Let $\mathrm{\π} \: E\to M$be a fiber bundle and let $$${\mathrm{\π}}^k\: J^k\left(E\right)\to M$ 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 $\mathrm{\ℓ}$. Then the generalized Lie bracket ${\left[X\, Y\right]}_{\mathrm{gen}}$ is the generalized vector field calculated by applying the $\mathrm{\ℓ}$-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, ${\left[X\, Y\right]}_{\mathrm{gen}} \= {\mathrm{pr}}^{\mathrm{\ℓ}}\left(X\right)\left(Y\right) - {\mathrm{pr}}^k\left(Y\right)\left(X\right)$.

The command GeneralizedLieBracket(X, Y) returns the generalized vector field ${\left[X\, Y\right]}_{\mathrm{gen}}$.

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(...).

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

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

$\mathrm{X1}≔\left({u\left[1,2,2,2\right]}^2\right)\&mult\mathrm{D\_u}\left[\right]$

$\mathrm{X1}≔u_{1,2,2,2}^2{\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{Y1}≔u\left[2,2\right]\&mult\mathrm{D\_u}\left[\right]$

$\mathrm{Y1}≔u_{2,2}{\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{Z1}≔\mathrm{GeneralizedLieBracket}\left(\mathrm{X1}\,\mathrm{Y1}\right)$

$\mathrm{Z1}≔2u_{1,2,2,2,2}^2{\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{prX1}≔\mathrm{Prolong}\left(\mathrm{X1}\,2\right)$

$\mathrm{prX1}≔u_{1,2,2,2}^2{\mathrm{D\_u}}_{\left[\right]}+2u_{1,2,2,2}{u}_{1,1,2,2,2}{\mathrm{D\_u}}_1+2u_{1,2,2,2}{u}_{1,2,2,2,2}{\mathrm{D\_u}}_2+\left(2u_{1,2,2,2}{u}_{1,1,1,2,2,2}+2u_{1,1,2,2,2}^2\right){\mathrm{D\_u}}_{1,1}+\left(2u_{1,2,2,2}{u}_{1,1,2,2,2,2}+2u_{1,2,2,2,2}{u}_{1,1,2,2,2}\right){\mathrm{D\_u}}_{1,2}+\left(2u_{1,2,2,2}{u}_{1,2,2,2,2,2}+2u_{1,2,2,2,2}^2\right){\mathrm{D\_u}}_{2,2}$

$\mathrm{term1}≔\mathrm{LieDerivative}\left(\mathrm{prX1}\,u\left[2,2\right]\right)$

$\mathrm{term1}≔2u_{1,2,2,2}{u}_{1,2,2,2,2,2}+2u_{1,2,2,2,2}^2$

$\mathrm{prY1}≔\mathrm{Prolong}\left(\mathrm{Y1}\,4\right)$

$\mathrm{prY1}≔u_{2,2}{\mathrm{D\_u}}_{\left[\right]}+u_{1,2,2}{\mathrm{D\_u}}_1+u_{2,2,2}{\mathrm{D\_u}}_2+u_{1,1,2,2}{\mathrm{D\_u}}_{1,1}+u_{1,2,2,2}{\mathrm{D\_u}}_{1,2}+u_{2,2,2,2}{\mathrm{D\_u}}_{2,2}+u_{1,1,1,2,2}{\mathrm{D\_u}}_{1,1,1}+u_{1,1,2,2,2}{\mathrm{D\_u}}_{1,1,2}+u_{1,2,2,2,2}{\mathrm{D\_u}}_{1,2,2}+u_{2,2,2,2,2}{\mathrm{D\_u}}_{2,2,2}+u_{1,1,1,1,2,2}{\mathrm{D\_u}}_{1,1,1,1}+u_{1,1,1,2,2,2}{\mathrm{D\_u}}_{1,1,1,2}+u_{1,1,2,2,2,2}{\mathrm{D\_u}}_{1,1,2,2}+u_{1,2,2,2,2,2}{\mathrm{D\_u}}_{1,2,2,2}+u_{2,2,2,2,2,2}{\mathrm{D\_u}}_{2,2,2,2}$

$\mathrm{term2}≔\mathrm{LieDerivative}\left(\mathrm{prY1}\,{u\left[1,2,2,2\right]}^2\right)$

$\mathrm{term2}≔2u_{1,2,2,2}{u}_{1,2,2,2,2,2}$

$\mathrm{term1}-\mathrm{term2}$

$2u_{1,2,2,2,2}^2$

$\mathrm{X2}≔\mathrm{evalDG}\left(u\left[2\right]x\mathrm{D\_x}+{u\left[1\right]}^2\mathrm{D\_u}\left[\right]\right)$

$\mathrm{X2}≔u_{2}x\mathrm{D\_x}+u_1^2{\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{Y2}≔\mathrm{evalDG}\left(u\left[1,2\right]\mathrm{D\_x}+y\mathrm{D\_u}\left[\right]\right)$

$\mathrm{Y2}≔u_{1,2}\mathrm{D\_x}+y{\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{GeneralizedLieBracket}\left(\mathrm{X2}\,\mathrm{Y2}\right)$

$-\left(x{u}_{1,1}{u}_{2,2}+x{u}_{1,2}^2+u_1{u}_{2,2}-2u_{1,1,2}{u}_1+2u_{1,2}{u}_2-2u_{1,1}{u}_{1,2}+x\right)\mathrm{D\_x}+2u_{1,1,2}{u}_1^2{\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{vars}≔x,y,u\left[1\right],u\left[2\right]\:$

$\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(F\left(\mathrm{vars}\right)\,G\left(\mathrm{vars}\right)\,\mathrm{quiet}\right)$

$\mathrm{X3}≔F\left(\mathrm{vars}\right)\&mult\mathrm{D\_u}\left[\right]$

$\mathrm{X3}≔F{\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{Y3}≔G\left(\mathrm{vars}\right)\&mult\mathrm{D\_u}\left[\right]$

$\mathrm{Y3}≔G{\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{GeneralizedLieBracket}\left(\mathrm{X3}\,\mathrm{Y3}\right)$

$\left(G_{u_1}{F}_x+G_{u_2}{F}_y-G_x{F}_{u_1}-G_y{F}_{u_2}\right){\mathrm{D\_u}}_{\left[\right]}$