The LAVF object is designed and created to represent a Lie algebras of vector field (LAVF). An LAVF is usually of the form before solving the symmetry determining system to get solution vector fields.

There are more than 60 methods that are available for a LAVF object, types of methods including (i) basic and solution properties of a LAVF, (ii) Lie algebraic and geometrical properties of a LAVF, (iii) Lie algebraic and geometrical relationships among LAVFs, (iv) utility methods for manipulating a LAVF (or LAVFs). Some Maple existing builtins are extended for allowing LAVF object.

A valid LAVF object is automatically in a rif-reduced form with respect to a ranking that is in total degree ranking. See methods IsRifReduced, IsTotalDegreeRanking and GetRanking of a LHPDE object.

The LAVF object is allowed to be either of finite or infinite type. And the form of its determining system can be un-solved, or solved, or any forms in between (i.e. partially solved).

All methods of the LAVF object become available only once a valid LAVF object is constructed successfully. To construct a LAVF object, see LieAlgebrasOfVectorFields[LAVF].

The LAVF object is the main Maple object exported by the LieAlgebrasOfVectorFields package. See Overview of the LieAlgebrasOfVectorFields package for more detail.

A LAVF object is mathematically represented by the formal vector field as a VectorField object and the determining system as a LHPDE object. Here the components of the formal vector field must be (part of) the dependent variables of the determining system.These two data attributes can be accessed via the GetVectorField and GetDeterminingSystem methods.

After a LAVF object L is successfully constructed, each method in L can be accessed by either the short form method(L, arguments) or the long form L:-method(L, arguments).

After a LAVF object is constructed, the following methods are available:

The following Maple builtins functions are extended so that they work for a LAVF object: type, has, hastype, indets. See LAVF Object Overloaded Builtins for more detail.

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

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

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

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

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

$\mathrm{E2Sys}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)+\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{E2Sys}≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x+{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{E2}≔\mathrm{LAVF}\left(V\,\mathrm{E2Sys}\right)$

$\mathrm{E2}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{type}\left(\mathrm{E2}\,'\mathrm{LAVF}'\right)$

$\mathrm{true}$

$\mathrm{exports}\left(\mathrm{E2}\,'\mathrm{static}'\right)$

$\mathrm{indets},\mathrm{has},\mathrm{hastype},\mathrm{type},\mathrm{GetVectorField},\mathrm{GetDeterminingSystem},\mathrm{ImplicitForm},\mathrm{SolutionDimension},\mathrm{IsFiniteType},\mathrm{IsTrivial},\mathrm{ParametricDerivatives},\mathrm{GetRanking},\mathrm{SetIDBasis},\mathrm{GetIDBasis},\mathrm{GetSpace},\mathrm{IsFlat},\mathrm{OrbitDistribution},\mathrm{OrbitDimension},\mathrm{InvariantCount},\mathrm{IsTransitive},\mathrm{Invariants},\mathrm{IsLieAlgebra},\mathrm{IsPerfect},\mathrm{DerivedAlgebra},\mathrm{IsSolvable},\mathrm{IsSoluble},\mathrm{DerivedSeries},\mathrm{SolvableRadical},\mathrm{SolubleRadical},\mathrm{Radical},\mathrm{IsNilpotent},\mathrm{Hypercentre},\mathrm{Hypercenter},\mathrm{NilRadical},\mathrm{Nilradical},\mathrm{LowerCentralSeries},\mathrm{UpperCentralSeries},\mathrm{IsAbelian},\mathrm{IsCommutative},\mathrm{Centre},\mathrm{Center},\mathrm{IsSemiSimple},\mathrm{IsReductive},\mathrm{NilpotentRadical},\mathrm{StructureConstants},\mathrm{StructureCoefficients},\mathrm{KillingRadical},\mathrm{KillingPolynomial},\mathrm{KillingForm},\mathrm{KillingOrthogonal},\mathrm{AdjointMatrix},\mathrm{AreCommuting},\mathrm{AreSame},\mathrm{AreSameSpace},\mathrm{Centraliser},\mathrm{Centralizer},\mathrm{Normaliser},\mathrm{CleanDependencies},\mathrm{Copy},\mathrm{DChange},\mathrm{dchange},\mathrm{Intersection},\mathrm{IsIdeal},\mathrm{IsInvariant},\mathrm{IsotropyRepresentation},\mathrm{IsSubspace},\mathrm{LAVFSolve},\mathrm{VectorSpaceSum},\mathrm{LieProduct},\mathrm{ProjectToSpace},\mathrm{Transporter},\mathrm{ModuleCopy},\mathrm{ModulePrint},\mathrm{ModuleApply}$

$X≔\mathrm{GetVectorField}\left(\mathrm{E2}\right)$

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

$S≔\mathrm{GetDeterminingSystem}\left(\mathrm{E2}\right)$

$S≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{type}\left(S\,'\mathrm{LHPDE}'\right)$

$\mathrm{true}$

$\mathrm{IsRifReduced}\left(S\right)$

$\mathrm{true}$

$\mathrm{IsTotalDegreeRanking}\left(S\right)$

$\mathrm{true}$

$\mathrm{GetRanking}\left(S\right)$

$\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{IsLieAlgebra}\left(\mathrm{E2}\right)$

