InvariantGeometricObjectFields - Maple Help
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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 .

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

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

 

Description

Examples

Description

 

• 

Let be a -dimensional Lie algebra of vector fields on a manifold . A vector field, differential form, tensor or connection  is said to be -invariant if the Lie derivative(*) for.

• 

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

• 

The procedure creates the general linear combination of the tensors in T (with coefficients which are functions of the coordinates on 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 -invariant functions on are computed.

• 

If connection = "yes", then 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

 

Define manifolds   with coordinates  and .

 

Example 1.

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

J > 

(2.1)
M > 

(2.2)

 

Invariant Functions:

M > 

(2.3)
M > 

(2.4)

 

Invariant 1-forms:

M > 

(2.5)
M > 

(2.6)

 

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

M > 

(2.7)

 

Invariant Metrics:

M > 

(2.8)
M > 

(2.9)

 

Invariant [1, 1] Tensors:

M > 

(2.10)
M > 

(2.11)
M > 

(2.12)

 

Example 2.

Find the vector fields which commute with the Lie algebra of vector fields .

M > 

M > 

(2.13)
M > 

(2.14)
M > 

(2.15)

 

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

J > 

(2.16)

 

Find the vector fields of the special form  + c(x)D_z which commute with .

M > 

(2.17)
M > 

(2.18)

 

Example 3.

Find the second and third order differential invariants for the infinitesimal Euclidean group acting on the plane.

M > 

J > 

(2.19)
J > 

(2.20)
J > 

(2.21)
J > 

(2.22)
J > 

(2.23)

 

Find the invariant Lagrangians on the 1-jet.

J > 

(2.24)
J > 

(2.25)

 

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

J > 

(2.26)
J > 

(2.27)

 

Example 4.

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

J > 

(2.28)
N > 

(2.29)
N > 

(2.30)
N > 

(2.31)
N > 

(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.

(2.33)

(2.34)
alg1 > 

(2.35)

 

Find the symmetric rank 2 tensors on alg1 which are invariant with respect to the subalgebra spanned by  

alg1 > 

(2.36)

See Also

DifferentialGeometry

GroupActions

JetCalculus

Tensor

GenerateForms

GenerateSymmetricTensors

GenerateTensors

pdsolve

LieDerivative

 


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