IsInvariant
check if a distribution is invariant under Lie group action
Calling Sequence
Parameters
Description
Examples
Compatibility
IsInvariant(dist, L)
dist
-
a Distribution object.
L
a LAVF object
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.
withLieAlgebrasOfVectorFields:
Typesetting:-Settingsuserep=true:
Building a LAVF object
Typesetting:-Suppressηx,y,ξx,y,ζz
X≔VectorFieldξx,yDx+ηx,yDy+ζzDz,space=x,y,z
X≔ξⅆⅆx+ηⅆⅆy+ζⅆⅆz
Sys≔LHPDEdiffξx,y,x=0,diffξx,y,y=1yξx,y,ηx,y=−xyξx,y,diffζz,z,z=0,indep=x,y,z,dep=ξ,η,ζ
Sys≔ξx=0,ξy=ξy,η=−xξy,ⅆ2ζⅆz2=0,indep=x,y,z,dep=ξ,η,ζ
L≔LAVFX,Sys
L≔ξⅆⅆx+ηⅆⅆy+ζⅆⅆz&whereⅆ2ζⅆz2=0,ηx=ηx,ηy=0,ξ=−ηyx
Building some Distribution objects
Tx≔VectorFieldDx,space=x,y,z
Tx≔ⅆⅆx
Ty≔VectorFieldDy,space=x,y,z
Ty≔ⅆⅆy
Σ≔DistributionTx,Ty
Σ≔ⅆⅆx,ⅆⅆy
Gamma≔DistributionTx
Γ≔ⅆⅆx
Now we test if the following distributions are invariant under L
IsInvariantΣ,L
true
IsInvariantGamma,L
false
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)
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