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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 fiber bundle

 

Description

Examples

Description

• 

 Let  π : EM be a fiber bundle and let πk: JkEM  be the k-th jet bundle of E. 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 X, Ygen is the generalized vector field 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, X, Ygen =  prℓXY  prkYX.

• 

The command GeneralizedLieBracket(X, Y) returns the generalized vector field X, Ygen.

• 

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

withDifferentialGeometry:withJetCalculus:

 

Example 1.

First initialize the jet space for 2 independent variables x, y and 1 dependent variable u and prolong it to order 4.

DGsetupx,y,u,E1,4:

 

Define 2 vector fields X1 and Y1.

E1 > 

X1u1,2,2,22 &mult D_u[]

X1u1,2,2,22D_u

(2.1)
E1 > 

Y1u2,2 &mult D_u[]

Y1u2,2D_u

(2.2)

 

Compute the generalized Lie bracket X1, Y1gen.

E1 > 

Z1GeneralizedLieBracketX1,Y1

Z12u1,2,2,2,22D_u

(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 X1

E1 > 

prX1ProlongX1,2

prX1u1,2,2,22D_u+2u1,2,2,2u1,1,2,2,2D_u1+2u1,2,2,2u1,2,2,2,2D_u2+2u1,2,2,2u1,1,1,2,2,2+2u1,1,2,2,22D_u1,1+2u1,2,2,2u1,1,2,2,2,2+2u1,2,2,2,2u1,1,2,2,2D_u1,2+2u1,2,2,2u1,2,2,2,2,2+2u1,2,2,2,22D_u2,2

(2.4)
E1 > 

term1LieDerivativeprX1,u2,2

term12u1,2,2,2u1,2,2,2,2,2+2u1,2,2,2,22

(2.5)

 

Next prolong Y1 to the order of the coefficient in X1 (namely 4). Apply the prolonged vector field to the coefficient of Y1.

E1 > 

prY1ProlongY1,4

prY1u2,2D_u+u1,2,2D_u1+u2,2,2D_u2+u1,1,2,2D_u1,1+u1,2,2,2D_u1,2+u2,2,2,2D_u2,2+u1,1,1,2,2D_u1,1,1+u1,1,2,2,2D_u1,1,2+u1,2,2,2,2D_u1,2,2+u2,2,2,2,2D_u2,2,2+u1,1,1,1,2,2D_u1,1,1,1+u1,1,1,2,2,2D_u1,1,1,2+u1,1,2,2,2,2D_u1,1,2,2+u1,2,2,2,2,2D_u1,2,2,2+u2,2,2,2,2,2D_u2,2,2,2

(2.6)
E1 > 

term2LieDerivativeprY1,u1,2,2,22

term22u1,2,2,2u1,2,2,2,2,2

(2.7)

 

The difference between term1 and term2 gives the coefficient of the generalized Lie bracket X1, Y1gen.

E1 > 

term1term2

2u1,2,2,2,22

(2.8)

 

Example 2.

The generalized Lie bracket is not restricted to evolutionary (vertical) generalized vector fields.

E1 > 

X2evalDGu2xD_x+u12D_u[]

X2u2xD_x+u12D_u

(2.9)
E1 > 

Y2evalDGu1,2D_x+yD_u[]

Y2u1,2D_x+yD_u

(2.10)
E1 > 

GeneralizedLieBracketX2,Y2

xu1,1u2,2+xu1,22+u1u2,22u1,1,2u1+2u1,2u22u1,1u1,2+xD_x+2u1,1,2u12D_u

(2.11)

 

Example 3.

The generalized Lie bracket for a pair of 1st order evolutionary vector fields coincides with the Jacobi bracket. For example:

E1 > 

varsx,y,u1,u2:

E1 > 

PDEtools[declare]Fvars,Gvars,quiet

E1 > 

X3Fvars &mult D_u[]

X3FD_u

(2.12)
E1 > 

Y3Gvars &mult D_u[]

Y3GD_u

(2.13)
E1 > 

GeneralizedLieBracketX3,Y3

Gu1Fx+Gu2FyGxFu1GyFu2D_u

(2.14)

See Also

DifferentialGeometry

JetCalculus

AssignVectorType

LieBracket

LieDerivative

Prolong

 


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