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LieAlgebrasOfVectorFields

  

LAVF

  

construct a LAVF object

 

Calling Sequence

Parameters

Description

Examples

Compatibility

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

withLieAlgebrasOfVectorFields:

Typesetting:-Settingsuserep=true:

Typesetting:-Suppressξx,y,ηx,y:

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

VVectorFieldξx,y,x,ηx,y,y,space=x,y

Vξx+ηy

(1)

E2SysLHPDEdiffξx,y,y,y=0,diffηx,y,x+diffξx,y,y=0,diffηx,y,y=0,diffξx,y,x=0,dep=ξ,η

E2Sysξy,y=0,ηx+ξy=0,ηy=0,ξx=0,indep=x,y,dep=ξ,η

(2)

E2LAVFV,E2Sys

E2ξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(3)

Data attributes of E2 can be obtained by...

GetVectorFieldE2

ξx+ηy

(4)

GetDeterminingSystemE2

ξy,y=0,ξx=0,ηx=ξy,ηy=0,indep=x,y,dep=ξ,η

(5)

A valid LAVF object then has access to various methods. See Overview of the LAVF object for more information.

exportsE2,static

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

(6)

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

LAVFV,trivial

ξx+ηy&whereξ=0,η=0

(7)

Similarly, construct a universal LAVF:

LAVFV,universal

ξx+ηy&where

(8)

Compatibility

• 

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

• 

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

See Also

LieAlgebrasOfVectorFields (Package overview)

LAVF (Object overview)

VectorField (Object overview)

LHPDE (Object overview)

LieAlgebrasOfVectorFields[VectorField]

LieAlgebrasOfVectorFields[LHPDE]