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

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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

•	This command uses pdsolve to find all geometric objects t in the span of T (over the functions on M) which satisfy LieDerivative(X, t) = 0 for all X in Gamma.

•	If T = [1], then the Gamma invariant functions are computed.

•	If connection = "yes", then Gamma 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 LieDerivative(X, t) = 0 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 = {a_1, a_2 ... , a_k} 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 M, N, J with coordinates [x, y, z], [x, y] and [x], [u].

Example 1.

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

J >

(2.1)

M >

(2.2)

Invariant Functions:

M >

(2.3)

M >

(2.4)

Invariant 1-forms:

M >

(2.5)

M >

[_DG([["form", M, 1], [[[1], x*(-z^2)^(1/2)/z], [[2], y*(-z^2)^(1/2)/z], [[3], (-z^2)^(1/2)]]])]

(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 vector field Gamma3.

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 >

[diff(A(x, y, z), x), diff(B(x, y, z), x), diff(C(x, y, z), x), exp(-y)*(diff(A(x, y, z), z)), exp(-y)*(diff(B(x, y, z), z)), exp(-y)*(B(x, y, z)+diff(C(x, y, z), z)), -A(x, y, z)+x*(diff(A(x, y, z), x))+diff(A(x, y, z), y), x*(diff(B(x, y, z), x))+diff(B(x, y, z), y), x*(diff(C(x, y, z), x))+diff(C(x, y, z), y)], [A(x, y, z), B(x, y, z), C(x, y, z)]

(2.16)

Find the vector fields of the special form Z = a(x)*D_x + b(x, y)*D_y + c(x, y, z)*D_z which commute with Gamma3.

M >

(2.17)

M >

(2.18)

Example 3.

Find the 2nd and 3rd order differential invariants for the infinitesimal Euclidean group acting on the [x, u] plane.

M >

J >

(2.19)

J >

(2.20)

J >

_F1(u[1, 1]/(1+u[1]^2)^(3/2))

(2.21)

J >

[_DG([["vector", J, ["projectable", 3]], [[[1], 1]]]), _DG([["vector", J, ["projectable", 3]], [[[2], 1]]]), _DG([["vector", J, ["point", 3]], [[[1], u[]], [[2], -x], [[3], -1-u[1]^2], [[4], -3*u[1]*u[1, 1]], [[5], -3*u[1, 1]^2-4*u[1]*u[1, 1, 1]]]])]

(2.22)

J >

_F1(u[1, 1]/(1+u[1]^2)^(3/2), (u[1, 1, 1]*u[1]^2-3*u[1, 1]^2*u[1]+u[1, 1, 1])/(1+u[1]^2)^3)

(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 >

_DG([["biform", J, [1, 1]], [[[1, 2], _F1(u[1, 1]/(1+u[1]^2)^(3/2))]]])

(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 >

$_DG([["form", N, 1], [[[2], _F1(y)]]]), _DG([["form", N, 1], [[[1], _C2*y^(-1/alpha)], [[2], _C1/y]]]), [{alpha = 0}, {alpha = alpha}]$

(2.31)

N >

$[_DG([["form", N, 1], [[[2], 1]]])], [_DG([["form", N, 1], [[[2], 1/y]]]), _DG([["form", N, 1], [[[1], y^(-1/alpha)]]])], [{alpha = 0}, {alpha = alpha}]$