DifferentialGeometry - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : DifferentialGeometry : DifferentialGeometry/LieBracket

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

X1y2D_x+x2D_y

(1)

Y1evalDGxD_x

Y1xD_x

(2)

Calculate the Lie bracket of X1 and Y1.

ZLieBracketX1,Y1

Zy2D_x2x2D_y

(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+x32yxFx,yxy22x2Fx,y+2xyFx,y+x22yxFx,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

X2ax,yD_x+bx,yD_y

(7)

Y2evalDGcx,yD_x+dx,yD_y

Y2cx,yD_x+dx,yD_y

(8)

LieBracketX2,Y2

cx,yxax,ydx,yyax,y+ax,yxcx,y+bx,yycx,yD_x+cx,yxbx,ydx,yybx,y+ax,yxdx,y+bx,yydx,yD_y

(9)

 

Example 3.  

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

X3evalDGxD_x+yD_y

X3xD_x+yD_y

(10)

Y3evalDGyx2x2+y2D_x+x3D_yx2+y2

Y3yx2D_xx2+y2+x3D_yx2+y2

(11)

LieBracketX3,Y3

0D_x

(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

XsinxD_x+lnxyD_y

(13)

YevalDGcosyD_x+ⅇxD_y

YcosyD_x+ⅇxD_y

(14)

ZevalDGxy3D_xyx3D_y

Zy3xD_xyx3D_y

(15)

LieBracketLieBracketX,Y,Z &plus LieBracketLieBracketZ,X,Y &plus LieBracketLieBracketY,Z,X

0D_x

(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| 0e32e1e2| e302e2e3| 2e12e20

(19)

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

LieBrackete1+e2,e2+e3

2e1+2e2+e3

(20)
alg > 

See Also

DifferentialGeometry

ExteriorDerivative

LieDerivative

Physics[ExteriorDerivative]

Physics[LieDerivative]

Physics[LieBracket]

Physics

 


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam