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GroupActions[InfinitesimalSymmetriesOfGeometricObjectFields] - find the infinitesimal symmetries (vector fields) for a collection of vector fields, differential forms tensors, or connections

Calling Sequences

     InfinitesimalSymmetriesOfGeometricObjectFields(T, option)

Parameters

     T         - a list of vector fields, differential forms, tensors, connections, list of vector fields, list of differential forms, list of tensors

     option    - output = "list", output = "pde", auxiliaryequations = [Delta1, Delta2,..] coefficientvariables = [x1, x2, ...], ansatz = X, unknowns = [F1, F2, ...], parameters = {a1, a2}

Description

• 

Let M be a manifold and let T1, T2, ...TN be a list of tensor fields on M. Then the Lie algebra Γof infinitesimal symmetries of the list of tensors Ti is the Lie algebra of vector fields X on M such that the Lie derivatives ℒX Ti = 0 for i = 1, 2, ... , N.   

• 

If the tensors Ti all have the same tensorial type, say Ti TsrM, then let 𝒯 = spanT1, T2, ...TN. Then the Lie algebra Γ of infinitesimal symmetries of the tensor space 𝒯 is Lie algebra of vector fields X on M such that ℒX Ti 𝒯 for i = 1, 2, ... , N. 

• 

The command InfinitesimalSymmetriesOfGeometricObjectFields(T) calculates the Lie algebra of infinitesimal symmetries of the tensors and tensor spaces in the list T. For example, ifT1, T2, T3,T4 are 4 tensor fields and T = T1, T2, T3,T4, then InfinitesimalSymmetriesOfGeometricObjectFields(T) will return the Lie algebra of vector fields X such that X T1 = 0 , X T1 = 0 , ℒX T3  span T3,T4, X T4  span T3,T4.  

• 

The procedure InfinitesimalSymmetriesOfGeometricObjectFields creates an arbitrary vector field X on M and generates a system of first order PDE for the coefficients of X from the Lie derivative equations X Ti = 0 and ℒX Ti 𝒯. These PDE are solved using pdsolve .

• 

If the (real) Lie algebra Γ of infinitesimal symmetries for a given collection of geometric object fields is finite dimensional (so that the most general infinitesimal symmetry depends only upon arbitrary constants), then the optional argument output = "list" will return a basis for Γ.

• 

With the option output = "pde", just the determining differential equations for the symmetries are returned.

• 

The variables appearing in the coefficients of the vector field X can be specified with the option coefficientvariables = [x1, x2, ...].

• 

The exact form of the infinitesimal symmetries to be found can be specified with the option ansatz = X. With this option, the unknown coefficients to be solved for must be explicitly identified with the option unknowns = [F1, F2, ...].

• 

Additional constraints on the symmetry vector field X can be specified with the optional argument auxiliaryequations = [Delta1, Delta2,..], where Delta1, Delta2,.. are differential equations whose unknowns are the coefficients of the vector field X.

• 

If the given geometric object fields T depend upon parameters {a1, a2, ...}, then the optional argument parameters = {a1, a2, ...} will invoke the case splitting capabilities of pdsolve. Exceptional parameter values will be determined and a sequence of lists of infinitesimal symmetries, one list for each set of parameter values, will be returned.

• 

Other optional arguments for pdsolve may be passed through the command InvariantGeometricObjectFields.

• 

If pdsolve is unable to explicitly solve the pde system for the infinitesimal symmetries, then NULL is returned.

• 

The command InfinitesimalSymmetriesOfGeometricObjectFields is part of the DifferentialGeometry:-GroupActions package. It can be used in the form InfinitesimalSymmetriesOfGeometricObjectFields(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-InfinitesimalSymmetriesOfGeometricObjectFields(...).

Examples

withDifferentialGeometry:withTensor:withGroupActions:

 

We define a manifold M with coordinates x, y, z.

J > 

DGsetupx,y,z,M

frame name: M

(2.1)

Example 1.

Find all vector fields which commute with the vector field Y = Dx.

M > 

Y:=D_x

Y:=D_x

(2.2)
M > 

InfinitesimalSymmetriesOfGeometricObjectFieldsY

_F3y,zD_x+_F2y,zD_y+_F1y,zD_z

(2.3)

 

Find all vector fields whose coefficients depend only on x which commute with the vector field Y = Dx.

M > 

InfinitesimalSymmetriesOfGeometricObjectFieldsY,coefficientvariables=x

D_x_C3+D_y_C2+D_z_C1

(2.4)

 

Example 2.

Find the infinitesimal symmetries for the metric g = dx2 +dy2 +dz2.

M > 

g:=evalDGdx &t dx+dy &t dy+dz &t dz

g:=dxdx+dydy+dzdz

(2.5)
M > 

InfinitesimalSymmetriesOfGeometricObjectFieldsg,output=list

D_yz+D_zy,D_z,D_xz+D_zx,D_xy+D_yx,D_y,D_x

(2.6)

 

Show the defining differential equations for these symmetries. Here we explicitly define the general form of the symmetry vector and specify the unknowns.

