ZigZag - Maple Help

JetCalculus[ZigZag] - lift a ${d}_{H}$-closed form on a jet space to a $d$-closed form

Calling Sequences

ZigZag(${\mathbf{ω}}$)

Parameters

$\mathrm{ω}$     - a differential bi-form on the jet space of a fiber bundle

Description

 • Let be a fiber bundle, with base dimension $n$ and fiber dimension $m$ and let be the infinite jet bundle of $E$. The space of $p$-forms on decomposes as a direct sum of bi-forms

Let be a bi-form of degree $\left(r,s\right)$. I f suppose that  or, if , that . See HorizontalExteriorDerivative, VerticalExteriorDerivative, and IntegrationByParts for the definitions of the space ${\mathrm{\Omega }}^{\left(r,s\right)}\left({J}^{\infty }\left(E\right)\right)$, the horizontal exterior derivative ${d}_{H}$, the vertical exterior derivative and the integration by parts operator

Given that  or define a degree form  by

where and ${d}_{V}\left({\mathrm{θ}}_{i}\right)={d}_{H}\left({\mathrm{θ}}_{i+1}\right).$

The forms ${\mathrm{θ}}_{i}$ are of bi-degree The forms can be calculated inductively using the horizontal homotopy operators . The fundamental property of this construction is that the form is always closed with respect to the standard exterior derivative, that is,

 • If $\mathrm{ω}$ is a bi-form of degree $\left(r,s\right),$then ZigZag(${\mathbf{\omega }}$) returns the differential form of degree $r+s$.
 • The command ZigZag is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form ZigZag(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-ZigZag(...).

Examples

 > with(DifferentialGeometry): with(JetCalculus):

Example 1.

Create the jet space ${J}^{3}\left(E\right)$for the bundle with coordinates $\left(x,y,u\right)\to \left(x,y\right)$.

 > DGsetup([x, y], [u], E, 3):

Define a type (1, 0) form and show that it is ${d}_{H}$ -closed.

 E > omega1 := evalDG((u[1, 2]*u[1, 1, 1] + u[1 ,1]*u[1, 1, 2])*Dx + (u[1, 2]*u[1 ,1, 2] + u[1, 1]*u[1, 2, 2])*Dy);
 ${\mathrm{ω1}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{{u}}_{{1}{,}{1}}{}{{u}}_{{1}{,}{1}{,}{2}}{+}{{u}}_{{1}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{1}}\right]{,}\left[\left[{2}\right]{,}{{u}}_{{1}{,}{1}}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{{u}}_{{1}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{2}}\right]\right]\right]\right)$ (2.1)
 E > HorizontalExteriorDerivative(omega1);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{0}\right]\right]\right]\right)$ (2.2)

Apply the ZigZag command to ${\mathrm{ω}}_{1}$ to obtain a form ${\mathrm{θ}}_{1}$.

 E > theta1 := ZigZag(omega1);
 ${\mathrm{θ1}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{E}{,}{1}\right]{,}\left[\left[\left[{6}\right]{,}{{u}}_{{1}{,}{2}}\right]{,}\left[\left[{7}\right]{,}{{u}}_{{1}{,}{1}}\right]\right]\right]\right)$ (2.3)

Check that ${\mathrm{θ}}_{1}$ is $d$-closed and that its [1, 0] component matches ${\mathrm{ω}}_{1}$.

 E > ExteriorDerivative(theta1);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{E}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{0}\right]\right]\right]\right)$ (2.4)
 E > convert(theta1, DGbiform, [1, 0]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{{u}}_{{1}{,}{1}}{}{{u}}_{{1}{,}{1}{,}{2}}{+}{{u}}_{{1}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{1}}\right]{,}\left[\left[{2}\right]{,}{{u}}_{{1}{,}{1}}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{{u}}_{{1}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{2}}\right]\right]\right]\right)$ (2.5)

Example 2.

Define a type (2, 0) form ${\mathrm{ω}}_{2}$ and show that its Euler-Lagrange form is 0.

 E > omega2 := evalDG((- u[2, 2]*u[1, 2] - u[2]*u[1, 2, 2] - u[1]*u[1, 2, 2] - u[1, 2]^2)*Dx &w Dy);
 ${\mathrm{ω2}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{{u}}_{{1}}{}{{u}}_{{1}{,}{2}{,}{2}}{-}{{u}}_{{2}}{}{{u}}_{{1}{,}{2}{,}{2}}{-}{{u}}_{{1}{,}{2}}^{{2}}{-}{{u}}_{{2}{,}{2}}{}{{u}}_{{1}{,}{2}}\right]\right]\right]\right)$ (2.6)
 E > EulerLagrange(omega2);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{0}\right]\right]\right]\right)$ (2.7)

Apply the ZigZag command to ${\mathrm{ω}}_{2}$ to obtain a 2-form ${\mathrm{θ}}_{2}$.

