GeneratingFunctionToContactVector - Maple Help

Home : Support : Online Help : Mathematics : DifferentialGeometry : JetCalculus : GeneratingFunctionToContactVector

JetCalculus[GeneratingFunctionToContactVector] - find the contact vector field defined by a generating function

Calling Sequences

GeneratingFunctionToContactVector(S)

Parameters

S         - a Maple expression

Description

 • Let be a fiber bundle with 1-dimensional fiber and let  be 1st order jet space of In terms of the usual coordinates on the contact form on ${J}^{1}\left(E\right)$ is . A vector field on which preserves the contact form in the sense that is called an infinitesimal contact transformation or a contact vector field. There is a formula which assigns to each locally defined real-valued function on a contact vector field The function $S$ is called the generating function for the contact vector field ${X}_{S}$. In terms of the local coordinates , we have and

.

For further details see P. J. Olver,  Equivalence, Invariants and Symmetry, page 131.

 • 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(...).

Examples

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

Example 1.

The formula for the contact vector field in terms of the generating function with 1 independent variable.

 > $\mathrm{DGsetup}\left(\left[x\right],\left[u\right],\mathrm{J11},1\right):$
 J11 > $\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(S\left(x,u\left[\right],u\left[1\right]\right),\mathrm{quiet}\right)$
 J11 > $\mathrm{GeneratingFunctionToContactVector}\left(S\left(x,u\left[\right],u\left[1\right]\right)\right)$
 ${-}{{S}}_{{{u}}_{{1}}}{}{\mathrm{D_x}}{+}\left({-}{{u}}_{{1}}{}{{S}}_{{{u}}_{{1}}}{+}{S}\right){}{{\mathrm{D_u}}}_{\left[\right]}{+}\left({{S}}_{{x}}{+}{{u}}_{{1}}{}{{S}}_{{{u}}_{\left[\right]}}\right){}{{\mathrm{D_u}}}_{{1}}$ (2.1)

The formula for the contact vector field in terms of the generating function with 2 independent variables.

 J11 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u\right],\mathrm{J21},1\right):$
 J21 > $\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(S\left(x,y,u\left[\right],u\left[1\right],u\left[2\right]\right),\mathrm{quiet}\right)$
 J21 > $\mathrm{GeneratingFunctionToContactVector}\left(S\left(x,y,u\left[\right],u\left[1\right],u\left[2\right]\right)\right)$
 ${-}{{S}}_{{{u}}_{{1}}}{}{\mathrm{D_x}}{-}{{S}}_{{{u}}_{{2}}}{}{\mathrm{D_y}}{+}\left({-}{{u}}_{{2}}{}{{S}}_{{{u}}_{{2}}}{-}{{u}}_{{1}}{}{{S}}_{{{u}}_{{1}}}{+}{S}\right){}{{\mathrm{D_u}}}_{\left[\right]}{+}\left({{S}}_{{x}}{+}{{u}}_{{1}}{}{{S}}_{{{u}}_{\left[\right]}}\right){}{{\mathrm{D_u}}}_{{1}}{+}\left({{S}}_{{y}}{+}{{u}}_{{2}}{}{{S}}_{{{u}}_{\left[\right]}}\right){}{{\mathrm{D_u}}}_{{2}}$ (2.2)

The formula for the contact vector field in terms of the generating function with 3 independent variables.

 J21 > $\mathrm{DGsetup}\left(\left[x,y,z\right],\left[u\right],\mathrm{J31},1\right):$
 J31 > $\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(S\left(x,y,z,u\left[\right],u\left[1\right],u\left[2\right],u\left[3\right]\right),\mathrm{quiet}\right)$
 J31 > $\mathrm{GeneratingFunctionToContactVector}\left(S\left(x,y,z,u\left[\right],u\left[1\right],u\left[2\right],u\left[3\right]\right)\right)$
 ${-}{{S}}_{{{u}}_{{1}}}{}{\mathrm{D_x}}{-}{{S}}_{{{u}}_{{2}}}{}{\mathrm{D_y}}{-}{{S}}_{{{u}}_{{3}}}{}{\mathrm{D_z}}{+}\left({-}{{u}}_{{3}}{}{{S}}_{{{u}}_{{3}}}{-}{{u}}_{{2}}{}{{S}}_{{{u}}_{{2}}}{-}{{u}}_{{1}}{}{{S}}_{{{u}}_{{1}}}{+}{S}\right){}{{\mathrm{D_u}}}_{\left[\right]}{+}\left({{S}}_{{x}}{+}{{u}}_{{1}}{}{{S}}_{{{u}}_{\left[\right]}}\right){}{{\mathrm{D_u}}}_{{1}}{+}\left({{S}}_{{y}}{+}{{u}}_{{2}}{}{{S}}_{{{u}}_{\left[\right]}}\right){}{{\mathrm{D_u}}}_{{2}}{+}\left({{S}}_{{z}}{+}{{u}}_{{3}}{}{{S}}_{{{u}}_{\left[\right]}}\right){}{{\mathrm{D_u}}}_{{3}}$ (2.3)

Example 2.

