JetCalculus[GeneralizedLieBracket] - find the Lie bracket of two generalized vector fields
Calling Sequences
GeneralizedLieBracket(X, Y)
Parameters
X,Y - generalized vector fields on a bundle E-> M
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Description
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Let X be a generalized vector field of order k and let Y be a generalized vector field of order l. Then the generalized Lie bracket of X with Y is calculated by applying the l-th prolongation of the vector X to (the coefficients of) Y and subtracting the k-th prolongation of the vector Y applied to (the coefficients of) X, that is, pr^l(X)(Y) - pr^k(Y)(X).
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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 2 dependent variables and prolong it to order 4.
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Define 2 vector fields X1 and Y1.
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E1 >
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Compute the generalized Lie bracket of X1 and Y1.
E1 >
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| (2.3) |
We show how this result is obtained. First prolong X1 to the order of the coefficient in Y1, namely 2. Apply the prolonged vector field to the coefficient of Y1.
E1 >
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| (2.4) |
E1 >
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Next prolong Y1 to the order of the coefficient in X1, namely 4. Apply the prolonged vector field to the coefficient of Y1.
E1 >
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| (2.6) |
E1 >
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| (2.7) |
The difference between term1 and term2 gives the coefficient of the generalized Lie bracket of X1 and Y1.
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| (2.8) |
Example 2.
The generalized Lie bracket is not restricted to evolutionary (vertical) generalized vector fields.
E1 >
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E1 >
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| (2.10) |
E1 >
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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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E1 >
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E1 >
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| (2.12) |
E1 >
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| (2.13) |
E1 >
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| (2.14) |
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