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GroupActions[InvariantGeometricObjectFields] - find the vector fields, differential forms, tensors or connections which are invariant with respect to a Lie algebra of vector fields

Calling Sequences

     InvariantGeometricObjectFields(Gamma, T, options)

Parameters

     Gamma     - a list of vector fields on a manifold M.

     T         - a list of vector fields, differential forms, or tensors on M

     options   - output = "list", output = "pde", connection = "yes"/"no", coefficientvariables = [x1, x2, ...], unknowns = [F1, F2, ...], ansatz = t , parameters = P

 

Description

Examples

Description

 

• 

Let Γ = X1, X2, ... , Xr be a r-dimensional Lie algebra of vector fields on a manifold M. A vector field, differential form, tensor or connection S is said to be Γ-invariant if the Lie derivative ℒXiS = 0 (*) for i = 1,2, ... , r.

• 

The procedure InvariantGeometricObjectFields(Gamma, T) calculates the Γinvariant geometric object fields which are in the span (over the functions on M) of the geometric object fields given by the second argument T.

• 

The procedure creates the general linear combination S of the tensors in T (with coefficients which are functions of the coordinates on M) and then generates the system of first order PDE for the coefficients arising from the invariance conditions (*).The command pdsolve is used to solve these PDE.

• 

If T = [1], then the Γ-invariant functions on M are computed.

• 

If connection = "yes", then Γ invariant connections are computed.

• 

With output = "list", the program returns a basis for the invariant tensors, over the ring of invariant functions. This option is not available when connection = "yes".

• 

With output = "pde", the pde system defined by the equations (*) is returned.

• 

The exact form for the geometric object fields can be specified by ansatz = t. With this option, the unknown functions in t must be explicitly listed with the unknowns option.

• 

If P = {a1, a2, ... } is a set of parameters appearing in Gamma, then the optional argument parameters = P will invoke the case splitting capabilities of pdsolve. Exceptional parameter values will be determined and a sequence of lists of invariant geometry object fields, 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 defined by LieDerivative(X, t) = 0, then NULL is returned.  

• 

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

Examples

withDifferentialGeometry:withGroupActions:withJetCalculus:withTensor:

 

Define manifolds M, N, J   with coordinates x, y, z, x,y and x, u.

DGsetupx,y,z,M:DGsetupx,y,N:DGsetupx,u,J,2:

 

Example 1.

Find all invariant functions, 1-forms, metrics and invariant type [1, 1] tensors for the infinitesimal group of rotations on M

J > 

ChangeFrameM

J

(2.1)
M > 

Γ1evalDGxD_yyD_x,xD_zzD_x,yD_zzD_y

Γ1:=D_xy+D_yx,D_xz+D_zx,D_yz+D_zy

(2.2)

 

Invariant Functions:

M > 

InvariantGeometricObjectFieldsΓ1,1

_F1x2+y2+z2

(2.3)
M > 

InvariantGeometricObjectFieldsΓ1,1,output=list

x2+y2+z2

(2.4)

 

Invariant 1-forms:

M > 

Tdx,dy,dz

T:=dx,dy,dz

(2.5)
M > 

InvariantGeometricObjectFieldsΓ1,T,output=list

z2xdxz+z2ydyz+z2dz

(2.6)

 

Note that the format of the answer can be improved with the assuming command.

M > 

InvariantGeometricObjectFields&Gamma;1&comma;T&comma;output&equals;listassuming0<z

xdxz&plus;ydyz&plus;dz

(2.7)

 

Invariant Metrics:

M > 

TGenerateSymmetricTensorsdx&comma;dy&comma;dz&comma;2

T:=dxdx&comma;12dxdy&plus;12dydx&comma;12dxdz&plus;12dzdx&comma;dydy&comma;12dydz&plus;12dzdy&comma;dzdz

(2.8)
M > 

InvariantGeometricObjectFields&Gamma;1&comma;T&comma;output&equals;list

dxdx&plus;dydy&plus;dzdz&comma;xzdxdz&plus;x2dxdx&plus;xydxdy&plus;xydydx&plus;y2dydy&plus;zydydz&plus;xzdzdx&plus;zydzdy&plus;z2dzdz

(2.9)

 

Invariant [1, 1] Tensors:

M > 

TGenerateTensorsdx&comma;dy&comma;dz&comma;D_x&comma;D_y&comma;D_z

T:=dxD_x&comma;dxD_y&comma;dxD_z&comma;dyD_x&comma;dyD_y&comma;dyD_z&comma;dzD_x&comma;dzD_y&comma;dzD_z

(2.10)
M > 

_EnvExplicittrue

_EnvExplicit:=true

(2.11)
M > 

InvInvariantGeometricObjectFields&Gamma;1&comma;T&comma;output&equals;listassuming0<z

Inv:=D_xdx&plus;D_ydy&plus;D_zdz&comma;D_xdxzD_ydyz&plus;D_zdxx&plus;D_zdyy&comma;D_xdyz&plus;D_ydxzD_zdxy&plus;D_zdyx

(2.12)

 

Example 2.

