The Xcommutator command receives two generators of one-parameter Lie groups, either in the form of a pair of infinitesimals [xi, eta] or in the form of differential operators, and the dependent variable y(x), and returns the commutator of these generators.

If k is given, the k extension of the generators X1 and X2 is calculated at first, and the command returns the commutator of these extended generators (that is, another extended generator; see eta_k and infgen ).

If the given generators are in the form of a list containing the infinitesimals, the result is returned as a list; otherwise, if X1 and X2 are given as differential operators (mappings) then the result is returned as a mapping (see examples).

This function is part of the DEtools package, and so it can be used in the form Xcommutator(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[Xcommutator](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{X1}≔\left[x\,-y\right]$

$\mathrm{X1}≔\left[x\,-y\right]$

$\mathrm{X2}≔\left[ax\,cy+y^2\right]$

$\mathrm{X2}≔\left[ax\,cy+y^2\right]$

$\mathrm{Xcommutator}\left(\mathrm{X1}\,\mathrm{X2}\,y\left(x\right)\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=-y^2\right]$

$\mathrm{Xcommutator}\left(\mathrm{X1}\,\mathrm{X2}\,y\left(x\right)\,3\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=-y^2\,{\mathrm{\_\η}}_1=-2\mathrm{\_y1}y\,{\mathrm{\_\η}}_2=-2{\mathrm{\_y1}}^2-2\mathrm{\_y2}y\,{\mathrm{\_\η}}_3=-6\mathrm{\_y1}\mathrm{\_y2}-2\mathrm{\_y3}y\right]$

$\mathrm{X1}$

$\left[x\,-y\right]$

$\mathrm{G1}≔\mathrm{infgen}\left(\mathrm{X1}\,y\left(x\right)\right)$

$\mathrm{G1}≔\mathrm{\_F1}\→x\left(\frac{\partial }{\partial x}\mathrm{\_F1}\right)-y\left(\frac{\partial }{\partial y}\mathrm{\_F1}\right)$

$\mathrm{X2}$

$\left[ax\,cy+y^2\right]$

$\mathrm{G2}≔\mathrm{infgen}\left(\mathrm{X2}\,y\left(x\right)\right)$

$\mathrm{G2}≔\mathrm{\_F1}\→ax\left(\frac{\partial }{\partial x}\mathrm{\_F1}\right)\+\left(cy\+y^2\right)\left(\frac{\partial }{\partial y}\mathrm{\_F1}\right)$

$\mathrm{Xcommutator}\left(\mathrm{G1}\,\mathrm{G2}\,y\left(x\right)\right)$

$\mathrm{\_F2}\→-\left(\frac{\partial }{\partial y}\mathrm{\_F2}\right)y^2$

$\mathrm{Xcommutator}\left(\mathrm{G1}\,\mathrm{G2}\,y\left(x\right)\,3\right)$

$\mathrm{\_F2}\→-2\left(\frac{\partial }{\partial \mathrm{\_y1}}\mathrm{\_F2}\right)\mathrm{\_y1}y\+\left(-2{\mathrm{\_y1}}^2-2\mathrm{\_y2}y\right)\left(\frac{\partial }{\partial \mathrm{\_y2}}\mathrm{\_F2}\right)\+\left(-6\mathrm{\_y1}\mathrm{\_y2}-2\mathrm{\_y3}y\right)\left(\frac{\partial }{\partial \mathrm{\_y3}}\mathrm{\_F2}\right)-\left(\frac{\partial }{\partial y}\mathrm{\_F2}\right)y^2$