Given a list with the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator, or a list of n of these lists or operators, possibly representing an n-dimensional symmetry group, Invariants returns a sequence of $m$ mathematical expressions ${\mathrm{\phi }}_k$ simultaneously invariant under all the $G_j$ symmetry generators ($1\le j\le n$) related to the given $n$ infinitesimals. The ${\mathrm{\phi }}_k$ are a complete set of differential invariants, so they are functions of the independent and dependent variables of the problem (DepVars), as well as of their partial derivatives up to order (default is 1), and satisfy $G_j\left({\mathrm{\phi }}_k\right)=0$ for all $j$ and $k$, where the number k depends on the problem.

Note that in the multidimensional case, depending on the form of the infinitesimals given, the problem may have no solution. Also the ${\mathrm{\phi }}_k$ are all independent of each other and their number is unique, but one can still rewrite the ${\mathrm{\phi }}_k$ in different ways, by combining invariants to construct equivalent but algebraically different invariants of the same order.

Optionally, a simplifier can be specified to be used instead of the default which is simplify/size. For that purpose use the optional argument simplifier = .... By default, the output of Invariants is in jet notation. You can change that also by passing the optional argument jetnotation = ... where the right-hand-side can be false or any of the jet notations available.

The invariants returned by Invariants can also be used as optional arguments for the group InvariantSolutions command when computing solutions to PDE systems.

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.

$\mathrm{with}\left(\mathrm{PDEtools}\right)\:$

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

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

$G≔\mathrm{InfinitesimalGenerator}\left(S\,u\left(x\,t\right)\,\mathrm{prolongation}=1\,\mathrm{expanded}\right)$

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

$\mathrm{\Phi }≔\mathrm{Invariants}\left(S\,u\left(x\,t\right)\right)$

$\mathrm{\Phi }≔\frac{u}{x},-\ln \left(x\right)+t,u_x,\frac{u_t}{x}$

$G\left(\mathrm{\Phi }\left[1\right]\right)$

$0$

$\mathrm{simplify}\left(\mathrm{map}\left(G\,\left[\mathrm{\Phi }\right]\right)\right)$

$\left[0\,0\,0\,0\right]$

$\mathrm{Invariants}\left(S\,u\left(x\,t\right)\,2\right)$

$\frac{u}{x},-\ln \left(x\right)+t,u_x,\frac{u_t}{x},u_{x,t},u_{x,x}x,\frac{u_{t,t}}{x}$

$\mathrm{Invariants}\left(S\,u\left(x\,t\right)\,2\,\mathrm{jetnotation}=\mathrm{false}\right)$

$-\ln \left(x\right)+t,\frac{u\left(x\,t\right)}{x},\frac{\partial }{\partial x}u\left(x\,t\right),\frac{\frac{\partial }{\partial t}u\left(x\,t\right)}{x},\frac{{\partial }^2}{\partial t\partial x}u\left(x\,t\right),\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)\right)x,\frac{\frac{{\partial }^2}{\partial t^2}u\left(x\,t\right)}{x}$

$\mathrm{Invariants}\left(S\,u\left(x\,t\right)\,2\,\mathrm{jetnotation}=\mathrm{jetnumbers}\right)$

$-\ln \left(x\right)+t,\frac{u\left[\right]}{x},u_1,\frac{u_2}{x},u_{1,2},u_{1,1}x,\frac{u_{2,2}}{x}$

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

$\left[-\ln \left(x\right)+t\,\frac{u\left[\right]}{x}\,u_1\,\frac{u_2}{x}\,u_{1,2}\,u_{1,1}x\,\frac{u_{2,2}}{x}\right]$

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

$\left[-\ln \left(x\right)+t\,\frac{u\left(x\,t\right)}{x}\,\frac{\partial }{\partial x}u\left(x\,t\right)\,\frac{\frac{\partial }{\partial t}u\left(x\,t\right)}{x}\,\frac{{\partial }^2}{\partial t\partial x}u\left(x\,t\right)\,\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)\right)x\,\frac{\frac{{\partial }^2}{\partial t^2}u\left(x\,t\right)}{x}\right]$

$\mathrm{DepVars}≔\left[u\,v\right]\left(x\,t\right)$

$\mathrm{DepVars}≔\left[u\left(x\,t\right)\,v\left(x\,t\right)\right]$

$L≔\left[\left[\mathrm{\_\ξ}\left[x\right]=0\,\mathrm{\_\ξ}\left[t\right]=1\,\mathrm{\_\η}\left[u\right]=0\,\mathrm{\_\η}\left[v\right]=0\right]\,\left[\mathrm{\_\ξ}\left[x\right]=0\,\mathrm{\_\ξ}\left[t\right]=0\,\mathrm{\_\η}\left[u\right]=1\,\mathrm{\_\η}\left[v\right]=0\right]\,\left[\mathrm{\_\ξ}\left[x\right]=1\,\mathrm{\_\ξ}\left[t\right]=0\,\mathrm{\_\η}\left[u\right]=2t\,\mathrm{\_\η}\left[v\right]=0\right]\,\left[\mathrm{\_\ξ}\left[x\right]=t\,\mathrm{\_\ξ}\left[t\right]=0\,\mathrm{\_\η}\left[u\right]=t^2+x\,\mathrm{\_\η}\left[v\right]=0\right]\,\left[\mathrm{\_\ξ}\left[x\right]=\frac{1}{2}x+\frac{3}{2}{t}^2\,\mathrm{\_\ξ}\left[t\right]=t\,\mathrm{\_\η}\left[u\right]=t^3+3tx\,\mathrm{\_\η}\left[v\right]=-v\right]\right]$

