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DEtools

 eta_k
 determines the k-prolongation of the infinitesimals of a one-parameter Lie group

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

 • eta_k receives a pair of infinitesimals; k, the order of the required prolongation; and the dependent variable y(x), and returns the k-prolongation of eta (see infgen).
 • This command also works with dynamical symmetries, in which case the ODE that is 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 prolongation of eta.
 • If the infinitesimal generator of a Lie group is written as
 > X = (F -> xi(x,y)*diff(F,x) + eta(x,y)*diff(F,y));
 ${X}{=}\left({F}{→}{\mathrm{ξ}}{}\left({x}{,}{y}\right){}\left(\frac{{\partial }}{{\partial }{x}}{}{F}\right){+}{\mathrm{η}}{}\left({x}{,}{y}\right){}\left(\frac{{\partial }}{{\partial }{y}}{}{F}\right)\right)$ (1)
 where y(x) plays the role of the dependent variable, the k prolongation of eta is obtained as follows.
 1) The linear operator A corresponding to the given ODE is built
 > A = (F -> diff(F,x) + _y1*diff(F,y) + _y2*diff(F,y,y) + ... + rhs_ODE * diff(y(x),x$(n-1)));  ${A}{=}\left({F}{→}\frac{{\partial }}{{\partial }{x}}{}{F}{+}{\mathrm{_y1}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{F}\right){+}{\mathrm{_y2}}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}{}{F}\right){+}{\mathrm{...}}{+}{\mathrm{rhs_ODE}}{}\left(\frac{{{ⅆ}}^{{n}{-}{1}}}{{ⅆ}{{x}}^{{n}{-}{1}}}{}{y}{}\left({x}\right)\right)\right)$ (2)  (rhs_ODE means the right hand side of the ODE).  2) The expression  > eta[n] = A(eta[n-1])- diff(y(x),x$n)*A(xi);
 ${{\mathrm{η}}}_{{n}}{=}{A}{}\left({{\mathrm{η}}}_{{n}{-}{1}}\right){-}\left(\frac{{{ⅆ}}^{{n}}}{{ⅆ}{{x}}^{{n}}}{}{y}{}\left({x}\right)\right){}{A}{}\left({\mathrm{ξ}}\right)$ (3)
 is evaluated recursively, and in the case of dynamical symmetries, the highest order derivative is replaced by the right hand side of the ODE.
 • This function is part of the DEtools package, and so it can be used in the form eta_k(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[eta_k](..).

Examples

Find the infinitesimals corresponding to the one-parameter rotation group (rotations in the plane), and the first, second, and third prolongations of eta.

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{eta_k},\mathrm{infgen}\right)$
 $\left[{\mathrm{eta_k}}{,}{\mathrm{infgen}}\right]$ (4)
 > $X≔\left[-y,x\right]$
 ${X}{≔}\left[{-}{y}{,}{x}\right]$ (5)
 > ${\mathrm{η}}_{1}=\mathrm{eta_k}\left(X,1,y\left(x\right)\right)$
 ${{\mathrm{η}}}_{{1}}{=}{{\mathrm{_y1}}}^{{2}}{+}{1}$ (6)
 > ${\mathrm{η}}_{2}=\mathrm{eta_k}\left(X,2,y\left(x\right)\right)$
 ${{\mathrm{η}}}_{{2}}{=}{3}{}{\mathrm{_y2}}{}{\mathrm{_y1}}$ (7)
 > ${\mathrm{η}}_{3}=\mathrm{eta_k}\left(X,3,y\left(x\right)\right)$
 ${{\mathrm{η}}}_{{3}}{=}{4}{}{\mathrm{_y1}}{}{\mathrm{_y3}}{+}{3}{}{{\mathrm{_y2}}}^{{2}}$ (8)

These prolongations of eta enter the expression of the extended related infinitesimal generator (a differential operator, see infgen).

