GeneralizedLieBracket - Maple Help
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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

 • Let be a fiber bundle and let  be the $k$-th jet bundle of Let  be a generalized vector field of order and let be a generalized vector field of order $\mathrm{ℓ}$. Then the generalized Lie bracket  is the generalized vector field calculated by applying the $\mathrm{ℓ}$-th prolongation of the vector  to (the coefficients of) and subtracting the $k$-th prolongation of the vector $Y$ applied to (the coefficients of) $X$, that is, .
 • The command GeneralizedLieBracket(X, Y) returns the generalized vector field .
 • 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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{JetCalculus}\right):$

Example 1.

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

 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u\right],\mathrm{E1},4\right):$

Define 2 vector fields $\mathrm{X1}$ and $\mathrm{Y1}$.

 E1 > $\mathrm{X1}≔\left({u\left[1,2,2,2\right]}^{2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_u}\left[\right]$
 ${\mathrm{X1}}{≔}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}^{{2}}{}{{\mathrm{D_u}}}_{\left[\right]}$ (2.1)
 E1 > $\mathrm{Y1}≔u\left[2,2\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_u}\left[\right]$
 ${\mathrm{Y1}}{≔}{{u}}_{{2}{,}{2}}{}{{\mathrm{D_u}}}_{\left[\right]}$ (2.2)

Compute the generalized Lie bracket .

 E1 > $\mathrm{Z1}≔\mathrm{GeneralizedLieBracket}\left(\mathrm{X1},\mathrm{Y1}\right)$
 ${\mathrm{Z1}}{≔}{2}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}{,}{2}}^{{2}}{}{{\mathrm{D_u}}}_{\left[\right]}$ (2.3)

We show how this result is obtained.  First prolong to the order of the coefficient in ${Y}_{1}$ namely 2. Apply the prolonged vector field to the coefficient of ${X}_{1}$

 E1 > $\mathrm{prX1}≔\mathrm{Prolong}\left(\mathrm{X1},2\right)$
 ${\mathrm{prX1}}{≔}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}^{{2}}{}{{\mathrm{D_u}}}_{\left[\right]}{+}{2}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{2}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{1}}{+}{2}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{2}}{+}\left({2}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{1}{,}{2}{,}{2}{,}{2}}{+}{2}{}{{u}}_{{1}{,}{1}{,}{2}{,}{2}{,}{2}}^{{2}}\right){}{{\mathrm{D_u}}}_{{1}{,}{1}}{+}\left({2}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{2}{,}{2}{,}{2}{,}{2}}{+}{2}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{2}{,}{2}{,}{2}}\right){}{{\mathrm{D_u}}}_{{1}{,}{2}}{+}\left({2}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}}{+}{2}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}{,}{2}}^{{2}}\right){}{{\mathrm{D_u}}}_{{2}{,}{2}}$ (2.4)
 E1 > $\mathrm{term1}≔\mathrm{LieDerivative}\left(\mathrm{prX1},u\left[2,2\right]\right)$
 ${\mathrm{term1}}{≔}{2}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}}{+}{2}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}{,}{2}}^{{2}}$ (2.5)

Next prolong ${Y}_{1}$ to the order of the coefficient in (namely 4). Apply the prolonged vector field to the coefficient of ${Y}_{1}$.

