Example 1.
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![[["Doubrov", 1], ["Gong", 1], ["Gonzalez-Lopez", 1], ["HawkingEllis", 1], ["Kamke", 1], ["Morozov", 1], ["Mubarakyzanov", 1], ["Mubarakyzanov", 2], ["Mubarakyzanov", 3], ["Olver", 1], ["Petrov", 1], ["Stephani", 1], ["Turkowski", 1], ["Turkowski", 2], ["USU", 2], ["Winternitz", 1]]](/support/helpjp/helpview.aspx?si=6657/file05778/math117.png)
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Example 2.
We retrieve a 5-dimensional Lie algebra appearing in the paper by \011J. Patera, R. T. Sharp, and P. Winternitz, Invariants of real low dimensional Lie algebras, Journal of Mathematical Physics, Vol 17, No 6 (1976), 966--994.
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At this point we can immediately initialize this Lie algebra and perform computations. For example, using the LeviDecomposition command we find that the algebra [5, 40] in ["Winternitz", 1] admits a non-trivial Levi decomposition with a 2 dimensional radical [e4, e5] and a semi-simple subalgebra [e1, e2, e3].
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Alg >
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Example 3.
We retrieve a 5 dimensional Lie algebra of vector fields in the plane appearing in the paper by A. Gonzalez-Lopex, N. Kamran and P. J. Olver, Lie algebras of vector fields in the real plane , Proc. London Math Soc. Vol 64 (1992), 339--368
Alg >
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Alg >
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We can immediately do computations with this 5-dimensional Lie algebra of vector fields. For example, let us prolong these vector fields to the 1st order jet spaces J^1(R, R) with the Prolong command and compute the isotropy subalgebra (with the IsotropySubalgebra command) for this prolonged infinitesimal action at a generic point [ x = a, y[ ] = b, y[1] = c].
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![[_DG([["vector", M, []], [[[1], (-y[]*b+2*y[]*a*c-a*c^2*x-2*b*c*a+a^2*c^2+b^2)/(-2*b*c*a+a^2*c^2+b^2)], [[2], c^2*(-b*x+a*y[])/(-2*b*c*a+a^2*c^2+b^2)], [[3], (-b*c^2+y[1]^2*b-2*y[1]^2*a*c+2*a*c^2*y[1])/(-2*b*c*a+a^2*c^2+b^2)]]]), _DG([["vector", M, []], [[[1], -(-b*x+a*y[])/(-2*b*c*a+a^2*c^2+b^2)], [[2], (-a*c^2*x+2*c*x*b-y[]*b-2*b*c*a+a^2*c^2+b^2)/(-2*b*c*a+a^2*c^2+b^2)], [[3], -(a*c^2-2*b*c-a*y[1]^2+2*y[1]*b)/(-2*b*c*a+a^2*c^2+b^2)]]])]](/support/helpjp/helpview.aspx?si=6657/file05778/math227.png)
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Since the infinitesimal isotropy of this 5 dimension Lie algebra of vector fields is 2 dimensional, the generic orbits for the corresponding group action are 3 dimensional and therefore, since J^1(R, R) has dimension 3, there are no differential invariants for this action on the 1-jets.
Example 4.
The DifferentialGeometry Library contains the lists of ordinary differential equations from the book by Kamke. Let us retrieve one such equation, use the procedure PDEtools:-Infinitesimals to find the infinitesimal symmetries of this equation, and then use the LieAlgebras command LieAlgebraData to calculate the structure equations for this Lie algebra.
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 = x, _eta[1](x, y) = 0], [_xi[1](x, y) = 1, _eta[1](x, y) = 0], [_xi[1](x, y) = 0, _eta[1](x, y) = y], [_xi[1](x, y) = 0, _eta[1](x, y) = y^(3/2)], [_xi[1](x, y) = 0, _eta[1](x, y) = x*y^(3/2)], [_xi[1](x, y) = 0, _eta[1](x, y) = (1/2)*x^2*y^(3/2)], [_xi[1](x, y) = (1/2)*x^2, _eta[1](x, y) = -2*x*y]](/support/helpjp/helpview.aspx?si=6657/file05778/math271.png)
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We use the convert command with the keyword DGvector to convert the output of the PDEtools:-Infinitesimals programs to the Differential Geometry vector field format.
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![_DG([["LieAlgebra", K7_8, [7]], [[[1, 2, 2], -1], [[1, 5, 5], 1], [[1, 6, 6], 2], [[1, 7, 7], 1], [[2, 5, 4], 1], [[2, 6, 5], 1], [[2, 7, 1], 1], [[2, 7, 3], -2], [[3, 4, 4], 1/2], [[3, 5, 5], 1/2], [[3, 6, 6], 1/2], [[4, 7, 5], 1], [[5, 7, 6], 1]]])](/support/helpjp/helpview.aspx?si=6657/file05778/math296.png)
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Let us find the Levi decomposition for this 6 dimensional symmetry algebra.
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K7_8 >
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Example 5.
The DifferentialGeometry Library contains detailed information on some of the space-time metrics found in the books by Hawking and Ellis and Stephani, Kramer et. al.
First define a spacetime manifold.
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Retrieve the metric and other fields (if present):
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![[_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], -exp(x)*cos(3^(1/2)*x)/_k^2], [[1, 4], -exp(x)*sin(3^(1/2)*x)/_k^2], [[2, 2], 1/_k^2], [[3, 3], exp(-2*x)/_k^2], [[4, 1], -exp(x)*sin(3^(1/2)*x)/_k^2], [[4, 4], exp(x)*cos(3^(1/2)*x)/_k^2]]])]](/support/helpjp/helpview.aspx?si=6657/file05778/math353.png)
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Retrieve a null tetrad for the space-time.
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Retrieve the Petrov type.
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Retrieve the Killing vectors
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The keyword argument output accepts a list of table indices chosen from ["Authors", "BasePoints", "Comments", "Coordinates", "CosmologicalConstant", "Domains", "Fields", "IsometryDimension , "KillingEquations", "KillingVectors", "NewtonConstant", "NullTetrad", "OrthonormalTetrad", "Parameters", "PetrovType", "PlebanskiPetrovType", "PrimaryDescription", "OrbitDimension", "OrbitType", "Reference", "SegreType", "SideConditionsAssuming", "SideConditionsSimplify", "SecondaryDescription", "TertiaryDescription" ].
Example 6.
Many table entries in the DifferentialGeometry library contain arbitrary parameters. These parameters can be assigned specific values by first adding the optional argument parameters = "yes" to the calling sequence for the Retrieve command and using the elements of the returned parameter list in conjunction with the eval command.
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This Lie algebra depends upon parameters [p, q, s]. Use the following command to assign the values p =3, q = 7, s= K in the structure equations L.
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