We choose some specific generating functions and calculate the resulting contact vector fields.

 J31 > $\mathrm{ChangeFrame}\left(\mathrm{J21}\right):$
 J21 > $S≔x+3y:$
 J21 > $\mathrm{GeneratingFunctionToContactVector}\left(S\right)$
 $\left({x}{+}{3}{}{y}\right){}{{\mathrm{D_u}}}_{\left[\right]}{+}{{\mathrm{D_u}}}_{{1}}{+}{3}{}{{\mathrm{D_u}}}_{{2}}$ (2.4)
 J21 > $S≔u\left[\right]:$
 J21 > $\mathrm{GeneratingFunctionToContactVector}\left(S\right)$
 ${{u}}_{\left[\right]}{}{{\mathrm{D_u}}}_{\left[\right]}{+}{{u}}_{{1}}{}{{\mathrm{D_u}}}_{{1}}{+}{{u}}_{{2}}{}{{\mathrm{D_u}}}_{{2}}$ (2.5)
 J21 > $S≔au\left[1\right]+bu\left[2\right]:$
 J21 > $\mathrm{GeneratingFunctionToContactVector}\left(S\right)$
 ${-}{a}{}{\mathrm{D_x}}{-}{b}{}{\mathrm{D_y}}$ (2.6)

Example 3.

Check the properties of the vector field obtained from  .

 J21 > $S≔u\left[\right]{u\left[2\right]}^{2}:$
 J21 > $X≔\mathrm{GeneratingFunctionToContactVector}\left(S\right)$
 ${X}{≔}{-}{2}{}{{u}}_{\left[\right]}{}{{u}}_{{2}}{}{\mathrm{D_y}}{-}{{u}}_{\left[\right]}{}{{u}}_{{2}}^{{2}}{}{{\mathrm{D_u}}}_{\left[\right]}{+}{{u}}_{{1}}{}{{u}}_{{2}}^{{2}}{}{{\mathrm{D_u}}}_{{1}}{+}{{u}}_{{2}}^{{3}}{}{{\mathrm{D_u}}}_{{2}}$ (2.7)

$X$ preserves the contact 1-form.

 J21 > $\mathrm{LieDerivative}\left(X,\mathrm{Cu}\left[\right]\right)$
 ${{u}}_{{2}}^{{2}}{}{{\mathrm{Cu}}}_{\left[\right]}$ (2.8)

$X$ is the prolongation of its projection to the space of independent and dependent variables.

 J21 > $\mathrm{\Phi }≔\mathrm{ProjectionTransformation}\left(1,0\right)$
 ${\mathrm{\Phi }}{≔}\left[{x}{=}{x}{,}{y}{=}{y}{,}{{u}}_{\left[\right]}{=}{{u}}_{\left[\right]}\right]$ (2.9)
 J21 > $Y≔\mathrm{Pushforward}\left(\mathrm{\Phi },X\right)$
 ${Y}{≔}{-}{2}{}{{u}}_{\left[\right]}{}{{u}}_{{2}}{}{\mathrm{D_y}}{-}{{u}}_{\left[\right]}{}{{u}}_{{2}}^{{2}}{}{{\mathrm{D_u}}}_{\left[\right]}$ (2.10)
 J21 > $\mathrm{Y1}≔\mathrm{Prolong}\left(Y,1\right)$
 ${\mathrm{Y1}}{≔}{-}{2}{}{{u}}_{\left[\right]}{}{{u}}_{{2}}{}{\mathrm{D_y}}{-}{{u}}_{\left[\right]}{}{{u}}_{{2}}^{{2}}{}{{\mathrm{D_u}}}_{\left[\right]}{+}{{u}}_{{1}}{}{{u}}_{{2}}^{{2}}{}{{\mathrm{D_u}}}_{{1}}{+}{{u}}_{{2}}^{{3}}{}{{\mathrm{D_u}}}_{{2}}$ (2.11)
 J21 > $\mathrm{Y1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}X$
 ${0}{}{\mathrm{D_x}}$ (2.12)

Example 4.

We use the commands GeneratingFunctionToContactVector and Flow to find a contact transformation.