M > 

X:=evalDGRx,y,zD_x+Sx,y,zD_y+Tx,y,zD_z

X:=Rx,y,zD_x+Sx,y,zD_y+Tx,y,zD_z

(2.7)
M > 

U:=Rx,y,z,Sx,y,z,Tx,y,z

U:=Rx,y,z,Sx,y,z,Tx,y,z

(2.8)
M > 

InfinitesimalSymmetriesOfGeometricObjectFieldsg,output=pde,unknowns=U,ansatz=X

2xRx,y,z,xSx,y,z+yRx,y,z,xTx,y,z+zRx,y,z,xSx,y,z+yRx,y,z,2ySx,y,z,yTx,y,z+zSx,y,z,xTx,y,z+zRx,y,z,yTx,y,z+zSx,y,z,2zTx,y,z,0,Rx,y,z,Sx,y,z,Tx,y,z

(2.9)

 

 We can use the auxilaryequations option to find the symmetries X of the metric g for which R + S+T = 0. 

M > 

Δ:=Rx,y,z+Sx,y,z+Tx,y,z=0

Δ:=Rx,y,z+Sx,y,z+Tx,y,z=0

(2.10)
M > 

InfinitesimalSymmetriesOfGeometricObjectFieldsg,output=list,auxiliaryequations=Δ,unknowns=U,ansatz=X

D_x+D_z,yzD_x+xzD_yxyD_z,D_x+D_y

(2.11)

 

Example 3.

Find the joint infinitesimal symmetries for the 0 connection C and the volume form dx dy  dz.

M > 

C:=Connection0 &mult D_x &tensor dx &tensor dx

C:=0D_xdxdx

(2.12)
M > 

μ:=evalDGdx &w dy &w dz

μ:=dxdydz

(2.13)
M > 

InfinitesimalSymmetriesOfGeometricObjectFieldsμ,C

_C2x+_C8x_C9zy_C11_C10D_x+_C5x+_C6z+_C8y+_C7D_y+_C1x+_C2z+_C4y+_C3D_z

(2.14)

 

Example 4.

Here is a famous calculation due to E. Cartan. See Fulton and Harris Representation Theory page 357. We find the linear infinitesimal symmetries of the 3-form ω defined on the 7-manifold N with coordinates v1, v3, v4, w1,w3,w4, u.

M > 

DGsetupv1,v3,v4,w1,w3,w4,u,N

frame name: N

(2.15)
N > 

ω:=evalDGdw3 &w du &w dv3+dv4 &w du &w dw4+dw1 &w du &w dv1+2dv1 &w dv3 &w dw4+2dw1 &w dw3 &w dv4

ω:=2dv1dv3dw4+dv1dw1du+dv3dw3du+2dv4dw1dw3dv4dw4du

(2.16)
N > 

A:=Matrix7,7,i,j→a||i||j

A:=a11a12a13a14a15a16a17a21a22a23a24a25a26a27a31a32a33a34a35a36a37a41a42a43a44a45a46a47a51a52a53a54a55a56a57a61a62a63a64a65a66a67a71a72a73a74a75a76a77

(2.17)
N > 

X:=convertA,DGvector

X:=a11v1+a12v3+a13v4+a14w1+a15w3+a16w4+a17uD_v1+a21v1+a22v3+a23v4+a24w1+a25w3+a26w4+a27uD_v3+a31v1+a32v3+a33v4+a34w1+a35w3+a36w4+a37uD_v4+a41v1+a42v3+a43v4+a44w1+a45w3+a46w4+a47uD_w1+a51v1+a52v3+a53v4+a54w1+a55w3+a56w4+a57uD_w3+a61v1+a62v3+a63v4+a64w1+a65w3+a66w4+a67uD_w4+a71v1+a72v3+a73v4+a74w1+a75w3+a76w4+a77uD_u

(2.18)
N > 

vars:=convertA,set

vars:=a11,a12,a13,a14,a15,a16,a17,a21,a22,a23,a24,a25,a26,a27,a31,a32,a33,a34,a35,a36,a37,a41,a42,a43,a44,a45,a46,a47,a51,a52,a53,a54,a55,a56,a57,a61,a62,a63,a64,a65,a66,a67,a71,a72,a73,a74,a75,a76,a77

(2.19)
N > 

Y:=InfinitesimalSymmetriesOfGeometricObjectFieldsω,ansatz=X,unknowns=vars

Y:=_C1v1+_C2v3+_C3v4+_C4w3+_C5w4+_C6uD_v1_C4w1_C7v1_C8v3_C9v4u_C11w4_C10D_v3+_C1v4+_C4u_C5w1+_C6v3+_C8v4v1_C11w3_C10D_v4_C1w1+_C7w3_C9uv3_C12v4_C13w4_C11D_w1_C2w1+_C3u+_C6w4+_C8w3+v1_C12v4_C14D_w3_C1w4+_C3w1+_C8w4+_C9w3+u_C12+v1_C13+v3_C14D_w42_C3v32_C4w42_C6w12_C9v1+2v4_C122w3_C11D_u

