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
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Description
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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.
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If T = [1], then the Gamma invariant functions are computed.
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If connection = "yes", then Gamma invariant connections are computed.
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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".
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With output = "pde", the pde system defined by the equations LieDerivative(X, t) = 0 is returned.
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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.
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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.
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Other optional arguments for pdsolve may be passed through the command InvariantGeometricObjectFields.
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If pdsolve is unable to explicitly solve the pde system defined by LieDerivative(X, t) = 0, then NULL is returned.
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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(...).
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Examples
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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 >
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M >
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Invariant Functions:
M >
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M >
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Invariant 1-forms:
M >
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M >
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Note that the format of the answer can be improved with the assuming command.
M >
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Invariant Metrics:
M >
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M >
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![[_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], y^2+z^2], [[1, 2], -y*x], [[1, 3], -z*x], [[2, 1], -y*x], [[2, 2], x^2+z^2], [[2, 3], -z*y], [[3, 1], -z*x], [[3, 2], -z*y], [[3, 3], x^2+y^2]]]), _DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], 1], [[2, 2], 1], [[3, 3], 1]]])]](/support/helpjp/helpview.aspx?si=6642/file05743/math237.png)
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Invariant [1, 1] Tensors:
M >
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M >
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M >
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![[_DG([["tensor", M, [["cov_bas", "con_bas"], []]], [[[1, 1], 1], [[2, 2], 1], [[3, 3], 1]]]), _DG([["tensor", M, [["cov_bas", "con_bas"], []]], [[[1, 1], -z], [[1, 3], x], [[2, 2], -z], [[2, 3], y]]]), _DG([["tensor", M, [["cov_bas", "con_bas"], []]], [[[1, 2], z], [[1, 3], -y], [[2, 1], -z], [[2, 3], x]]])]](/support/helpjp/helpview.aspx?si=6642/file05743/math262.png)
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Example 2.
Find the vector fields which commute with the vector field Gamma3.
M >
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M >
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M >
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| (2.14) |
M >
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Give the partial differential equations which were solved to calculate the commuting vectors in the list Z.
J >
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![[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)]](/support/helpjp/helpview.aspx?si=6642/file05743/math306.png)
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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 >
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M >
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Example 3.
Find the 2nd and 3rd order differential invariants for the infinitesimal Euclidean group acting on the [x, u] plane.
M >
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J >
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J >
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J >
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J >
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![[_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]]]])]](/support/helpjp/helpview.aspx?si=6642/file05743/math368.png)
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J >
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Find the invariant Lagrangians on the 1-jet.
J >
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J >
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Find the invariant "source" forms on the 2-jet.
J >
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J >
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Example 4.
Find the invariant 1-forms for a list of vector fields depending on a parameter alpha.
J >
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N >
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N >
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N >
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N >
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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.
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alg1 >
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alg1 >
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Find the symmetric rank 2 tensors on alg1 which are invariant with respect to [e1, e2].
alg1 >
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