 E1 > $\mathrm{prY1}≔\mathrm{Prolong}\left(\mathrm{Y1},4\right)$
 ${\mathrm{prY1}}{≔}{{u}}_{{2}{,}{2}}{}{{\mathrm{D_u}}}_{\left[\right]}{+}{{u}}_{{1}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{1}}{+}{{u}}_{{2}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{2}}{+}{{u}}_{{1}{,}{1}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{1}{,}{1}}{+}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{1}{,}{2}}{+}{{u}}_{{2}{,}{2}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{2}{,}{2}}{+}{{u}}_{{1}{,}{1}{,}{1}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{1}{,}{1}{,}{1}}{+}{{u}}_{{1}{,}{1}{,}{2}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{1}{,}{1}{,}{2}}{+}{{u}}_{{1}{,}{2}{,}{2}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{1}{,}{2}{,}{2}}{+}{{u}}_{{2}{,}{2}{,}{2}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{2}{,}{2}{,}{2}}{+}{{u}}_{{1}{,}{1}{,}{1}{,}{1}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{1}{,}{1}{,}{1}{,}{1}}{+}{{u}}_{{1}{,}{1}{,}{1}{,}{2}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{1}{,}{1}{,}{1}{,}{2}}{+}{{u}}_{{1}{,}{1}{,}{2}{,}{2}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{1}{,}{1}{,}{2}{,}{2}}{+}{{u}}_{{1}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{1}{,}{2}{,}{2}{,}{2}}{+}{{u}}_{{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{2}{,}{2}{,}{2}{,}{2}}$ (2.6)
 E1 > $\mathrm{term2}≔\mathrm{LieDerivative}\left(\mathrm{prY1},{u\left[1,2,2,2\right]}^{2}\right)$
 ${\mathrm{term2}}{≔}{2}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}}$ (2.7)

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

 E1 > $\mathrm{term1}-\mathrm{term2}$
 ${2}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}{,}{2}}^{{2}}$ (2.8)

Example 2.

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

 E1 > $\mathrm{X2}≔\mathrm{evalDG}\left(u\left[2\right]x\mathrm{D_x}+{u\left[1\right]}^{2}\mathrm{D_u}\left[\right]\right)$
 ${\mathrm{X2}}{≔}{{u}}_{{2}}{}{x}{}{\mathrm{D_x}}{+}{{u}}_{{1}}^{{2}}{}{{\mathrm{D_u}}}_{\left[\right]}$ (2.9)
 E1 > $\mathrm{Y2}≔\mathrm{evalDG}\left(u\left[1,2\right]\mathrm{D_x}+y\mathrm{D_u}\left[\right]\right)$
 ${\mathrm{Y2}}{≔}{{u}}_{{1}{,}{2}}{}{\mathrm{D_x}}{+}{y}{}{{\mathrm{D_u}}}_{\left[\right]}$ (2.10)
 E1 > $\mathrm{GeneralizedLieBracket}\left(\mathrm{X2},\mathrm{Y2}\right)$
 ${-}\left({x}{}{{u}}_{{1}{,}{1}}{}{{u}}_{{2}{,}{2}}{+}{x}{}{{u}}_{{1}{,}{2}}^{{2}}{+}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}}{-}{2}{}{{u}}_{{1}{,}{1}{,}{2}}{}{{u}}_{{1}}{+}{2}{}{{u}}_{{1}{,}{2}}{}{{u}}_{{2}}{-}{2}{}{{u}}_{{1}{,}{1}}{}{{u}}_{{1}{,}{2}}{+}{x}\right){}{\mathrm{D_x}}{+}{2}{}{{u}}_{{1}{,}{1}{,}{2}}{}{{u}}_{{1}}^{{2}}{}{{\mathrm{D_u}}}_{\left[\right]}$ (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 > $\mathrm{vars}≔x,y,u\left[1\right],u\left[2\right]:$
 E1 > $\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(F\left(\mathrm{vars}\right),G\left(\mathrm{vars}\right),\mathrm{quiet}\right)$
 E1 > $\mathrm{X3}≔F\left(\mathrm{vars}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_u}\left[\right]$
 ${\mathrm{X3}}{≔}{F}{}{{\mathrm{D_u}}}_{\left[\right]}$ (2.12)
 E1 > $\mathrm{Y3}≔G\left(\mathrm{vars}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_u}\left[\right]$
 ${\mathrm{Y3}}{≔}{G}{}{{\mathrm{D_u}}}_{\left[\right]}$ (2.13)
 E1 > $\mathrm{GeneralizedLieBracket}\left(\mathrm{X3},\mathrm{Y3}\right)$
 $\left({{G}}_{{{u}}_{{1}}}{}{{F}}_{{x}}{+}{{G}}_{{{u}}_{{2}}}{}{{F}}_{{y}}{-}{{G}}_{{x}}{}{{F}}_{{{u}}_{{1}}}{-}{{G}}_{{y}}{}{{F}}_{{{u}}_{{2}}}\right){}{{\mathrm{D_u}}}_{\left[\right]}$ (2.14)

 See Also