Let $\mathrm{\π}\:E \to M$be a fiber bundle with 1-dimensional fiber and let ${\mathrm{\π}}^1\: J^1\left(E\right) \to M$ be 1st order jet space of $E\.$In terms of the usual coordinates $\left[x^i\, u\, u_i\right]$on $J^1\left(E\right)\,$the contact form on $J^1\left(E\right)$ is $C \= \mathrm{du} - u_i{\mathrm{dx}}^i$. A vector field $X$on $J^1\left(E\right)$which preserves the contact form $C\,$in the sense that ${\mathrm{\ℒ}}_X\left(C\right)\= \mathrm{\λ} C\,$is called an infinitesimal contact transformation or a contact vector field. There is a formula which assigns to each locally defined real-valued function $S$on $J^1\left(E\right)$a contact vector field $X_S\.$The function $S$ is called the generating function for the contact vector field $X_S$. In terms of the local coordinates $\left[x^i\, u\, u_i\right]$, we have $S \= S\left(x^i\, u\, u_i\right)$and

The command GeneratingFunctionToContactVector(S) returns the contact vector field defined by the function S.

The command GeneratingFunctionToContactVector is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form GeneratingFunctionToContactVector(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-GeneratingFunctionToContactVector(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{JetCalculus}\right)\:$

$\mathrm{DGsetup}\left(\left[x\right]\,\left[u\right]\,\mathrm{J11}\,1\right)\:$

$\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(S\left(x\,u\left[\right]\,u\left[1\right]\right)\,\mathrm{quiet}\right)$

$\mathrm{GeneratingFunctionToContactVector}\left(S\left(x\,u\left[\right]\,u\left[1\right]\right)\right)$

$-S_{u_1}\mathrm{D\_x}+\left(-u_1{S}_{u_1}+S\right){\mathrm{D\_u}}_{\left[\right]}+\left(S_x+u_1{S}_{u_{\left[\right]}}\right){\mathrm{D\_u}}_1$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\right]\,\mathrm{J21}\,1\right)\:$

$\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(S\left(x\,y\,u\left[\right]\,u\left[1\right]\,u\left[2\right]\right)\,\mathrm{quiet}\right)$

$\mathrm{GeneratingFunctionToContactVector}\left(S\left(x\,y\,u\left[\right]\,u\left[1\right]\,u\left[2\right]\right)\right)$

$-S_{u_1}\mathrm{D\_x}-S_{u_2}\mathrm{D\_y}+\left(-u_2{S}_{u_2}-u_1{S}_{u_1}+S\right){\mathrm{D\_u}}_{\left[\right]}+\left(S_x+u_1{S}_{u_{\left[\right]}}\right){\mathrm{D\_u}}_1+\left(S_y+u_2{S}_{u_{\left[\right]}}\right){\mathrm{D\_u}}_2$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,\left[u\right]\,\mathrm{J31}\,1\right)\:$

$\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(S\left(x\,y\,z\,u\left[\right]\,u\left[1\right]\,u\left[2\right]\,u\left[3\right]\right)\,\mathrm{quiet}\right)$

$\mathrm{GeneratingFunctionToContactVector}\left(S\left(x\,y\,z\,u\left[\right]\,u\left[1\right]\,u\left[2\right]\,u\left[3\right]\right)\right)$

$-S_{u_1}\mathrm{D\_x}-S_{u_2}\mathrm{D\_y}-S_{u_3}\mathrm{D\_z}+\left(-u_3{S}_{u_3}-u_2{S}_{u_2}-u_1{S}_{u_1}+S\right){\mathrm{D\_u}}_{\left[\right]}+\left(S_x+u_1{S}_{u_{\left[\right]}}\right){\mathrm{D\_u}}_1+\left(S_y+u_2{S}_{u_{\left[\right]}}\right){\mathrm{D\_u}}_2+\left(S_z+u_3{S}_{u_{\left[\right]}}\right){\mathrm{D\_u}}_3$

$\mathrm{ChangeFrame}\left(\mathrm{J21}\right)\:$

$S≔x+3y\:$

$\mathrm{GeneratingFunctionToContactVector}\left(S\right)$

$\left(x+3y\right){\mathrm{D\_u}}_{\left[\right]}+{\mathrm{D\_u}}_1+3{\mathrm{D\_u}}_2$

$S≔u\left[\right]\:$

$\mathrm{GeneratingFunctionToContactVector}\left(S\right)$

$u_{\left[\right]}{\mathrm{D\_u}}_{\left[\right]}+u_1{\mathrm{D\_u}}_1+u_2{\mathrm{D\_u}}_2$

