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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
Description
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).
For applications to the generalized symmetries of integrable evolution equations such as the KdV equation, see the tutorial titled Recursion Operators For Integrable Evolution Equations.
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(...).
Examples
Example 1.
First initialize the jet space for 2 independent variables and 2 dependent variables and prolong it to order 4.
Define 2 vector fields X1 and Y1.
Compute the generalized Lie bracket of X1 and Y1.
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.
Next prolong Y1 to the order of the coefficient in X1, namely 4. Apply the prolonged vector field to the coefficient of Y1.
The difference between term1 and term2 gives the coefficient of the generalized Lie bracket of X1 and Y1.
Example 2.
The generalized Lie bracket is not restricted to evolutionary (vertical) generalized vector fields.
Example 3.
The generalized Lie bracket for a pair of 1st order evolutionary vector fields coincides with the Jacobi bracket. For example:
See Also
DifferentialGeometry, JetCalculus, AssignVectorType, LieBracket, LieDerivative, Prolong
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