$\mathrm{true}$

$\mathrm{IsAbelian}\left(\mathrm{E2}\right)$

$\mathrm{false}$

$\mathrm{IsSolvable}\left(\mathrm{E2}\right)$

$\mathrm{true}$

$C≔\mathrm{StructureConstants}\left(\mathrm{E2}\right)$

$\mathrm{DisplayStructure}\left(C\,'Z'\right)$

$\left[\begin{array}{ccc}0& 0& Z_2\\ 0& 0& -Z_1\\ -Z_2& Z_1& 0\end{array}\right]$

$\mathrm{Centre}\left(\mathrm{E2}\right)$

$\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\mathrm{\xi }=0\,\mathrm{\eta }=0\right]\right\}$

$\mathrm{SolvableRadical}\left(\mathrm{E2}\right)$

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

$\mathrm{DerivedSeries}\left(\mathrm{E2}\right)$

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

$\mathrm{VFPDO}\left(\mathrm{E2}\right)$

$X\mapsto \left[\frac{{\partial }^2}{\partial y^2}X\left(x\right)\,\frac{\ⅆ}{\ⅆx}X\left(x\right)\,\frac{\partial }{\partial x}X\left(y\right)+\frac{\partial }{\partial y}X\left(x\right)\,\frac{\ⅆ}{\ⅆy}X\left(y\right)\right]$

$\mathrm{LAVFSolve}\left(\mathrm{E2}\right)$

$\left(-\mathrm{c\_\_1}y+\mathrm{c\_\_3}\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\left(\mathrm{c\_\_1}x+\mathrm{c\_\_2}\right)\left(\frac{\ⅆ}{\ⅆy}\right)$

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

$\mathrm{R1}≔\mathrm{VectorField}\left(x\mathrm{D}\left[y\right]-y\mathrm{D}\left[x\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$\mathrm{R1}≔-y\left(\frac{\ⅆ}{\ⅆx}\right)+x\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{R2}≔\mathrm{VectorField}\left(x\mathrm{D}\left[z\right]-z\mathrm{D}\left[x\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$\mathrm{R2}≔-z\left(\frac{\ⅆ}{\ⅆx}\right)+x\left(\frac{\ⅆ}{\ⅆz}\right)$

$\mathrm{R3}≔\mathrm{VectorField}\left(y\mathrm{D}\left[z\right]-z\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$\mathrm{R3}≔-z\left(\frac{\ⅆ}{\ⅆy}\right)+y\left(\frac{\ⅆ}{\ⅆz}\right)$

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

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

$\mathrm{SO3}≔\mathrm{EliminationLAVF}\left(X\,\left[\mathrm{R1}\,\mathrm{R2}\,\mathrm{R3}\right]\right)$

$\mathrm{SO3}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)+\mathrm{\zeta }\left(\frac{\ⅆ}{\ⅆz}\right)\right]\&where\left\{\left[\mathrm{\xi }=\frac{-\mathrm{\zeta }z-\mathrm{\eta }y}{x}\,{\mathrm{\eta }}_x=\frac{\left({\mathrm{\zeta }}_y\right)z+\mathrm{\eta }}{x}\,{\mathrm{\eta }}_y=0\,{\mathrm{\eta }}_z=-{\mathrm{\zeta }}_y\,{\mathrm{\zeta }}_{y,y}=0\,{\mathrm{\zeta }}_x=\frac{-\left({\mathrm{\zeta }}_y\right)y+\mathrm{\zeta }}{x}\,{\mathrm{\zeta }}_z=0\right]\right\}$

$\mathrm{OrbitDimension}\left(\mathrm{SO3}\right)$

$2$

$\mathrm{IsTransitive}\left(\mathrm{SO3}\right)$

$\mathrm{false}$

$\mathrm{InvariantCount}\left(\mathrm{SO3}\right)$

$1$

$\mathrm{Invariants}\left(\mathrm{SO3}\right)$

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

$\mathrm{OD}≔\mathrm{OrbitDistribution}\left(\mathrm{SO3}\right)$

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

$\mathrm{type}\left(\mathrm{OD}\,'\mathrm{Distribution}'\right)$

$\mathrm{true}$

$\mathrm{exports}\left(\mathrm{OD}\,'\mathrm{static}'\right)$

$\mathrm{GetSpace},\mathrm{GetVectorFields},\mathrm{GetAnnihilator},\mathrm{AreSameSpace},\mathrm{IsTrivial},\mathrm{Dimension},\mathrm{Codimension},\mathrm{IsInvolutive},\mathrm{IsIntegrable},\mathrm{Integrals},\mathrm{IsSubspace},\mathrm{Intersection},\mathrm{VectorSpaceSum},\mathrm{IsInvariant},\mathrm{DerivedDistribution},\mathrm{CauchyDistribution},\mathrm{dchange},\mathrm{DChange},\mathrm{indets},\mathrm{has},\mathrm{hastype},\mathrm{type},\mathrm{convert}$