 J21 > $S≔{u\left[1\right]}^{2}+{x}^{2}:$
 J21 > $X≔\mathrm{GeneratingFunctionToContactVector}\left(S\right)$
 ${X}{≔}{-}{2}{}{{u}}_{{1}}{}{\mathrm{D_x}}{+}\left({-}{{u}}_{{1}}^{{2}}{+}{{x}}^{{2}}\right){}{{\mathrm{D_u}}}_{\left[\right]}{+}{2}{}{x}{}{{\mathrm{D_u}}}_{{1}}$ (2.13)
 J21 > $\mathrm{\Phi }≔\mathrm{Flow}\left(X,t\right)$
 ${\mathrm{\Phi }}{≔}\left[{x}{=}{-}{{u}}_{{1}}{}{\mathrm{sin}}{}\left({2}{}{t}\right){+}{x}{}{\mathrm{cos}}{}\left({2}{}{t}\right){,}{y}{=}{y}{,}{{u}}_{\left[\right]}{=}{-}{{u}}_{{1}}^{{2}}{}\left(\frac{{\mathrm{cos}}{}\left({2}{}{t}\right){}{\mathrm{sin}}{}\left({2}{}{t}\right)}{{4}}{+}\frac{{t}}{{2}}\right){+}{{u}}_{{1}}{}{{\mathrm{cos}}{}\left({2}{}{t}\right)}^{{2}}{}{x}{-}{{x}}^{{2}}{}\left({-}\frac{{\mathrm{cos}}{}\left({2}{}{t}\right){}{\mathrm{sin}}{}\left({2}{}{t}\right)}{{4}}{+}\frac{{t}}{{2}}\right){+}{{u}}_{{1}}^{{2}}{}\left({-}\frac{{\mathrm{cos}}{}\left({2}{}{t}\right){}{\mathrm{sin}}{}\left({2}{}{t}\right)}{{4}}{+}\frac{{t}}{{2}}\right){+}{{x}}^{{2}}{}\left(\frac{{\mathrm{cos}}{}\left({2}{}{t}\right){}{\mathrm{sin}}{}\left({2}{}{t}\right)}{{4}}{+}\frac{{t}}{{2}}\right){-}{{u}}_{{1}}{}{x}{+}{{u}}_{\left[\right]}{,}{{u}}_{{1}}{=}{{u}}_{{1}}{}{\mathrm{cos}}{}\left({2}{}{t}\right){+}{x}{}{\mathrm{sin}}{}\left({2}{}{t}\right){,}{{u}}_{{2}}{=}{{u}}_{{2}}\right]$ (2.14)

Check that $\mathrm{Φ}$ is a contact transformation.

 J21 > $\mathrm{Pullback}\left(\mathrm{\Phi },\mathrm{du}\left[\right]-u\left[1\right]\mathrm{dx}-u\left[2\right]\mathrm{dy}\right)$
 ${-}{{u}}_{{1}}{}{\mathrm{dx}}{-}{{u}}_{{2}}{}{\mathrm{dy}}{+}{{\mathrm{du}}}_{\left[\right]}$ (2.15)

We note that $\mathrm{Φ}$ takes on a simple form for  and that it linearizes the Monge-Ampere equation .

 J21 > $\mathrm{Φ1}≔\mathrm{eval}\left(\mathrm{\Phi },t=\frac{\mathrm{\pi }}{4}\right)$
 ${\mathrm{Φ1}}{≔}\left[{x}{=}{-}{{u}}_{{1}}{,}{y}{=}{y}{,}{{u}}_{\left[\right]}{=}{-}{{u}}_{{1}}{}{x}{+}{{u}}_{\left[\right]}{,}{{u}}_{{1}}{=}{x}{,}{{u}}_{{2}}{=}{{u}}_{{2}}\right]$ (2.16)
 J21 > $\mathrm{Φ2}≔\mathrm{Prolong}\left(\mathrm{Φ1},2\right)$
 ${\mathrm{Φ2}}{≔}\left[{x}{=}{-}{{u}}_{{1}}{,}{y}{=}{y}{,}{{u}}_{\left[\right]}{=}{-}{{u}}_{{1}}{}{x}{+}{{u}}_{\left[\right]}{,}{{u}}_{{1}}{=}{x}{,}{{u}}_{{2}}{=}{{u}}_{{2}}{,}{{u}}_{{1}{,}{1}}{=}{-}\frac{{1}}{{{u}}_{{1}{,}{1}}}{,}{{u}}_{{1}{,}{2}}{=}{-}\frac{{{u}}_{{1}{,}{2}}}{{{u}}_{{1}{,}{1}}}{,}{{u}}_{{2}{,}{2}}{=}{-}\frac{{{u}}_{{1}{,}{2}}^{{2}}}{{{u}}_{{1}{,}{1}}}{+}{{u}}_{{2}{,}{2}}\right]$ (2.17)
 J21 > $\mathrm{\Delta }≔\mathrm{Pullback}\left(\mathrm{Φ2},u\left[1,1\right]u\left[2,2\right]-{u\left[1,2\right]}^{2}-1\right)$
 ${\mathrm{\Delta }}{≔}{-}\frac{{-}\frac{{{u}}_{{1}{,}{2}}^{{2}}}{{{u}}_{{1}{,}{1}}}{+}{{u}}_{{2}{,}{2}}}{{{u}}_{{1}{,}{1}}}{-}\frac{{{u}}_{{1}{,}{2}}^{{2}}}{{{u}}_{{1}{,}{1}}^{{2}}}{-}{1}$ (2.18)
 J21 > $\mathrm{simplify}\left(\mathrm{\Delta }\right)$
 ${-}\frac{{{u}}_{{2}{,}{2}}{+}{{u}}_{{1}{,}{1}}}{{{u}}_{{1}{,}{1}}}$ (2.19)