(2.20)
N > 

c:=Tools:-DGinfoY,NonJetIndets

c:=_C1,_C2,_C3,_C4,_C5,_C6,_C7,_C8,_C9,_C10,_C11,_C12,_C13,_C14

(2.21)
N > 

Γ:=seqTools:-DGmap1,diff,Y,v,v=c

Γ:=v1D_v1+v4D_v4w1D_w1w4D_w4,v3D_v1w1D_w3,2D_uv3uD_w3+v4D_v1w1D_w4,2D_uw4+uD_v4w1D_v3+w3D_v1,w1D_v4+w4D_v1,2D_uw1+uD_v1+v3D_v4w4D_w3,v1D_v3w3D_w1,v3D_v3+v4D_v4w3D_w3w4D_w4,2D_uv1+uD_w1+v4D_v3w3D_w4,w3D_v4+w4D_v3,2D_uw3+uD_v3v1D_v4+w4D_w1,2D_uv4uD_w4v1D_w3+v3D_w1,v1D_w4+v4D_w1,v3D_w4+v4D_w3

(2.22)
N > 

nopsΓ

14

(2.23)

 

It is a simple matter to use the package LieAlgebras to check that this Lie algebra is indecomposable and simple and is a realization of the exceptional Lie algebra g2.

 

Example 5.

Find the point symmetries of the Lagrangian for the (2 +1) wave equation. The result is a 8-dimensional Lie algebra.

N > 

DGsetupx,y,t,u,J,1

frame name: J

(2.24)
J > 

λ:=evalDGu12+u22u32Dx &w Dy &w Dt

λ:=u12+u22u32DxDyDt

(2.25)
J > 

Γ:=InfinitesimalSymmetriesOfGeometricObjectFieldsλ,output=list

Γ:=2D_tt2D_xx2D_yy+D_u[]u[],D_u[],D_ty+D_yt,D_t,D_tx+D_xt,D_xy+D_yx,D_y,D_x

(2.26)
J > 

nopsΓ

8

(2.27)

 

Example 6.

Find the infinitesimal conformal symmetries of the metric g = dx2 +dy2 +dz2.  These are the vector fields X such that ℒXg = λg or ℒXg spang.

J > 

DGsetupx,y,z,M

frame name: M

(2.28)
M > 

g:=evalDGdx &t dx+dy &t dy+dz &t dz

g:=dxdx+dydy+dzdz

(2.29)

 

Note that the first argument is now a list of a list.

M > 

ConSym:=InfinitesimalSymmetriesOfGeometricObjectFieldsg,output=list

ConSym:=14y214z2+14x2D_x+12xyD_y+12xzD_z,12xzD_x+12yzD_y14z2+14x2+14y2D_z,12xD_x+12yD_y+12zD_z,12xyD_x14z214y2+14x2D_y+12yzD_z,D_yz+D_zy,D_z,D_xz+D_zx,D_xy+D_yx,D_y,D_x

(2.30)

 

The conformal symmetries of 𝔤 define a 10-dimensional Lie algebra.

M > 

nopsConSym

10

(2.31)

 

Example 7.

Find the  infinitesimal symmetries of a distribution of vector fields Δ. These are the vector fields X such that ℒX(Y)  Δ for each Y Δ.

M > 

DGsetupx1,x2,x3,x4,x5,Q

frame name: Q

(2.32)
Q > 

Δ:=evalDGD_x1+x3D_x2+x4D_x3+x43D_x5,D_x4:

Q > 

InfinitesimalSymmetriesOfGeometricObjectFieldsΔ,output=list

x2D_x2x3D_x3x4D_x43x5D_x5,12x42D_x116x5+12x3x42D_x213x43D_x315x45D_x5,x1D_x12x2D_x2x3D_x3x5D_x5,D_x5,x1D_x2+D_x3,D_x2,D_x1

(2.33)

 

Example 8.

Find the symmetries of a metric which depend upon 2 parameters α, β, where α 0.

Q > 

g:=evalDGdx &t dx+ⅇαxdy &t dy+βy+1dz &t dz

g:=dxdx+ⅇαxdydy+yβ+1dzdz

(2.34)
M > 

InfinitesimalSymmetriesOfGeometricObjectFieldsg,output=list,parameters=α,β,auxiliaryequations=α0

D_z,2D_xα+yD_y,D_y,yαD_x14y2α2ⅇαxD_y,4D_xα2yβ+1D_yβ+zD_z,D_z,α=α,β=0,α=α,β=β

(2.35)

 

Example 9.

The command InfinitesimalSymmetriesOfGeometricObjectFields can also be used to calculate the symmetries of a tensor T defined on a Lie algebra.

LD:=Library:-RetrieveWinternitz,1,4,10,alg1

LD:=e2,e3=e1,e2,e4=e3,e3,e4=e2

(2.36)