Find the vector fields which commute with the Lie algebra of vector fields &Gamma;2.

M > 

ChangeFrameM&colon;

M > 

&Gamma;2D_x&comma;&ExponentialE;yD_z&comma;xD_x&plus;D_y

&Gamma;2:=D_x&comma;&ExponentialE;yD_z&comma;D_xx&plus;D_y

(2.13)
M > 

TD_x&comma;D_y&comma;D_z

T:=D_x&comma;D_y&comma;D_z

(2.14)
M > 

ZInvariantGeometricObjectFields&Gamma;2&comma;T&comma;output&equals;list

Z:=D_zzD_y&comma;D_z&comma;&ExponentialE;yD_x

(2.15)

 

Give the partial differential equations which were solved to calculate the commuting vectors in the list Z.

J > 

InvariantGeometricObjectFields&Gamma;2&comma;T&comma;unknowns&equals;Ax&comma;y&comma;z&comma;Bx&comma;y&comma;z&comma;Cx&comma;y&comma;z&comma;output&equals;pde

xAx&comma;y&comma;z&comma;xBx&comma;y&comma;z&comma;xCx&comma;y&comma;z&comma;&ExponentialE;yzAx&comma;y&comma;z&comma;&ExponentialE;yzBx&comma;y&comma;z&comma;&ExponentialE;yBx&comma;y&comma;z&plus;zCx&comma;y&comma;z&comma;Ax&comma;y&comma;z&plus;xxAx&comma;y&comma;z&plus;yAx&comma;y&comma;z&comma;xxBx&comma;y&comma;z&plus;yBx&comma;y&comma;z&comma;xxCx&comma;y&comma;z&plus;yCx&comma;y&comma;z&comma;0&comma;Ax&comma;y&comma;z&comma;Bx&comma;y&comma;z&comma;Cx&comma;y&comma;z

(2.16)

 

Find the vector fields of the special form Z &equals; axD_x &plus; bxD_y  + c(x)D_z which commute with &Gamma;2.

M > 

ZevalDGaxD_x&plus;bx&comma;yD_y&plus;cx&comma;y&comma;zD_z

Z:=axD_x&plus;bx&comma;yD_y&plus;cx&comma;y&comma;zD_z

(2.17)
M > 

InvariantGeometricObjectFields&Gamma;2&comma;Z&comma;unknowns&equals;ax&comma;bx&comma;y&comma;cx&comma;y&comma;z&comma;output&equals;list

D_zzD_y&comma;D_z

(2.18)

 

Example 3.

Find the second and third order differential invariants for the infinitesimal Euclidean group acting on the x&comma; u plane.

M > 

ChangeFrameJ&colon;

J > 

&Gamma;3evalDGD_x&comma;D_u&lsqb;&rsqb;&comma;u&lsqb;&rsqb;D_xxD_u&lsqb;&rsqb;

&Gamma;3:=D_x&comma;D_u&lsqb;&rsqb;&comma;D_xu&lsqb;&rsqb;xD_u&lsqb;&rsqb;

(2.19)
J > 

Gamma3amapProlong&comma;&Gamma;3&comma;2

Gamma3a:=D_x&comma;D_u&lsqb;&rsqb;&comma;u&lsqb;&rsqb;D_xxD_u&lsqb;&rsqb;u12&plus;1D_u13u1u1&comma;1D_u1&comma;1

(2.20)
J > 

InvariantGeometricObjectFieldsGamma3a&comma;1&comma;coefficientvariables&equals;x&comma;u&lsqb;&rsqb;&comma;u1&comma;u1&comma;1

_F1u1&comma;1u12&plus;13&sol;2

(2.21)
J > 

Gamma2bmapProlong&comma;Gamma3a&comma;3

Gamma2b:=D_x&comma;D_u&lsqb;&rsqb;&comma;u&lsqb;&rsqb;D_xxD_u&lsqb;&rsqb;u12&plus;1D_u13u1u1&comma;1D_u1&comma;14u1u1&comma;1&comma;1&plus;3u1&comma;12D_u1&comma;1&comma;1

(2.22)
J > 

InvariantGeometricObjectFieldsGamma2b&comma;1

_F1u1&comma;1u12&plus;13&sol;2&comma;u12u1&comma;1&comma;13u1u1&comma;12&plus;u1&comma;1&comma;1u12&plus;13

(2.23)

 

Find the invariant Lagrangians on the 1-jet.

