Xcommutator - Maple Help

DEtools

 Xcommutator
 calculate the commutator of two generators of one-parameter Lie groups

 Calling Sequence Xcommutator(X1, X2, k, y(x), ODE)

Parameters

 X1, X2 - lists of the coefficients of the symmetry generators (infinitesimals) as in [xi, eta] y(x) - 'dependent variable'; it can be any indeterminate function of one variable k - (optional) extension of the generators entering the commutator ODE - (optional) right hand side is used to replace the highest derivative in the result; required if dynamical symmetries are given

Description

 • 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{X1}≔\left[x,-y\right]$
 ${\mathrm{X1}}{≔}\left[{x}{,}{-}{y}\right]$ (1)
 > $\mathrm{X2}≔\left[ax,cy+{y}^{2}\right]$
 ${\mathrm{X2}}{≔}\left[{a}{}{x}{,}{c}{}{y}{+}{{y}}^{{2}}\right]$ (2)

The commutator of X1 with X2

 > $\mathrm{Xcommutator}\left(\mathrm{X1},\mathrm{X2},y\left(x\right)\right)$
 $\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{-}{{y}}^{{2}}\right]$ (3)

The commutator of the third extensions of X1 and X2 involves four elements.

 > $\mathrm{Xcommutator}\left(\mathrm{X1},\mathrm{X2},y\left(x\right),3\right)$
 $\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{-}{{y}}^{{2}}{,}{{\mathrm{_η}}}_{{1}}{=}{-}{2}{}{\mathrm{_y1}}{}{y}{,}{{\mathrm{_η}}}_{{2}}{=}{-}{2}{}{{\mathrm{_y1}}}^{{2}}{-}{2}{}{\mathrm{_y2}}{}{y}{,}{{\mathrm{_η}}}_{{3}}{=}{-}{6}{}{\mathrm{_y1}}{}{\mathrm{_y2}}{-}{2}{}{\mathrm{_y3}}{}{y}\right]$ (4)

The generators G1 and G2 associated to the lists X1 and X2

 > $\mathrm{X1}$
 $\left[{x}{,}{-}{y}\right]$ (5)
 > $\mathrm{G1}≔\mathrm{infgen}\left(\mathrm{X1},y\left(x\right)\right)$
 ${\mathrm{G1}}{≔}{\mathrm{_F1}}{→}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{_F1}}\right){-}{y}{}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{_F1}}\right)$ (6)
 > $\mathrm{X2}$
 $\left[{a}{}{x}{,}{c}{}{y}{+}{{y}}^{{2}}\right]$ (7)
 > $\mathrm{G2}≔\mathrm{infgen}\left(\mathrm{X2},y\left(x\right)\right)$
 ${\mathrm{G2}}{≔}{\mathrm{_F1}}{→}{a}{}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{_F1}}\right){+}\left({c}{}{y}{+}{{y}}^{{2}}\right){}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{_F1}}\right)$ (8)

If G1 and G2 are differential operators (mappings), the commutator of G1 and G2 is returned as a mapping as well:

 > $\mathrm{Xcommutator}\left(\mathrm{G1},\mathrm{G2},y\left(x\right)\right)$
 ${\mathrm{_F2}}{→}{-}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{_F2}}\right){}{{y}}^{{2}}$ (9)

The commutator of the third extensions of G1 and G2

 > $\mathrm{Xcommutator}\left(\mathrm{G1},\mathrm{G2},y\left(x\right),3\right)$
 ${\mathrm{_F2}}{→}{-}{2}{}\left(\frac{{\partial }}{{\partial }{\mathrm{_y1}}}{}{\mathrm{_F2}}\right){}{\mathrm{_y1}}{}{y}{+}\left({-}{2}{}{{\mathrm{_y1}}}^{{2}}{-}{2}{}{\mathrm{_y2}}{}{y}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{_y2}}}{}{\mathrm{_F2}}\right){+}\left({-}{6}{}{\mathrm{_y1}}{}{\mathrm{_y2}}{-}{2}{}{\mathrm{_y3}}{}{y}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{_y3}}}{}{\mathrm{_F2}}\right){-}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{_F2}}\right){}{{y}}^{{2}}$ (10)