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IsInvariant

check if a distribution is invariant under Lie group action

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

IsInvariant(dist, L)

Parameters

dist

-

a Distribution object.

L

-

a LAVF object

Description

• 

The IsInvariant method decides whether Distribution object dist is invariant under the action of the Lie transformation group generated by the vector fields from a LAVF object L. It returns the values true or false.

• 

For this method to make sense, Distribution dist must be in involution (see IsInvolutive), and L must specify a Lie algebra (see IsLieAlgebra).  An involutive distribution Σ is invariant under a Lie group action if the foliation induced by Σ maps to itself (i.e. leaves map to leaves).

• 

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

Examples

withLieAlgebrasOfVectorFields:

Typesetting:-Settingsuserep=true:

Building a LAVF object

Typesetting:-Suppressηx,y,ξx,y,zetaz

XVectorFieldξx,yDx+ηx,yDy+zetazDz,space=x,y,z

Xξx+ηy+ζz

(1)

SysLHPDEdiffξx,y,x=0,diffξx,y,y=1yξx,y,ηx,y=xyξx,y,diffzetaz,z,z=0,indep=x,y,z,dep=ξ,η,zeta

Sysξx=0,ξy=ξy,η=xξy,ⅆ2ζⅆz2=0,indep=x,y,z,dep=ξ,η,ζ

(2)

LLAVFX,Sys

Lξx+ηy+ζz&whereⅆ2ζⅆz2=0,ηx=ηx,ηy=0,ξ=ηyx

(3)

Building some Distribution objects

TxVectorFieldDx,space=x,y,z

Txx

(4)

TyVectorFieldDy,space=x,y,z

Tyy

(5)

ΣDistributionTx,Ty

Σx,y

(6)

GammaDistributionTx

Γx

(7)

Now we test if the following distributions are invariant under L

IsInvariantΣ,L

true

(8)

IsInvariantGamma,L

false

(9)

Compatibility

• 

The IsInvariant command was introduced in Maple 2020.

• 

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

See Also

Distribution (Object overview)

LieAlgebrasOfVectorFields[Distribution]

VectorField (Object overview)

LieAlgebrasOfVectorFields[VectorField]

LieAlgebrasOfVectorFields (Package overview)