 E > theta2 := ZigZag(omega2);
 ${\mathrm{θ2}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{E}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{-}\frac{{{u}}_{{1}{,}{2}}}{{3}}{-}\frac{{{u}}_{{2}{,}{2}}}{{6}}\right]{,}\left[\left[{1}{,}{5}\right]{,}{-}\frac{{2}{}{{u}}_{{1}{,}{2}}}{{3}}\right]{,}\left[\left[{1}{,}{7}\right]{,}{-}\frac{{{u}}_{{1}}}{{3}}{-}\frac{{2}{}{{u}}_{{2}}}{{3}}\right]{,}\left[\left[{1}{,}{8}\right]{,}{-}\frac{{{u}}_{{1}}}{{6}}\right]{,}\left[\left[{2}{,}{4}\right]{,}\frac{{{u}}_{{2}{,}{2}}}{{3}}\right]{,}\left[\left[{2}{,}{5}\right]{,}\frac{{{u}}_{{1}{,}{2}}}{{3}}{+}\frac{{{u}}_{{2}{,}{2}}}{{6}}\right]{,}\left[\left[{2}{,}{7}\right]{,}\frac{{{u}}_{{2}}}{{3}}\right]{,}\left[\left[{2}{,}{8}\right]{,}\frac{{{u}}_{{2}}}{{6}}{+}\frac{{{u}}_{{1}}}{{3}}\right]{,}\left[\left[{3}{,}{7}\right]{,}{-}\frac{{1}}{{3}}\right]{,}\left[\left[{3}{,}{8}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{4}{,}{5}\right]{,}\frac{{1}}{{3}}\right]\right]\right]\right)$ (2.8)

Check that ${\mathrm{θ}}_{2}$ is $d-$closed and that its [2, 0] component matches ${\mathrm{ω}}_{2}$.

 E > ExteriorDerivative(theta2);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{E}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{0}\right]\right]\right]\right)$ (2.9)
 E > convert(theta2, DGbiform, [2, 0]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{{u}}_{{1}}{}{{u}}_{{1}{,}{2}{,}{2}}{-}{{u}}_{{2}}{}{{u}}_{{1}{,}{2}{,}{2}}{-}{{u}}_{{1}{,}{2}}^{{2}}{-}{{u}}_{{2}{,}{2}}{}{{u}}_{{1}{,}{2}}\right]\right]\right]\right)$ (2.10)

Example 3.

Define a type (2, 1) form ${\mathrm{ω}}_{3}$ and show that .

 E > omega3 := EulerLagrange(u[1]*u[2]^2*Dx &w Dy);
 ${\mathrm{ω3}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}}{-}{4}{}{{u}}_{{2}}{}{{u}}_{{1}{,}{2}}\right]\right]\right]\right)$ (2.11)
 E > IntegrationByParts(VerticalExteriorDerivative(omega3));
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{2}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{4}\right]{,}{0}\right]\right]\right]\right)$ (2.12)

Apply the ZigZag command to ${\mathrm{ω}}_{3}$ to obtain a form ${\mathrm{θ}}_{3}$.

 E > theta3 := ZigZag(omega3);
 ${\mathrm{θ3}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{E}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{4}\right]{,}{-}{2}{}{{u}}_{{2}}^{{2}}\right]{,}\left[\left[{1}{,}{2}{,}{5}\right]{,}{-}{4}{}{{u}}_{{1}}{}{{u}}_{{2}}\right]{,}\left[\left[{1}{,}{3}{,}{4}\right]{,}{2}{}{{u}}_{{2}}\right]{,}\left[\left[{1}{,}{3}{,}{5}\right]{,}{2}{}{{u}}_{{1}}\right]{,}\left[\left[{2}{,}{3}{,}{5}\right]{,}{-}{2}{}{{u}}_{{2}}\right]\right]\right]\right)$ (2.13)

Check that ${\mathrm{θ}}_{3}$ is $d-$closed and that its [2, 1] component matches ${\mathrm{ω}}_{3}.$

 E > ExteriorDerivative(theta3);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{E}{,}{4}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{4}\right]{,}{0}\right]\right]\right]\right)$ (2.14)
 E > convert(theta3, DGbiform, [2, 1]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}}{-}{4}{}{{u}}_{{2}}{}{{u}}_{{1}{,}{2}}\right]\right]\right]\right)$ (2.15)