J > 

SDx

S:=Dx

(2.24)
J > 

InvariantGeometricObjectFieldsGamma2b&comma;S&comma;coefficientvariables&equals;x&comma;u&lsqb;&rsqb;&comma;u1

_C1u12&plus;1Dx

(2.25)

 

Find the invariant "source" forms on the 2-jet.

J > 

SDx &wedge Cu&lsqb;&rsqb;

S:=DxCu&lsqb;&rsqb;

(2.26)
J > 

InvariantGeometricObjectFieldsGamma2b&comma;S&comma;coefficientvariables&equals;x&comma;u&lsqb;&rsqb;&comma;u1&comma;u1&comma;1

_F1u1&comma;1u12&plus;13&sol;2DxCu&lsqb;&rsqb;

(2.27)

 

Example 4.

Find the invariant 1-forms for a list of vector fields &Gamma;4 depending on a parameter alpha.

J > 

ChangeFrameN

J

(2.28)
N > 

&Gamma;4evalDGD_x&comma;xD_x&plus;&alpha;yD_y

&Gamma;4:=D_x&comma;D_yy&alpha;&plus;D_xx

(2.29)
N > 

Fdx&comma;dy

F:=dx&comma;dy

(2.30)
N > 

InvariantGeometricObjectFields&Gamma;4&comma;F&comma;parameters&equals;&alpha;

_F1ydy&comma;_C2y1&alpha;dx&plus;_C1dyy&comma;&alpha;&equals;0&comma;&alpha;&equals;&alpha;

(2.31)
N > 

InvariantGeometricObjectFields&Gamma;4&comma;F&comma;output&equals;list&comma;parameters&equals;&alpha;

dy&comma;dyy&comma;y1&alpha;dx&comma;&alpha;&equals;0&comma;&alpha;&equals;&alpha;

(2.32)

 

Example 5.

The command InvariantGeometricObjectFields can also be used to calculate tensors on a Lie algebra which are invariant with respect to a subalgebra.

 

 Retrieve a Lie algebra from the DifferentialGeometry library.

LDLibrary:-RetrieveWinternitz&comma;1&comma;4&comma;10&comma;alg1

LD:=e2&comma;e3&equals;e1&comma;e2&comma;e4&equals;e3&comma;e3&comma;e4&equals;e2

(2.33)

DGsetupLD

Lie algebra: alg1

(2.34)
alg1 > 

SGenerateSymmetricTensors&theta;1&comma;&theta;2&comma;&theta;3&comma;&theta;4&comma;2

S:=&theta;1&theta;1&comma;12&theta;1&theta;2&plus;12&theta;2&theta;1&comma;12&theta;1&theta;3&plus;12&theta;3&theta;1&comma;12&theta;1&theta;4&plus;12&theta;4&theta;1&comma;&theta;2&theta;2&comma;12&theta;2&theta;3&plus;12&theta;3&theta;2&comma;12&theta;2&theta;4&plus;12&theta;4&theta;2&comma;&theta;3&theta;3&comma;12&theta;3&theta;4&plus;12&theta;4&theta;3&comma;&theta;4&theta;4

(2.35)

 

Find the symmetric rank 2 tensors on alg1 which are invariant with respect to the subalgebra spanned by e1&comma; e2&period; 

alg1 > 

InvariantGeometricObjectFieldse1&comma;e2&comma;S&comma;output&equals;list

12&theta;1&theta;4&plus;12&theta;3&theta;3&plus;12&theta;4&theta;1&comma;&theta;2&theta;2&comma;12&theta;2&theta;4&plus;12&theta;4&theta;2&comma;&theta;4&theta;4

(2.36)

See Also

DifferentialGeometry

GroupActions

JetCalculus

Tensor

GenerateForms

GenerateSymmetricTensors

GenerateTensors

pdsolve

LieDerivative

 


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