JetCalculus[GeneralizedLieBracket] - find the Lie bracket of two generalized vector fields
Calling Sequences
GeneralizedLieBracket(X, Y)
Parameters
X,Y - generalized vector fields on a fiber bundle
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Description
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Let be a fiber bundle and let be the -th jet bundle of Let be a generalized vector field of order and let be a generalized vector field of order . Then the generalized Lie bracket is the generalized vector field calculated by applying the -th prolongation of the vector to (the coefficients of) and subtracting the -th prolongation of the vector applied to (the coefficients of) , that is, .
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The command GeneralizedLieBracket(X, Y) returns the generalized vector field .
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The command GeneralizedLieBracket is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form GeneralizedLieBracket(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-GeneralizedLieBracket(...).
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Examples
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Example 1.
First initialize the jet space for 2 independent variables and 1 dependent variable and prolong it to order 4.
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Define 2 vector fields and .
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Compute the generalized Lie bracket .
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We show how this result is obtained. First prolong to the order of the coefficient in namely 2. Apply the prolonged vector field to the coefficient of
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| (2.4) |
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| (2.5) |
Next prolong to the order of the coefficient in (namely 4). Apply the prolonged vector field to the coefficient of .
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| (2.6) |
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| (2.7) |
The difference between term1 and term2 gives the coefficient of the generalized Lie bracket .
Example 2.
The generalized Lie bracket is not restricted to evolutionary (vertical) generalized vector fields.
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| (2.9) |
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| (2.11) |
Example 3.
The generalized Lie bracket for a pair of 1st order evolutionary vector fields coincides with the Jacobi bracket. For example:
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| (2.14) |
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