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Define a 2-dimensional manifold M..
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Example 1.
Define a pair of vector fields X1 and Y1.
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Calculate the Lie bracket of X1 and Y1.
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Let's check this result against the commutator definition of the Lie bracket acting on functions. To apply a vector field to a function we use the LieDerivative command.
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Example 2.
Here is the general coordinate formula for the Lie bracket of two vector fields defined on a 2-dimensional manifold.
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Example 3.
Two vector fields are said to commute if their Lie bracket is 0. For example:
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Example 4.
The Lie bracket satisfies the Jacobi identity [[X, Y], Z] + [[Z, X], Y] + [[Y, Z], X] = 0. For example:
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Example 5.
Use LieAlgebraData and DGsetup to initialize a Lie algebra.
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Calculate the Lie bracket of 2 vectors in this Lie algebra.
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