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LieAlgebrasOfVectorFields

 LAVF
 construct a LAVF object

 Calling Sequence LAVF(vf, dq) LAVF(vf, str)

Parameters

 vf - a VectorField object dq - a LHPDE object str - a string: either "trivial" or "universal"

Description

 • The command LAVF(...) is for constructing a LAVF object. A valid LAVF object then has access to at least 60 methods which allow it to be manipulated and its contents queried. For more detail, see Overview of the LAVF object.
 • In the first calling sequence, the input argument vf must be a type VectorField whose components are indeterminant functionals (as infinitesimals), and dq must be a type LHPDE object whose dependent variables include all components of vf.
 • For convenience the second calling sequence is a special constructor for either a trivial LAVF object or a universal LAVF object. A trivial LAVF object means its determining system is trivial (i.e. only the zero solution). For example, let V be a VectorField object containing indeterminant infinitesimals, a call LAVF(V,"trivial") is equal to the call LAVF(V, LHPDE("trivial", dep = GetComponents(V), indep = GetSpace(V))). And a universal LAVF object has empty system (i.e. no restriction on solutions).
 • 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 LAVF(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used in the form :-LieAlgebrasOfVectorFields:-LAVF(...).

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{ξ}\left(x,y\right),\mathrm{η}\left(x,y\right)\right]\right):$

We first construct a vector field and a LHPDE object for representing the determining system for E(2).

 > $V≔\mathrm{VectorField}\left(\left[\left[\mathrm{ξ}\left(x,y\right),x\right],\left[\mathrm{η}\left(x,y\right),y\right]\right],\mathrm{space}=\left[x,y\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (1)
 > $\mathrm{E2Sys}≔\mathrm{LHPDE}\left(\left[\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{η}\left(x,y\right)+\frac{\partial }{\partial y}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial y}\mathrm{η}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{ξ}\left(x,y\right)=0\right],\mathrm{dep}=\left[\mathrm{ξ},\mathrm{η}\right]\right)$
 ${\mathrm{E2Sys}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{+}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)
 > $\mathrm{E2}≔\mathrm{LAVF}\left(V,\mathrm{E2Sys}\right)$
 ${\mathrm{E2}}{≔}\left[{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (3)

Data attributes of E2 can be obtained by...

 > $\mathrm{GetVectorField}\left(\mathrm{E2}\right)$
 ${\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (4)
 > $\mathrm{GetDeterminingSystem}\left(\mathrm{E2}\right)$
 $\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (5)

 > $\mathrm{exports}\left(\mathrm{E2},'\mathrm{static}'\right)$
 ${\mathrm{indets}}{,}{\mathrm{has}}{,}{\mathrm{hastype}}{,}{\mathrm{type}}{,}{\mathrm{GetVectorField}}{,}{\mathrm{GetDeterminingSystem}}{,}{\mathrm{ImplicitForm}}{,}{\mathrm{SolutionDimension}}{,}{\mathrm{IsFiniteType}}{,}{\mathrm{IsTrivial}}{,}{\mathrm{ParametricDerivatives}}{,}{\mathrm{GetRanking}}{,}{\mathrm{SetIDBasis}}{,}{\mathrm{GetIDBasis}}{,}{\mathrm{GetSpace}}{,}{\mathrm{IsFlat}}{,}{\mathrm{OrbitDistribution}}{,}{\mathrm{OrbitDimension}}{,}{\mathrm{InvariantCount}}{,}{\mathrm{IsTransitive}}{,}{\mathrm{Invariants}}{,}{\mathrm{IsLieAlgebra}}{,}{\mathrm{IsPerfect}}{,}{\mathrm{DerivedAlgebra}}{,}{\mathrm{IsSolvable}}{,}{\mathrm{IsSoluble}}{,}{\mathrm{DerivedSeries}}{,}{\mathrm{SolvableRadical}}{,}{\mathrm{SolubleRadical}}{,}{\mathrm{Radical}}{,}{\mathrm{IsNilpotent}}{,}{\mathrm{Hypercentre}}{,}{\mathrm{Hypercenter}}{,}{\mathrm{NilRadical}}{,}{\mathrm{Nilradical}}{,}{\mathrm{LowerCentralSeries}}{,}{\mathrm{UpperCentralSeries}}{,}{\mathrm{IsAbelian}}{,}{\mathrm{IsCommutative}}{,}{\mathrm{Centre}}{,}{\mathrm{Center}}{,}{\mathrm{IsSemiSimple}}{,}{\mathrm{IsReductive}}{,}{\mathrm{NilpotentRadical}}{,}{\mathrm{StructureConstants}}{,}{\mathrm{StructureCoefficients}}{,}{\mathrm{KillingRadical}}{,}{\mathrm{KillingPolynomial}}{,}{\mathrm{KillingForm}}{,}{\mathrm{KillingOrthogonal}}{,}{\mathrm{AdjointMatrix}}{,}{\mathrm{AreCommuting}}{,}{\mathrm{AreSame}}{,}{\mathrm{AreSameSpace}}{,}{\mathrm{Centraliser}}{,}{\mathrm{Centralizer}}{,}{\mathrm{Normaliser}}{,}{\mathrm{CleanDependencies}}{,}{\mathrm{Copy}}{,}{\mathrm{DChange}}{,}{\mathrm{dchange}}{,}{\mathrm{Intersection}}{,}{\mathrm{IsIdeal}}{,}{\mathrm{IsInvariant}}{,}{\mathrm{IsotropyRepresentation}}{,}{\mathrm{IsSubspace}}{,}{\mathrm{LAVFSolve}}{,}{\mathrm{VectorSpaceSum}}{,}{\mathrm{LieProduct}}{,}{\mathrm{ProjectToSpace}}{,}{\mathrm{Transporter}}{,}{\mathrm{ModuleCopy}}{,}{\mathrm{ModulePrint}}{,}{\mathrm{ModuleApply}}$ (6)

A simple way to construct a LAVF object whose determining system has trivial solution.

 > $\mathrm{LAVF}\left(V,"trivial"\right)$
 $\left[{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{=}{0}{,}{\mathrm{\eta }}{=}{0}\right]\right\}$ (7)

Similarly, construct a universal LAVF:

 > $\mathrm{LAVF}\left(V,"universal"\right)$
 $\left[{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[\right]\right\}$ (8)

Compatibility

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