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DifferentialGeometry

  

LieBracket

  

calculate the Lie bracket of two vector fields or 2 vectors in a Lie algebra

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

LieBracket(X, Y)

Parameters

X, Y

-

vector fields, defined on the same manifold or Lie algebra

Description

• 

If X is a vector field on a manifold M and f is a real-valued function on M, then X may be applied to f to give a new real valued function.  In coordinates, X(f) is the directional derivative of f with respect to X.  The Lie bracket of two vector fields X, Y , defined on a manifold M, is the vector field Z defined by the commutator rule Z(f) = X(Y(f)) - Y(X(f)).  The standard notation for the Lie bracket is Z = [X, Y].

• 

The LieBracket command is also used to calculate brackets in an abstract Lie algebra.

• 

This command is part of the DifferentialGeometry package, and so can be used in the form LieBracket(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-LieBracket.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Define a 2-dimensional manifold M..

DGsetupx,y,M:

 

Example 1.

Define a pair of vector fields X1 and Y1.

X1evalDGy2D_x+x2D_y

_DGvector,M,,1,y2,2,x2

(1)

Y1evalDGxD_x

_DGvector,M,,1,x

(2)

Calculate the Lie bracket of X1 and Y1.

ZLieBracketX1,Y1

_DGvector,M,,1,y2,2,2x2

(3)

Let's check this result against the commutator definition of the Lie bracket acting on functions.  To apply a vector field to a function we use the LieDerivative command.

LHSofDefinitionLieDerivativeZ,Fx,y

LHSofDefinitiony2xFx,y2x2yFx,y

(4)

RHSofDefinitionLieDerivativeX1,LieDerivativeY1,Fx,yLieDerivativeY1,LieDerivativeX1,Fx,y

RHSofDefinitiony2xFx,y+x2x2Fx,y+x32xyFx,yxy22x2Fx,y+2xyFx,y+x22xyFx,y

(5)

simplifyRHSofDefinition

y2xFx,y2x2yFx,y

(6)

 

Example 2.  

Here is the general coordinate formula for the Lie bracket of two vector fields defined on a 2-dimensional manifold.

X2evalDGax,yD_x+bx,yD_y

_DGvector,M,,1,ax,y,2,bx,y

(7)

Y2evalDGcx,yD_x+dx,yD_y

_DGvector,M,,1,cx,y,2,dx,y

(8)

LieBracketX2,Y2

_DGvector,M,,1,cx,ydiffax,y,xdx,ydiffax,y,y+ax,ydiffcx,y,x+bx,ydiffcx,y,y,2,cx,ydiffbx,y,xdx,ydiffbx,y,y+ax,ydiffdx,y,x+bx,ydiffdx,y,y

(9)

 

Example 3.  

Two vector fields are said to commute if their Lie bracket is 0.  For example:

X3evalDGxD_x+yD_y

_DGvector,M,,1,x,2,y

(10)

Y3evalDGyx2x2+y2D_x+x3x2+y2D_y

_DGvector,M,,1,yx2x2+y2,2,x3x2+y2

(11)

LieBracketX3,Y3

_DGvector,M,,1,0

(12)

 

Example 4.  

The Lie bracket satisfies the Jacobi identity [[X, Y], Z] + [[Z, X], Y] + [[Y, Z], X] = 0.  For example:

XevalDGsinxD_x+lnxyD_y

_DGvector,M,,1,sinx,2,lnxy

(13)

YevalDGcosyD_x+expxD_y

_DGvector,M,,1,cosy,2,expx

(14)

ZevalDGxy3D_xyx3D_y

_DGvector,M,,1,y3x,2,yx3

(15)

LieBracketLieBracketX,Y,Z&plusLieBracketLieBracketZ,X,Y&plusLieBracketLieBracketY,Z,X

_DGvector,M,,1,0

(16)

 

Example 5.  

Use LieAlgebraData and DGsetup to initialize a Lie algebra.  

LDLieAlgebraDatax1,x2=x3,x3,x1=2x1,x3,x2=2x2,x1,x2,x3,alg

LDe1,e2=e3,e1,e3=2e1,e2,e3=2e2

(17)

DGsetupLD

Lie algebra: alg

(18)

MultiplicationTableLieTable

| e1e2e3----------------e1| 0_DGvector,alg,,3,1_DGvector,alg,,1,2e2| _DGvector,alg,,3,10_DGvector,alg,,2,2e3| _DGvector,alg,,1,2_DGvector,alg,,2,20

(19)

Calculate the Lie bracket of 2 vectors in this Lie algebra.

LieBrackete1+e2,e2+e3

_DGvector,alg,,1,2,2,2,3,1

(20)
alg > 

See Also

DifferentialGeometry

ExteriorDerivative

LieDerivative

Physics[ExteriorDerivative]

Physics[LieDerivative]

Physics[LieBracket]

Physics