 DEtools - Maple Programming Help

DEtools

 infgen
 find the k-extension of the infinitesimal generator of a one-parameter Lie group

 Calling Sequence infgen([xi, eta], k, y(x)) infgen([xi, eta], k, ODE) infgen([xi, eta], k, y(x), ODE)

Parameters

 [xi, eta] - list of the coefficients of the symmetry generator (infinitesimals) k - positive integer indicating the order of the required prolongation y(x) - 'dependent variable'; it can be any indeterminate function of one variable ODE - ODE invariant under the given infinitesimals; required only if they represent dynamical symmetries

Description

 • The infgen command receives a pair of infinitesimals; k, the order of the required prolongation; and the dependent variable, say y(x). It returns the k-extension of the infinitesimal generator (see eta_k and symgen).
 • 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 infgen(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[infgen](..).

Examples

The infinitesimals xi and eta of the one-parameter rotation group and the first extension of the related infinitesimal generator

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $X≔\left[-y,x\right]$
 ${X}{≔}\left[{-}{y}{,}{x}\right]$ (1)
 > $\mathrm{infgen}\left(X,1,y\left(x\right)\right)$
 ${\mathrm{_F1}}{→}{-}{y}{}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{_F1}}\right){+}{x}{}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{_F1}}\right){+}\left({{\mathrm{_y1}}}^{{2}}{+}{1}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{_y1}}}{}{\mathrm{_F1}}\right)$ (2)

When an ODE is given as an argument, its right hand side is used to replace all occurrences of the highest derivative in the infinitesimal generator. To obtain a meaningful result, the ODE is invariant under the related symmetry group (or at least more general than the related invariant ODE). For example, this is the most general first order ODE invariant under rotations in the plane.

 > $\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right),x\right)=\frac{x+\mathrm{tan}\left(\mathrm{_F1}\left({x}^{2}+{y\left(x\right)}^{2}\right)\right)y\left(x\right)}{-y\left(x\right)+x\mathrm{tan}\left(\mathrm{_F1}\left({x}^{2}+{y\left(x\right)}^{2}\right)\right)}$
 ${\mathrm{ODE}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{x}{+}{\mathrm{tan}}{}\left({\mathrm{_F1}}{}\left({{x}}^{{2}}{+}{{y}{}\left({x}\right)}^{{2}}\right)\right){}{y}{}\left({x}\right)}{{-}{y}{}\left({x}\right){+}{x}{}{\mathrm{tan}}{}\left({\mathrm{_F1}}{}\left({{x}}^{{2}}{+}{{y}{}\left({x}\right)}^{{2}}\right)\right)}$ (3)
 > $\mathrm{symtest}\left(X,\mathrm{ODE}\right)$
 ${0}$ (4)

The first extension of the related infinitesimal generator is given by

 > $\mathrm{infgen}\left(X,1,\mathrm{ODE}\right)$
 ${\mathrm{_F2}}{→}{-}{y}{}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{_F2}}\right){+}{x}{}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{_F2}}\right){+}\frac{\left(\left({1}{+}{{\mathrm{tan}}{}\left({\mathrm{_F1}}{}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right)\right)}^{{2}}\right){}{{y}}^{{2}}{+}{{\mathrm{tan}}{}\left({\mathrm{_F1}}{}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right)\right)}^{{2}}{}{{x}}^{{2}}{+}{{x}}^{{2}}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{_y1}}}{}{\mathrm{_F2}}\right)}{{\left({-}{y}{+}{x}{}{\mathrm{tan}}{}\left({\mathrm{_F1}}{}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right)\right)\right)}^{{2}}}$ (5)

It was not necessary to also give y(x) above, since this information is already present in the ODE.

The most general case of a point symmetry and the first extension of the related infinitesimal generator

 > $X≔\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)\right]$
 ${X}{≔}\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]$ (6)
 > $\mathrm{infgen}\left(X,1,y\left(x\right)\right)$
 ${\mathrm{_F1}}{→}{\mathrm{ξ}}{}\left({x}{,}{y}\right){}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{_F1}}\right){+}{\mathrm{η}}{}\left({x}{,}{y}\right){}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{_F1}}\right){+}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{η}}{}\left({x}{,}{y}\right){+}\left({-}{\mathrm{_y1}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}{}\left({x}{,}{y}\right)\right){+}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{η}}{}\left({x}{,}{y}\right){-}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ξ}}{}\left({x}{,}{y}\right)\right)\right){}{\mathrm{_y1}}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{_y1}}}{}{\mathrm{_F1}}\right)$ (7)

The final example illustrates the most general case of a dynamical symmetry in the context of second order ODEs and the first extension of the related infinitesimal generator. When working with dynamical symmetries, the ODE itself is required as an argument.

 > $X≔\left[\mathrm{\xi }\left(x,y,\mathrm{_y1}\right),\mathrm{\eta }\left(x,y,\mathrm{_y1}\right)\right]$
 ${X}{≔}\left[{\mathrm{\xi }}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right){,}{\mathrm{\eta }}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)\right]$ (8)
 > $\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right),x,x\right)=F\left(x,y\left(x\right),\mathrm{diff}\left(y\left(x\right),x\right)\right)$
 ${\mathrm{ODE}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{F}{}\left({x}{,}{y}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (9)
 > $\mathrm{infgen}\left(X,1,\mathrm{ODE}\right)$
 ${\mathrm{_F1}}{→}{\mathrm{ξ}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right){}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{_F1}}\right){+}{\mathrm{η}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right){}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{_F1}}\right){+}\left({-}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)\right){}{{\mathrm{_y1}}}^{{2}}{+}\left({-}{F}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{_y1}}}{}{\mathrm{ξ}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)\right){+}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{η}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right){-}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ξ}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)\right)\right){}{\mathrm{_y1}}{+}{F}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{_y1}}}{}{\mathrm{η}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)\right){+}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{η}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{_y1}}}{}{\mathrm{_F1}}\right)$ (10)