 > $\mathrm{infgen}\left(X,3,y\left(x\right)\right)$
 ${\mathrm{_F1}}{→}{-}{y}{}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{_F1}}\right){+}{x}{}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{_F1}}\right){+}\left({{\mathrm{_y1}}}^{{2}}{+}{1}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{_y1}}}{}{\mathrm{_F1}}\right){+}{3}{}{\mathrm{_y2}}{}{\mathrm{_y1}}{}\left(\frac{{\partial }}{{\partial }{\mathrm{_y2}}}{}{\mathrm{_F1}}\right){+}\left({4}{}{\mathrm{_y1}}{}{\mathrm{_y3}}{+}{3}{}{{\mathrm{_y2}}}^{{2}}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{_y3}}}{}{\mathrm{_F1}}\right)$ (9)

You can also determine point symmetries for second order ODEs. The general case is as follows:

 > $\mathrm{ODE}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)=F\left(x,y\left(x\right),\frac{ⅆ}{ⅆx}y\left(x\right)\right)$
 ${\mathrm{ODE}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){=}{F}{}\left({x}{,}{y}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right)$ (10)
 > $X≔\left[\mathrm{ξ}\left(x,y\right),\mathrm{η}\left(x,y\right)\right]$
 ${X}{≔}\left[{\mathrm{ξ}}{}\left({x}{,}{y}\right){,}{\mathrm{η}}{}\left({x}{,}{y}\right)\right]$ (11)

The prolongation of order 0 is just eta(x,y):

 > ${\mathrm{η}}_{0}=\mathrm{eta_k}\left(X,0,\mathrm{ODE}\right)$
 ${{\mathrm{η}}}_{{0}}{=}{\mathrm{η}}{}\left({x}{,}{y}\right)$ (12)

The first and second prolongations of eta look like this:

 > ${\mathrm{η}}_{1}=\mathrm{eta_k}\left(X,1,\mathrm{ODE}\right)$
 ${{\mathrm{η}}}_{{1}}{=}{-}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}{}\left({x}{,}{y}\right)\right){}{{\mathrm{_y1}}}^{{2}}{+}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{η}}{}\left({x}{,}{y}\right){-}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ξ}}{}\left({x}{,}{y}\right)\right)\right){}{\mathrm{_y1}}{+}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{η}}{}\left({x}{,}{y}\right)$ (13)
 > ${\mathrm{η}}_{2}=\mathrm{eta_k}\left(X,2,\mathrm{ODE}\right)$
 ${{\mathrm{η}}}_{{2}}{=}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}{}{\mathrm{ξ}}{}\left({x}{,}{y}\right)\right){}{{\mathrm{_y1}}}^{{3}}{+}\left({-}{2}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{y}{}{\partial }{x}}{}{\mathrm{ξ}}{}\left({x}{,}{y}\right)\right){+}\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}{}{\mathrm{η}}{}\left({x}{,}{y}\right)\right){}{{\mathrm{_y1}}}^{{2}}{+}\left({-}{3}{}{F}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right){}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}{}\left({x}{,}{y}\right)\right){+}{2}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{y}{}{\partial }{x}}{}{\mathrm{η}}{}\left({x}{,}{y}\right)\right){-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}{}{\mathrm{ξ}}{}\left({x}{,}{y}\right)\right)\right){}{\mathrm{_y1}}{+}{F}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right){}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{η}}{}\left({x}{,}{y}\right)\right){-}{2}{}{F}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right){}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ξ}}{}\left({x}{,}{y}\right)\right){+}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}{}{\mathrm{η}}{}\left({x}{,}{y}\right)$ (14)

Finally, here is the general case of a dynamical symmetry in the framework of second order ODEs and the first prolongation of eta.

 > $X≔\left[\mathrm{ξ}\left(x,y,\mathrm{_y1}\right),\mathrm{η}\left(x,y,\mathrm{_y1}\right)\right]$
 ${X}{≔}\left[{\mathrm{ξ}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right){,}{\mathrm{η}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)\right]$ (15)
 > ${\mathrm{η}}_{1}=\mathrm{eta_k}\left(X,1,\mathrm{ODE}\right)$
 ${{\mathrm{η}}}_{{1}}{=}{-}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)\right){}{{\mathrm{_y1}}}^{{2}}{+}\left({-}{F}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{_y1}}}{}{\mathrm{ξ}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)\right){-}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ξ}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)\right){+}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{η}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)\right){}{\mathrm{_y1}}{+}{F}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right){}\left(\frac{{\partial }}{{\partial }{\mathrm{_y1}}}{}{\mathrm{η}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)\right){+}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{η}}{}\left({x}{,}{y}{,}{\mathrm{_y1}}\right)$ (16)