Xgauge - Maple Help

DEtools

 Xgauge
 transforms a given generator of a one-parameter Lie group to the indicated "gauge"

 Calling Sequence Xgauge([xi,eta], y(x), xi=expr) Xgauge([xi,eta], y(x), eta=expr) Xgauge([xi,eta], y(x), xi=expr1, eta=expr2)

Parameters

 [xi, eta] - list of the coefficients of the symmetry generator (infinitesimals) y(x) - 'dependent variable'; it can be any indeterminate function of one variable xi=expr - (optional) equation having xi or _xi on the left hand side and any algebraic expression on the right hand side eta=expr - (optional) equation having eta or _eta on the left hand side and any algebraic expression on the right hand side

Description

 • The Xgauge command receives a list of a pair of infinitesimals of a one-parameter Lie group, the dependent variable y(x), and optionally the "gauge" in the form of $\mathrm{\xi }=\mathrm{expr}$ or $\mathrm{\eta }=\mathrm{expr}$. It returns the given infinitesimals "gauged" to satisfy the indication received. If no gauge is given, the command returns the given generator rewritten in the most general form by introducing an arbitrary function.
 • This command also works with dynamical symmetries, in which case the ODE assumed to be invariant under the given infinitesimals is also required as an argument. The right hand side of the given nth order ODE is then used to replace the nth order derivatives of the dependent variable appearing in the infinitesimal generator.
 • This function is part of the DEtools package, and so it can be used in the form Xgauge(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[Xgauge](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $X≔\left[x,x+y\right]$
 ${X}{≔}\left[{x}{,}{x}{+}{y}\right]$ (1)

The general first order ODE invariant under X

 > $\mathrm{ODE}≔\mathrm{equinv}\left(X,y\left(x\right)\right)$
 ${\mathrm{ODE}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{\mathrm{ln}}{}\left({x}\right){+}{\mathrm{_F1}}{}\left({-}\frac{{x}{}{\mathrm{ln}}{}\left({x}\right){-}{y}{}\left({x}\right)}{{x}}\right)$ (2)
 > $\mathrm{symtest}\left(X,\mathrm{ODE}\right)$
 ${0}$ (3)

X rewritten in the gauge xi=0

 > $\mathrm{Xgauge}\left(X,y\left(x\right),\mathrm{\xi }=0\right)$
 $\left[{0}{,}{-}{\mathrm{_y1}}{}{x}{+}{x}{+}{y}\right]$ (4)

If the ODE is given, its right hand side is used to replace the highest derivative (in this example the ODE is of first order).

 > $\mathrm{X_xi=0}≔\mathrm{Xgauge}\left(X,\mathrm{ODE},\mathrm{\xi }=0\right)$
 ${\mathrm{X_xi=0}}{≔}\left[{0}{,}{y}{+}\left({-}{\mathrm{_F1}}{}\left({-}\frac{{x}{}{\mathrm{ln}}{}\left({x}\right){-}{y}}{{x}}\right){-}{\mathrm{ln}}{}\left({x}\right){+}{1}\right){}{x}\right]$ (5)

So, the above is just the original symmetry X rewritten in the gauge xi=0. Let's test this result.

 > $\mathrm{symtest}\left(\mathrm{X_xi=0},\mathrm{ODE}\right)$
 ${0}$ (6)

The general form in which X above can be rewritten is as follows:

 > $X$
 $\left[{x}{,}{x}{+}{y}\right]$ (7)
 > $\mathrm{X_general}≔\mathrm{Xgauge}\left(X,\mathrm{ODE}\right)$
 ${\mathrm{X_general}}{≔}\left[{\mathrm{_F2}}{}\left({x}{,}{y}\right){,}{y}{+}\left({-}{\mathrm{_F1}}{}\left({-}\frac{{x}{}{\mathrm{ln}}{}\left({x}\right){-}{y}}{{x}}\right){-}{\mathrm{ln}}{}\left({x}\right){+}{1}\right){}{x}{+}{\mathrm{_F2}}{}\left({x}{,}{y}\right){}{\mathrm{_F1}}{}\left({-}\frac{{x}{}{\mathrm{ln}}{}\left({x}\right){-}{y}}{{x}}\right){+}{\mathrm{_F2}}{}\left({x}{,}{y}\right){}{\mathrm{ln}}{}\left({x}\right)\right]$ (8)

where _F2(x, y) is an arbitrary function of its arguments, introduced by Xgauge. This result can be tested as well.

 > $\mathrm{symtest}\left(\mathrm{X_general},\mathrm{ODE}\right)$
 ${0}$ (9)