InfinitesimalSymmetriesOfGeometricObjectFields - Maple Help
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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

Examples

Description

• 

Let  be a manifold and let  be a list of tensor fields on . Then the Lie algebra of infinitesimal symmetries of the list of tensors  is the Lie algebra of vector fields on  such that the Lie derivatives for   

• 

If the tensors all have the same tensorial type, say then let span. Then the Lie algebra of infinitesimal symmetries of the tensor space is Lie algebra of vector fields on  such that for

• 

The command InfinitesimalSymmetriesOfGeometricObjectFields(T) calculates the Lie algebra of infinitesimal symmetries of the tensors and tensor spaces in the list T. For example, if are 4 tensor fields and T, then InfinitesimalSymmetriesOfGeometricObjectFields(T) will return the Lie algebra of vector fields such that span span .  

• 

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

• 

If the (real) Lie algebra  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 .

• 

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

 

We define a manifold with coordinates.

J > 

(2.1)

Example 1.

Find all vector fields which commute with the vector field .

M > 

(2.2)
M > 

(2.3)

 

Find all vector fields whose coefficients depend only on  which commute with the vector field .

M > 

(2.4)

 

Example 2.

Find the infinitesimal symmetries for the metric .

M > 

(2.5)
M > 

(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 > 

(2.7)
M > 

(2.8)
M > 

(2.9)

 

 We can use the auxilaryequations option to find the symmetries X of the metric g for which

M > 

(2.10)
M > 

(2.11)

 

Example 3.

Find the joint infinitesimal symmetries for the 0 connection  and the volume form .

M > 

(2.12)
M > 

(2.13)
M > 

(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  defined on the 7-manifold N with coordinates .

M > 

(2.15)
N > 

(2.16)
N > 

N > 

(2.17)
N > 

(2.18)
N > 

(2.19)
N > 

(2.20)
N > 

(2.21)
N > 

(2.22)

 

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 .

 

Example 5.

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

N > 

(2.23)
J > 

(2.24)
J > 

(2.25)
J > 

(2.26)

 

Example 6.

Find the infinitesimal conformal symmetries of the metric .  These are the vector fields  such that  or span.

J > 

(2.27)
M > 

(2.28)

 

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

M > 

(2.29)

 

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

M > 

(2.30)

 

Example 7.

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

M > 

(2.31)
Q > 

Q > 

(2.32)

 

Example 8.

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

Q > 

(2.33)
M > 

(2.34)

 

Example 9.

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

(2.35)

(2.36)
alg1 > 

(2.37)
alg1 > 

(2.38)

 

See Also

DifferentialGeometry

GroupActions

JetCalculus

Tensor

LieAlgebras

Connection

LieDerivative

DGinfo

PDEtools[Infinitesimals]

Physics[LieDerivative]

Physics

 


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