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Define manifolds with coordinates and .
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Example 1.
Find all invariant functions, 1-forms, metrics and invariant type [1, 1] tensors for the infinitesimal group of rotations on
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| (2.2) |
Invariant Functions:
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| (2.3) |
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Invariant 1-forms:
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| (2.5) |
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| (2.6) |
Note that the format of the answer can be improved with the assuming command.
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| (2.7) |
Invariant Metrics:
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| (2.8) |
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| (2.9) |
Invariant [1, 1] Tensors:
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| (2.10) |
| (2.11) |
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| (2.12) |
Example 2.
Find the vector fields which commute with the Lie algebra of vector fields .
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| (2.13) |
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| (2.14) |
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| (2.15) |
Give the partial differential equations which were solved to calculate the commuting vectors in the list Z.
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| (2.16) |
Find the vector fields of the special form + c(x)D_z which commute with .
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| (2.17) |
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| (2.18) |
Example 3.
Find the second and third order differential invariants for the infinitesimal Euclidean group acting on the plane.
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| (2.19) |
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| (2.20) |
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| (2.21) |
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| (2.22) |
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| (2.23) |
Find the invariant Lagrangians on the 1-jet.
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Find the invariant "source" forms on the 2-jet.
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| (2.27) |
Example 4.
Find the invariant 1-forms for a list of vector fields depending on a parameter alpha.
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| (2.29) |
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| (2.31) |
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| (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.
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| (2.33) |
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| (2.35) |
Find the symmetric rank 2 tensors on alg1 which are invariant with respect to the subalgebra spanned by
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| (2.36) |