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DifferentialGeometry

  

LieDerivative

  

calculate the Lie derivative of a vector field, differential form, tensor, or connection with respect to a vector field

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

LieDerivative(X, T)

Parameters

X

-

a vector field on a manifold M or a vector in a Lie algebra A

T

-

a vector field, a Maple expression, a differential form or a tensor field on the manifold M or the Lie algebra A

Description

• 

If T is a Maple expression, then LieDerivative(X, T) is the directional derivative X(T) of T in the direction of the vector field X.

• 

If T is a vector field, then LieDerivative(X, T) coincides with the Lie bracket [X, T] = LieBracket(X, T).

• 

If T is a differential 1-form, then alpha = LieDerivative(X, T) is the 1-form defined by alpha(Y) = X(alpha(Y)) - alpha([X,Y]), where Y is any vector field on M.

• 

The Lie derivative operator acts as a derivation with respect to both the wedge and tensor products.  If alpha and beta are differential forms and T and S are tensors, then LieDerivative(X, alpha &w beta) = LieDerivative(X, alpha) &w beta + alpha &w LieDerivative(X, beta), and LieDerivative(X, S &t T) = LieDerivative(X, S) &t T + S &w LieDerivative(X, T).

• 

The Lie derivative of a differential form can also be calculated from the Cartan formula, LieDerivative(X, alpha) = ExteriorDerivative(Hook(X, alpha)) + Hook(X, ExteriorDerivative(alpha))

• 

The Lie derivative of a connection nabla_Y(Z) is the type (1, 2) tensor field S = LieDerivative(X, nabla), defined (when viewed as mapping from pairs of vector fields to vector fields) by S(Y, Z) = LieDerivative(X, nabla_Y(Z)) - nabla_{LieDerivative(X, Y)}(Z) - nabla_X(LieDerivative(Y, Z)).

• 

For the definition of the Lie derivative of these geometric objects in terms of the flow of the vector field X see, for example, Spivak page 207-208.

• 

The Lie derivative of a tensor defined on a Lie algebra can also be computed.

• 

The first argument also be a list of vectors. The second argument can be a list of a vectors, Maple  expressions, a differential forms or tensors.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

First initialize a manifold M with local coordinates [x, y, z].

DGsetupx,y,z,M:

 

Example 1.

First we calculate the Lie derivative of a function f and note that it agrees with the directional derivative f.

XevalDGaD_x+bD_y+cD_z

_DGvector,M,,1,a,2,b,3,c

(1)

LieDerivativeX,fx,y,z

axfx,y,z+byfx,y,z+czfx,y,z

(2)

 

Example 2.

First we calculate the Lie derivative of a vector field and check that it coincides with the Lie bracket.

XevalDGxD_yy2zD_z

_DGvector,M,,2,x,3,zy2

(3)

YevalDGyD_x+z2D_y

_DGvector,M,,1,y,2,z2

(4)

LieDerivativeX,Y

_DGvector,M,,1,x,2,2y2z2y,3,2z3y

(5)

LieBracketX,Y

_DGvector,M,,1,x,2,2y2z2y,3,2z3y

(6)

 

Example 3.

First we calculate the Lie derivative of a differential form and check the result against Cartan's formula.

XevalDGz2D_xyD_z

_DGvector,M,,1,z2,3,y

(7)

ωevalDGydx&wdz

_DGform,M,2,1,3,y

(8)

LieDerivativeX,ω

_DGform,M,2,1,2,y

(9)

HookX,ExteriorDerivativeω&plusExteriorDerivativeHookX,ω

_DGform,M,2,1,2,y

(10)

 

Example 4.

We calculate the Lie derivative of a tensor field.

XevalDGz2D_xyD_z

_DGvector,M,,1,z2,3,y

(11)

TevalDGzD_x&tdy&tdz

_DGtensor,M,con_bas,cov_bas,cov_bas,,1,2,3,z

(12)

LieDerivativeX,T

_DGtensor,M,con_bas,cov_bas,cov_bas,,1,2,2,z,1,2,3,y

(13)

 

Example 5.

