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OrbitDimension

calculate the dimension of the orbit distribution of a LAVF object.

InvariantCount

calculate the count of invariant of a LAVF object.

IsTransitive

check if a LAVF object is transitive.

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

OrbitDimension( obj)

InvariantCount( obj, t)

IsTransitive( obj)

Parameters

obj

-

a LAVF object.

t

-

(optional) a string: "all", "essential", or "inessential"

Description

• 

The OrbitDimension method calculates the dimension of the orbit distribution of a LAVF object.

• 

The InvariantCount method calculates the count of scalar invariants of a LAVF object. By default (type="all"), all invariants are counted.

• 

If type="essential" is specified, then only essential invariants are counted. An invariant is essential, roughly speaking, if the group action cannot be expressed without it.

• 

Let L be a LAVF object. Then IsTransitive(L) returns true if and only if the action of L is transitive, that is, InvariantCount(L) = 0.

• 

Let L be a LAVF object and let OD be the orbit distribution of L. Then OrbitDimension(L) equals to Dimension(OD) and InvariantCount(L) equals to Codimension(OD). See Overview of the Distribution object for more detail.

• 

These methods are associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

withLieAlgebrasOfVectorFields:

Typesetting:-Settingsuserep=true:

Typesetting:-Suppressξx,y,z,ηx,y,z,zetax,y,z:

Example 1: Build vector fields associated with 3-d spatial rotations...

RxVectorFieldzDy+yDz,space=x,y,z

Rxzy+yz

(1)

RyVectorFieldxDz+zDx,space=x,y,z

Ryzxxz

(2)

RzVectorFieldyDx+xDy,space=x,y,z

Rzyx+xy

(3)

We now construct a LAVF object for SO(3) that are generated by these rotation vector fields.

VVectorFieldξx,y,zDx+ηx,y,zDy+zetax,y,zDz,space=x,y,z

Vξx+ηy+ζz

(4)

LEliminationLAVFV,Rx,Ry,Rz

Lξx+ηy+ζz&whereξ=ηyζzx,ηx=ζyz+ηx,ηy=0,ηz=ζy,ζy,y=0,ζx=ζyy+ζx,ζz=0

(5)

OrbitDimensionL

2

(6)

InvariantCountL

1

(7)

L is not transitive since SO(3) has one invariant.

IsTransitiveL

false

(8)

InvariantsL

x2+y2+z2

(9)

Example 2:

YVectorFieldayDx+bzDx,space=x,y,z

Yay+bzx

(10)

L2EliminationLAVFV,Y,coefficients=a,b

L2ξx+ηy+ζz&whereξz,z=0,ξx=0,ξy=ξzz+ξy,η=0,ζ=0

(11)

OrbitDimensionL2

1

(12)

IsTransitiveL2

false

(13)

InvariantCountL2

2

(14)

InvariantCountL2,essential

1

(15)

InvariantCountL2,inessential

1

(16)

The counts above are found directly from L2. Finding invariants involve integration...

InvariantsL2

y,z

(17)

InvariantsL2,essential

zy

(18)

Compatibility

• 

The OrbitDimension, InvariantCount and IsTransitive commands were introduced in Maple 2020.

• 

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

See Also

LieAlgebrasOfVectorFields (Package overview)

LAVF (Object overview)

Distribution (Object overview)

LieAlgebrasOfVectorFields[VectorField]

LieAlgebrasOfVectorFields[EliminationLAVF]

Invariants