{}{z}}{{x}}{,}{{\mathrm{\eta }}}_{{x}}{=}\frac{\left({{\mathrm{ζ}}}_{{y}}\right){}{z}{+}{\mathrm{\eta }}}{{x}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{z}}{=}{-}{{\mathrm{ζ}}}_{{y}}{,}{{\mathrm{ζ}}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{ζ}}}_{{x}}{=}\frac{{-}\left({{\mathrm{ζ}}}_{{y}}\right){}{y}{+}{\mathrm{ζ}}}{{x}}{,}{{\mathrm{ζ}}}_{{z}}{=}{0}\right]\right\}$ (5)

Solving the LAVF brings back these three vector fields.

 > $\mathrm{LAVFSolve}\left(\mathrm{SO3},'\mathrm{output}'="basis"\right)$
 $\left[{-}{z}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{y}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}{-}{z}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}{-}{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]$ (6)

Compatibility

 • The LieAlgebrasOfVectorFields[EliminationLAVF] command was introduced in Maple 2020.