We calculate the Lie derivative of the zero connection.

XevalDGz2D_xy2D_z

_DGvector,M,,1,z2,3,y2

(14)

TTensor:-Connection0&multD_x&tensordx&tensordx

_DGconnection,M,con_bas,cov_bas,cov_bas,,1,1,1,0

(15)

LieDerivativeX,T

_DGtensor,M,con_bas,cov_bas,cov_bas,,2,3,2,2,3,1,3,2

(16)

 

Example 6. 

The Lie derivative with respect to a list of vectors can be calculated simultaneously.

LieDerivativeD_x,D_y,D_z,xyzDz

yzDz,xzDz,xyDz

(17)

The Lie derivative of a list of tensors can be calculated simultaneously.

LieDerivativeD_x,D_x&tD_x,xD_x&tDy,x2D_x&tDz

_DGtensor,M,con_bas,con_bas,,1,1,0,_DGtensor,M,con_bas,,1,Dy,_DGtensor,M,con_bas,,1,2xDz

(18)

Both arguments to LieDerivative can be lists.

LieDerivativeD_x,D_y,D_x,xD_x,yD_x,xyDz

_DGvector,M,,1,0,_DGvector,M,,1,1,_DGvector,M,,1,0,yDz,_DGvector,M,,1,0,_DGvector,M,,1,0,_DGvector,M,,1,1,xDz

(19)

The Lie derivative of a Matrix of differential 2-forms can be calculated simultaneously.

TmapevalDG,Matrixy2dx&wdy,x2dy&wdz,xydx&wdz,z2dx&wdy

_DGform,M,2,1,2,y2_DGform,M,2,2,3,x2_DGform,M,2,1,3,xy_DGform,M,2,1,2,z2

(20)

LieDerivativexD_x+yD_y+zD_z,T

_DGform,M,2,1,2,4y2_DGform,M,2,2,3,4x2_DGform,M,2,1,3,4xy_DGform,M,2,1,2,4z2

(21)

 

Example 7. 

The Lie derivative can be calculated in anholonomic frames. Use FrameData to find the structure equations for an anholonomic frame and initialize with DGsetup.

FDFrameDataydx,dx+zdy,dz,P

FDdΘ1=Θ1Θ2yz,dΘ2=Θ1Θ3yzΘ2Θ3z,dΘ3=0

(22)

DGsetupFD,U,σ

frame name: P

(23)

LieDerivativeU1,σ1&wσ3

_DGform,P,2,2,3,1yz

(24)

LieDerivativexU1,U1&tσ3

_DGtensor,P,con_bas,cov_bas,,1,3,1y

(25)

 

Example 8. 

The Lie derivative can be calculated for abstract forms.

DGsetupω1,ω2,ω3,β1=dgform2,dω1=ω2&wω3,dω2=β1,N

frame name: N

(26)

LieDerivativeD_omega2,ω1

_DGform,N,1,3,1

(27)

LieDerivativeD_omega2,β1

_DGform,N,2,6,1,8,1

(28)

 

Example 9. 

The Lie derivative can be calculated for tensors on a Lie algebra. 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

(29)

DGsetupLD

Lie algebra: alg

(30)

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

(31)

Calculate the Killing form for the Lie algebra and show that its Lie derivative is zero for all vectors in the Lie algebra.

KKillingForm

_DGtensor,alg,cov_bas,cov_bas,,1,2,4,2,1,4,3,3,8

(32)

LieDerivativee1,e2,e3,K

_DGtensor,alg,cov_bas,cov_bas,,1,1,0,_DGtensor,alg,cov_bas,cov_bas,,1,1,0,_DGtensor,alg,cov_bas,cov_bas,,1,1,0

(33)
alg > 

See Also

DifferentialGeometry

Tensor

Connection

ExteriorDerivative

Hook

LieBracket

Physics[ExteriorDerivative]

Physics[LieDerivative]

Physics[LieBracket]

Physics