canoni - Maple Help
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DEtools

 canoni
 look for a pair of canonical coordinates for a given Lie group of symmetries

 Calling Sequence canoni([xi(x, y), eta(x, y)], y(x), s(r))

Parameters

 [xi(x, y), eta(x, y)] - list of the coefficients of the infinitesimal symmetry generator (infinitesimals) y(x) - original dependent variable s(r) - canonical dependent variable ode - first order ODE

Description

 • The canoni command tries to find a set of transformations from the original coordinates to the canonical coordinates by knowing the coefficients of the symmetry generator (infinitesimals) of a one-parameter Lie group.
 • If there is more than one derivative in the ODE, canoni requires an extra argument (anywhere in the calling sequence), say y(x), specifying the dependent variable.
 • This function is part of the DEtools package, and so it can be used in the form canoni(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[canoni](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{symgen},\mathrm{canoni}\right)$
 $\left[{\mathrm{symgen}}{,}{\mathrm{canoni}}\right]$ (1)
 > $\mathrm{with}\left(\mathrm{PDEtools},\mathrm{dchange}\right)$
 $\left[{\mathrm{dchange}}\right]$ (2)

An ODE with an arbitrary function F

 > $\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right),x\right)=\frac{2a}{{x}^{2}\left(-y\left(x\right)+2F\left(\frac{x{y\left(x\right)}^{2}-4a}{x}\right)a\right)}$
 ${\mathrm{ODE}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{2}{}{a}}{{{x}}^{{2}}{}\left({-}{y}{}\left({x}\right){+}{2}{}{F}{}\left(\frac{{x}{}{{y}{}\left({x}\right)}^{{2}}{-}{4}{}{a}}{{x}}\right){}{a}\right)}$ (3)

A pair of infinitesimals for it

 > $\mathrm{infinitesimals}≔\mathrm{symgen}\left(\mathrm{ODE}\right)$
 ${\mathrm{infinitesimals}}{≔}\left[{\mathrm{_ξ}}{=}{y}{}{{x}}^{{2}}{,}{\mathrm{_η}}{=}{-}{2}{}{a}\right]$ (4)

The canonical coordinates

 > $\mathrm{tr}≔\mathrm{canoni}\left(\mathrm{infinitesimals},y\left(x\right),s\left(r\right)\right)$
 ${\mathrm{tr}}{≔}\left\{{r}{=}{-}\frac{{-}{x}{}{{y}{}\left({x}\right)}^{{2}}{+}{4}{}{a}}{{x}}{,}{s}{}\left({r}\right){=}{-}\frac{{y}{}\left({x}\right)}{{2}{}{a}}\right\}$ (5)

The inverse transformation

 > $\mathrm{itr}≔\mathrm{op}\left(1,\left[\mathrm{solve}\left(\mathrm{tr},\left\{x,y\left(x\right)\right\}\right)\right]\right)$
 ${\mathrm{itr}}{≔}\left\{{x}{=}\frac{{4}{}{a}}{{4}{}{{s}{}\left({r}\right)}^{{2}}{}{{a}}^{{2}}{-}{r}}{,}{y}{}\left({x}\right){=}{-}{2}{}{s}{}\left({r}\right){}{a}\right\}$ (6)

The change of variables, using the canonical coordinates, reducing eq to a quadrature (see dchange)

 > $\mathrm{ODE2}≔\mathrm{dchange}\left(\mathrm{itr},\mathrm{ODE},\left[r,s\left(r\right)\right],\mathrm{simplify}\right):$
 > $\mathrm{ODE1}≔\mathrm{op}\left(\mathrm{solve}\left(\mathrm{ODE2},\left\{\mathrm{diff}\left(s\left(r\right),r\right)\right\}\right)\right)$
 ${\mathrm{ODE1}}{≔}\frac{{ⅆ}}{{ⅆ}{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{s}{}\left({r}\right){=}{-}\frac{{1}}{{8}{}{{a}}^{{2}}{}{F}{}\left({r}\right)}$ (7)