Given a list with the infinitesimals of a symmetry generator, or the corresponding infinitesimal generator differential operator, the CanonicalCoordinates command computes a transformation from the dependent and independent variables of the problem to the canonical coordinates of the symmetry. This is the change of variables transforming the given list of infinitesimals into one consisting of zeros but for one element equal to 1. The corresponding transformed infinitesimal generator is thus of the form

where $u_j$ is a canonical dependent variable for a problem with $m$ dependent variables.

There exist thus $m$ different sets of canonical coordinates leading to infinitesimals generators of the form above with different $u_j$. To choose among them use Canvars to tell the name of the dependent variable of the problem with respect to which the canonical variables are computed. Canvars is not required when there is only one dependent variable.

When there is only one dependent variable, DepVars and NewVars can be represented by a function; otherwise they must be a list of functions representing dependent variables. If NewVars are not given, CanonicalCoordinates will generate a list of globals to represent the canonical coordinates.

Optionally, a simplifier can be specified to be used instead of the default which is the composition of simplify and combine followed by a simplification in size.

To avoid having to remember the optional keywords, if you type the keyword misspelled, 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.

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

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

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

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

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

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

$\mathrm{CanonicalCoordinates}\left(S\,u\left(x\,t\right)\,v\left(r\,s\right)\right)$

$\left\{r=-\ln \left(x\right)+t\,s=\frac{u\left(x\,t\right)}{x}\,v\left(r\,s\right)=\ln \left(x\right)\right\}$

$\mathrm{CanonicalCoordinates}\left(G\,u\left(x\,t\right)\,v\left(r\,s\right)\right)$

$\left\{r=-\ln \left(x\right)+t\,s=\frac{u\left(x\,t\right)}{x}\,v\left(r\,s\right)=\ln \left(x\right)\right\}$

$\mathrm{solve}\left(\,\left\{t\,x\,u\left(x\,t\right)\right\}\right)$

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

$\mathrm{ChangeSymmetry}\left(\,S\,u\left(x\,t\right)\right)$

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

$\mathrm{ChangeSymmetry}\left(\,G\,u\left(x\,t\right)\right)$

$f\→\frac{\partial }{\partial v}f$

$\mathrm{CanonicalCoordinates}\left(S\,u\left(x\,t\right)\right)$

$\left\{\mathrm{\_t1}=-\ln \left(x\right)+t\,\mathrm{\_t2}=\frac{u\left(x\,t\right)}{x}\,\mathrm{\_u1}\left(\mathrm{\_t1}\,\mathrm{\_t2}\right)=\ln \left(x\right)\right\}$

$\mathrm{CanonicalCoordinates}\left(S\,u\left(x\,t\right)\,v\left(r\,s\right)\,\mathrm{jetnotation}\right)$

$\left\{r=-\ln \left(x\right)+t\,s=\frac{u}{x}\,v=\ln \left(x\right)\right\}$

$\mathrm{CanonicalCoordinates}\left(S\,u\left(x\,t\right)\,v\left(r\,s\right)\,\mathrm{jetnotation}=\mathrm{jetvariableswithbrackets}\right)$

$\left\{r=-\ln \left(x\right)+t\,s=\frac{u\left[\right]}{x}\,v\left[\right]=\ln \left(x\right)\right\}$