$L≔\left[\left[{\mathrm{\_\ξ}}_x=0\,{\mathrm{\_\ξ}}_t=1\,{\mathrm{\_\η}}_u=0\,{\mathrm{\_\η}}_v=0\right]\,\left[{\mathrm{\_\ξ}}_x=0\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=1\,{\mathrm{\_\η}}_v=0\right]\,\left[{\mathrm{\_\ξ}}_x=1\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=2t\,{\mathrm{\_\η}}_v=0\right]\,\left[{\mathrm{\_\ξ}}_x=t\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=t^2+x\,{\mathrm{\_\η}}_v=0\right]\,\left[{\mathrm{\_\ξ}}_x=\frac{x}{2}+\frac{3t^2}{2}\,{\mathrm{\_\ξ}}_t=t\,{\mathrm{\_\η}}_u=t^3+3tx\,{\mathrm{\_\η}}_v=-v\right]\right]$

$\mathrm{\Phi }≔\left[\mathrm{Invariants}\left(L\,\mathrm{DepVars}\right)\right]$

$\mathrm{\Phi }≔\left[\frac{v_x}{v^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\,\frac{\frac{u_x^2}{2}-2x+u_t}{v}\,\frac{u_x{v}_x+v_t}{v^2}\right]$

$G≔\mathrm{map}\left(\mathrm{InfinitesimalGenerator}\,L\,\mathrm{DepVars}\,\mathrm{prolongation}=1\right)$

$G≔\left[f\→\mathrm{add}\left({\mathrm{\ξ}}_{x_j}\left(\frac{\partial }{\partial x_j}f\right)\,j\=1..2\right)\+\mathrm{add}\left({\mathrm{\η}}_{u_m}\left(\frac{\partial }{\partial u_m}f\right)\+{\mathrm{\η}}_{u_m\,\left[x\right]}\left(\frac{\partial }{\partial {u_m}_x}f\right)\+{\mathrm{\η}}_{u_m\,\left[t\right]}\left(\frac{\partial }{\partial {u_m}_t}f\right)\,m\=1..2\right)\,f\→\mathrm{add}\left({\mathrm{\ξ}}_{x_j}\left(\frac{\partial }{\partial x_j}f\right)\,j\=1..2\right)\+\mathrm{add}\left({\mathrm{\η}}_{u_m}\left(\frac{\partial }{\partial u_m}f\right)\+{\mathrm{\η}}_{u_m\,\left[x\right]}\left(\frac{\partial }{\partial {u_m}_x}f\right)\+{\mathrm{\η}}_{u_m\,\left[t\right]}\left(\frac{\partial }{\partial {u_m}_t}f\right)\,m\=1..2\right)\,f\→\mathrm{add}\left({\mathrm{\ξ}}_{x_j}\left(\frac{\partial }{\partial x_j}f\right)\,j\=1..2\right)\+\mathrm{add}\left({\mathrm{\η}}_{u_m}\left(\frac{\partial }{\partial u_m}f\right)\+{\mathrm{\η}}_{u_m\,\left[x\right]}\left(\frac{\partial }{\partial {u_m}_x}f\right)\+{\mathrm{\η}}_{u_m\,\left[t\right]}\left(\frac{\partial }{\partial {u_m}_t}f\right)\,m\=1..2\right)\,f\→\mathrm{add}\left({\mathrm{\ξ}}_{x_j}\left(\frac{\partial }{\partial x_j}f\right)\,j\=1..2\right)\+\mathrm{add}\left({\mathrm{\η}}_{u_m}\left(\frac{\partial }{\partial u_m}f\right)\+{\mathrm{\η}}_{u_m\,\left[x\right]}\left(\frac{\partial }{\partial {u_m}_x}f\right)\+{\mathrm{\η}}_{u_m\,\left[t\right]}\left(\frac{\partial }{\partial {u_m}_t}f\right)\,m\=1..2\right)\,f\→\mathrm{add}\left({\mathrm{\ξ}}_{x_j}\left(\frac{\partial }{\partial x_j}f\right)\,j\=1..2\right)\+\mathrm{add}\left({\mathrm{\η}}_{u_m}\left(\frac{\partial }{\partial u_m}f\right)\+{\mathrm{\η}}_{u_m\,\left[x\right]}\left(\frac{\partial }{\partial {u_m}_x}f\right)\+{\mathrm{\η}}_{u_m\,\left[t\right]}\left(\frac{\partial }{\partial {u_m}_t}f\right)\,m\=1..2\right)\right]$

$$\begin{gathered} \mathbf{for}j\mathbf{to}5\mathbf{do} \\ 'G'\left[j\right]\left('\mathrm{\Phi }'\right)=\mathrm{simplify}\left(\mathrm{map}\left(G\left[j\right]\,\mathrm{\Phi }\right)\right) \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$G_1\left(\mathrm{\Phi }\right)=\left[0\,0\,0\right]$

$G_2\left(\mathrm{\Phi }\right)=\left[0\,0\,0\right]$

$G_3\left(\mathrm{\Phi }\right)=\left[0\,0\,0\right]$

$G_4\left(\mathrm{\Phi }\right)=\left[0\,0\,0\right]$

$G_5\left(\mathrm{\Phi }\right)=\left[0\,0\,0\right]$