calculate the normal form of the generators of a 2-D solvable Lie algebra - Maple Help

Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for ODEs : DEtools/normalG2

DEtools[normalG2] - calculate the normal form of the generators of a 2-D solvable Lie algebra

 Calling Sequence normalG2(X1, X2, y(x))

Parameters

 X1, X2 - lists of the coefficients of symmetry generators (pairs of infinitesimals) as in $\left[\mathrm{\xi },\mathrm{\eta }\right]$ y(x) - 'dependent variable'; it can be any indeterminate function of one variable

Description

 • The normalG2 command receives two pairs of infinitesimals, and an indication of the dependent variable y(x), and returns a sequence of infinitesimals $\mathrm{Y1},\mathrm{Y2}$, each one of the form $\left[\mathrm{\xi },\mathrm{\eta }\right]$, such that $\mathrm{Y1}$ and $\mathrm{Y2}$ are built using linear combinations of X1 and X2, and $\left[\mathrm{Y1},\mathrm{Y2}\right]=\mathrm{Y1}$, where $\left[\mathrm{Y1},\mathrm{Y2}\right]$ is the commutator of the two infinitesimals.
 • This command presently accepts only point symmetries, and when the given $\mathrm{X1},\mathrm{X2}$ do not form a solvable algebra (the problem has no solution), the command returns FAIL.
 • This function is part of the DEtools package, and so it can be used in the form normalG2(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[normalG2](..).

Examples

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

X1 and X2 are not in "normal form"; that is, their commutator is not equal to one of them:

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

The normalized $\mathrm{X1},\mathrm{X2}$

 > $Y:=\mathrm{normalG2}\left(\mathrm{X1},\mathrm{X2},y\left(x\right)\right)$
 ${Y}{:=}\left[{0}{,}{-}{{y}}^{{2}}\right]{,}\left[{-}\frac{{a}{}{x}}{{c}}{,}{-}\frac{{c}{}{y}{+}{{y}}^{{2}}}{{c}}\right]$ (4)

The commutator of the generators $Y$ satisfies $\left[{Y}_{1},{Y}_{2}\right]={Y}_{1}$.

 > $\mathrm{Xcommutator}\left({Y}_{1},{Y}_{2},y\left(x\right)\right)$
 $\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{-}{{y}}^{{2}}\right]$ (5)