LieProduct - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : LieAlgebrasOfVectorFields : LAVF : LieAlgebrasOfVectorFields/LAVF/LieProduct

LieProduct

find an LAVF object for the Lie product of the spaces defined by two LAVF objects

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

LieProduct(L, M, N)

Parameters

L, M, N

-

LAVF objects living on the same space and L, M commute mod N

Description

• 

Let L, M, N be LAVF objects on the same space, and L commutes with M mod N (i.e. AreCommuting(L,M,N) returns true. See AreCommuting). Then LieProduct(L, M, N) finds an LAVF object for the Lie product L,M of the spaces defined by L, M.

• 

The method only works where all spaces are finite dimensional.

• 

Some Lie algebraic structural methods (DerivedAlgebra, DerivedSeries, and LowerCentralSeries) are front-ends to LieProduct.

• 

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

Examples

withLieAlgebrasOfVectorFields:

Typesetting:-Settingsuserep=true:

Typesetting:-Suppressξx,y,ηx,y:

VVectorFieldξx,yDx+ηx,yDy,space=x,y

Vξx+ηy

(1)

E2LHPDEdiffξx,y,y,y=0,diffηx,y,x=diffξx,y,y,diffηx,y,y=0,diffξx,y,x=0,indep=x,y,dep=ξ,η

E2ξy,y=0,ηx=ξy,ηy=0,ξx=0,indep=x,y,dep=ξ,η

(2)

T2LHPDEdiffξx,y,x=0,diffξx,y,y=0,diffηx,y,x=0,diffηx,y,y=0,indep=x,y,dep=ξ,η

T2ξx=0,ξy=0,ηx=0,ηy=0,indep=x,y,dep=ξ,η

(3)

Construct a LAVF for the 2-dim Euclidean group E(2) and the 2-dim translation group T(2)

LLAVFV,E2

Lξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(4)

LT2LAVFV,T2

LT2ξx+ηy&whereξx=0,ηx=0,ξy=0,ηy=0

(5)

IsLieAlgebraL

true

(6)

IsLieAlgebraLT2

true

(7)

LieProductL,L,L

newAbsL,absLAVF6

ξx+ηy&whereξx=0,ηx=0,ξy=0,ηy=0

(8)

The above call is equivalent to finding the derived algebra of L. which is 2-dim translation group.

DADerivedAlgebraL

DAξx+ηy&whereξx=0,ηx=0,ξy=0,ηy=0

(9)

AreSameDA,LT2

true

(10)

LieProductLT2,LT2,L

newAbsL,absLAVF11

ξx+ηy&whereξ=0,η=0

(11)

Compatibility

• 

The LieProduct 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)

LieAlgebrasOfVectorFields[VectorField]

LieAlgebrasOfVectorFields[LHPDE]

LieAlgebrasOfVectorFields[LAVF]

AreCommuting

IsLieAlgebra

DerivedAlgebra

DerivedSeries

LowerCentralSeries