$S≔au\left[1\right]+bu\left[2\right]\:$

$\mathrm{GeneratingFunctionToContactVector}\left(S\right)$

$-a\mathrm{D\_x}-b\mathrm{D\_y}$

$S≔u\left[\right]{u\left[2\right]}^2\:$

$X≔\mathrm{GeneratingFunctionToContactVector}\left(S\right)$

$X≔-2u_{\left[\right]}{u}_2\mathrm{D\_y}-u_{\left[\right]}{u}_2^2{\mathrm{D\_u}}_{\left[\right]}+u_1{u}_2^2{\mathrm{D\_u}}_1+u_2^3{\mathrm{D\_u}}_2$

$\mathrm{LieDerivative}\left(X\,\mathrm{Cu}\left[\right]\right)$

$u_2^2{\mathrm{Cu}}_{\left[\right]}$

$\mathrm{\Phi }≔\mathrm{ProjectionTransformation}\left(1\,0\right)$

$\mathrm{\Phi }≔\left[x=x\,y=y\,u_{\left[\right]}=u_{\left[\right]}\right]$

$Y≔\mathrm{Pushforward}\left(\mathrm{\Phi }\,X\right)$

$Y≔-2u_{\left[\right]}{u}_2\mathrm{D\_y}-u_{\left[\right]}{u}_2^2{\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{Y1}≔\mathrm{Prolong}\left(Y\,1\right)$

$\mathrm{Y1}≔-2u_{\left[\right]}{u}_2\mathrm{D\_y}-u_{\left[\right]}{u}_2^2{\mathrm{D\_u}}_{\left[\right]}+u_1{u}_2^2{\mathrm{D\_u}}_1+u_2^3{\mathrm{D\_u}}_2$

$\mathrm{Y1}\&minusX$

$0\mathrm{D\_x}$

$S≔{u\left[1\right]}^2+x^2\:$

$X≔\mathrm{GeneratingFunctionToContactVector}\left(S\right)$

$X≔-2u_1\mathrm{D\_x}+\left(-u_1^2+x^2\right){\mathrm{D\_u}}_{\left[\right]}+2x{\mathrm{D\_u}}_1$

$\mathrm{\Phi }≔\mathrm{Flow}\left(X\,t\right)$

$\mathrm{\Phi }≔\left[x=-u_1\sin \left(2t\right)+x\cos \left(2t\right)\,y=y\,u_{\left[\right]}=-u_1^2\left(\frac{\cos \left(2t\right)\sin \left(2t\right)}{4}+\frac{t}{2}\right)+u_1{\cos \left(2t\right)}^{2}x-x^2\left(-\frac{\cos \left(2t\right)\sin \left(2t\right)}{4}+\frac{t}{2}\right)+u_1^2\left(-\frac{\cos \left(2t\right)\sin \left(2t\right)}{4}+\frac{t}{2}\right)+x^2\left(\frac{\cos \left(2t\right)\sin \left(2t\right)}{4}+\frac{t}{2}\right)-u_{1}x+u_{\left[\right]}\,u_1=u_1\cos \left(2t\right)+x\sin \left(2t\right)\,u_2=u_2\right]$

$\mathrm{Pullback}\left(\mathrm{\Phi }\,\mathrm{du}\left[\right]-u\left[1\right]\mathrm{dx}-u\left[2\right]\mathrm{dy}\right)$

$-u_1\mathrm{dx}-u_2\mathrm{dy}+{\mathrm{du}}_{\left[\right]}$

$\mathrm{\Φ1}≔\mathrm{eval}\left(\mathrm{\Phi }\,t=\frac{\mathrm{\pi }}{4}\right)$

$\mathrm{\Φ1}≔\left[x=-u_1\,y=y\,u_{\left[\right]}=-u_{1}x+u_{\left[\right]}\,u_1=x\,u_2=u_2\right]$

$\mathrm{\Φ2}≔\mathrm{Prolong}\left(\mathrm{\Φ1}\,2\right)$

$\mathrm{\Φ2}≔\left[x=-u_1\,y=y\,u_{\left[\right]}=-u_{1}x+u_{\left[\right]}\,u_1=x\,u_2=u_2\,u_{1,1}=-\frac{1}{u_{1,1}}\,u_{1,2}=-\frac{u_{1,2}}{u_{1,1}}\,u_{2,2}=-\frac{u_{1,2}^2}{u_{1,1}}+u_{2,2}\right]$

$\mathrm{\Delta }≔\mathrm{Pullback}\left(\mathrm{\Φ2}\,u\left[1,1\right]u\left[2,2\right]-{u\left[1,2\right]}^2-1\right)$

$\mathrm{\Delta }≔-\frac{-\frac{u_{1,2}^2}{u_{1,1}}+u_{2,2}}{u_{1,1}}-\frac{u_{1,2}^2}{u_{1,1}^2}-1$

$\mathrm{simplify}\left(\mathrm{\Delta }\right)$

$-\frac{u_{2,2}+u_{1,1}}{u_{1,1}}$