The ChangeSymmetry command performs changes of variables in a list of infinitesimals of a symmetry generator or its corresponding infinitesimal generator differential operator. This transformation takes into account that the infinitesimals are coefficients of differentiation operators which are also changed by the transformation, thus contributing to the resulting infinitesimals in the new variables.

To avoid having to remember the optional keywords if you misspell the keyword, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.

$\mathrm{with}\left(\mathrm{PDEtools}\,\mathrm{ChangeSymmetry}\,\mathrm{CanonicalCoordinates}\,\mathrm{InfinitesimalGenerator}\right)$

$\left[\mathrm{ChangeSymmetry}\,\mathrm{CanonicalCoordinates}\,\mathrm{InfinitesimalGenerator}\right]$

$S≔\left[\mathrm{\_\ξ}\left[x\right]=x\,\mathrm{\_\ξ}\left[t\right]=1\,\mathrm{\_\η}\left[u\right]=u\right]$

$S≔\left[{\mathrm{\_\ξ}}_x=x\,{\mathrm{\_\ξ}}_t=1\,{\mathrm{\_\η}}_u=u\right]$

$G≔\mathrm{InfinitesimalGenerator}\left(S\,u\left(x\,t\right)\right)$

$G≔f\→x\left(\frac{\partial }{\partial x}f\right)\+\frac{\partial }{\partial t}f\+u\left(\frac{\partial }{\partial u}f\right)$

$\mathrm{TR}≔\left\{t=r+v\left(r\,s\right)\,x=\exp \left(v\left(r\,s\right)\right)\,u\left(x\,t\right)=s\exp \left(v\left(r\,s\right)\right)\right\}$

$\mathrm{TR}≔\left\{t=r+v\left(r\,s\right)\,x={\ⅇ}^{v\left(r\,s\right)}\,u\left(x\,t\right)=s{\ⅇ}^{v\left(r\,s\right)}\right\}$

$\mathrm{ChangeSymmetry}\left(\mathrm{TR}\,S\right)$

$\left[{\mathrm{\_\ξ}}_r=0\,{\mathrm{\_\ξ}}_s=0\,{\mathrm{\_\η}}_v=1\right]$

$\mathrm{ChangeSymmetry}\left(\mathrm{TR}\,G\right)$

$f\→\frac{\partial }{\partial v}f$

$\mathrm{ChangeSymmetry}\left(\mathrm{TR}\,\left[\mathrm{\_\ξ}\left[x\right]=u\,\mathrm{\_\ξ}\left[t\right]=x\,\mathrm{\_\η}\left[u\right]=t\right]\right)$

$\left[{\mathrm{\_\ξ}}_r={\ⅇ}^v-s\,{\mathrm{\_\ξ}}_s=\frac{-{\ⅇ}^v{s}^2+r+v}{{\ⅇ}^v}\,{\mathrm{\_\η}}_v=s\right]$

$\mathrm{ChangeSymmetry}\left(\mathrm{TR}\,\left[u\,x\,t\right]\,\mathrm{jetnotation}=\mathrm{jetnumbers}\right)$

$\left[{\ⅇ}^{v\left[\right]}-s\,\frac{-{\ⅇ}^{v\left[\right]}{s}^2+r+v\left[\right]}{{\ⅇ}^{v\left[\right]}}\,s\right]$

$\mathrm{ChangeSymmetry}\left(\mathrm{TR}\,\left[u\,x\,t\right]\,\mathrm{jetnotation}=\mathrm{false}\right)$

$\left[{\ⅇ}^{v\left(r\,s\right)}-s\,\frac{-{\ⅇ}^{v\left(r\,s\right)}{s}^2+r+v\left(r\,s\right)}{{\ⅇ}^{v\left(r\,s\right)}}\,s\right]$