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);
 ${\mathrm{ω1}}{≔}\left({{u}}_{{1}{,}{1}}{}{{u}}_{{1}{,}{1}{,}{2}}{+}{{u}}_{{1}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{1}}\right){}{\mathrm{Dx}}{+}\left({{u}}_{{1}{,}{1}}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{{u}}_{{1}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{2}}\right){}{\mathrm{Dy}}$ (2.1)
 E > HorizontalExteriorDerivative(omega1);
 ${0}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}$ (2.2)

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

 E > theta1 := ZigZag(omega1);
 ${\mathrm{θ1}}{≔}{{u}}_{{1}{,}{2}}{}{{\mathrm{du}}}_{{1}{,}{1}}{+}{{u}}_{{1}{,}{1}}{}{{\mathrm{du}}}_{{1}{,}{2}}$ (2.3)

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

 E > ExteriorDerivative(theta1);
 ${0}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}$ (2.4)
 E > convert(theta1, DGbiform, [1, 0]);
 $\left({{u}}_{{1}{,}{1}}{}{{u}}_{{1}{,}{1}{,}{2}}{+}{{u}}_{{1}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{1}}\right){}{\mathrm{Dx}}{+}\left({{u}}_{{1}{,}{1}}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{{u}}_{{1}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{2}}\right){}{\mathrm{Dy}}$ (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);
 ${\mathrm{ω2}}{≔}{-}\left({{u}}_{{1}}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{{u}}_{{2}}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{{u}}_{{1}{,}{2}}^{{2}}{+}{{u}}_{{2}{,}{2}}{}{{u}}_{{1}{,}{2}}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}$ (2.6)
 E > EulerLagrange(omega2);
 ${0}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}$ (2.7)

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

 E > theta2 := ZigZag(omega2);
 ${\mathrm{θ2}}{≔}{-}\left(\frac{{{u}}_{{1}{,}{2}}}{{3}}{+}\frac{{{u}}_{{2}{,}{2}}}{{6}}\right){}{\mathrm{dx}}{}{\bigwedge }{}{{\mathrm{du}}}_{{1}}{-}\frac{{2}{}{{u}}_{{1}{,}{2}}}{{3}}{}{\mathrm{dx}}{}{\bigwedge }{}{{\mathrm{du}}}_{{2}}{-}\left(\frac{{{u}}_{{1}}}{{3}}{+}\frac{{2}{}{{u}}_{{2}}}{{3}}\right){}{\mathrm{dx}}{}{\bigwedge }{}{{\mathrm{du}}}_{{1}{,}{2}}{-}\frac{{{u}}_{{1}}}{{6}}{}{\mathrm{dx}}{}{\bigwedge }{}{{\mathrm{du}}}_{{2}{,}{2}}{+}\frac{{{u}}_{{2}{,}{2}}}{{3}}{}{\mathrm{dy}}{}{\bigwedge }{}{{\mathrm{du}}}_{{1}}{+}\left(\frac{{{u}}_{{1}{,}{2}}}{{3}}{+}\frac{{{u}}_{{2}{,}{2}}}{{6}}\right){}{\mathrm{dy}}{}{\bigwedge }{}{{\mathrm{du}}}_{{2}}{+}\frac{{{u}}_{{2}}}{{3}}{}{\mathrm{dy}}{}{\bigwedge }{}{{\mathrm{du}}}_{{1}{,}{2}}{+}\left(\frac{{{u}}_{{2}}}{{6}}{+}\frac{{{u}}_{{1}}}{{3}}\right){}{\mathrm{dy}}{}{\bigwedge }{}{{\mathrm{du}}}_{{2}{,}{2}}{-}\frac{{1}}{{3}}{}{{\mathrm{du}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{du}}}_{{1}{,}{2}}{+}\frac{{1}}{{6}}{}{{\mathrm{du}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{du}}}_{{2}{,}{2}}{+}\frac{{1}}{{3}}{}{{\mathrm{du}}}_{{1}}{}{\bigwedge }{}{{\mathrm{du}}}_{{2}}$ (2.8)

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

 E > ExteriorDerivative(theta2);
 ${0}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{}{\bigwedge }{}{{\mathrm{du}}}_{\left[\right]}$ (2.9)
 E > convert(theta2, DGbiform, [2, 0]);
 ${-}\left({{u}}_{{1}}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{{u}}_{{2}}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{{u}}_{{1}{,}{2}}^{{2}}{+}{{u}}_{{2}{,}{2}}{}{{u}}_{{1}{,}{2}}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}$ (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);
 ${\mathrm{ω3}}{≔}{-}\left({2}{}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}}{+}{4}{}{{u}}_{{2}}{}{{u}}_{{1}{,}{2}}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}$ (2.11)
 E > IntegrationByParts(VerticalExteriorDerivative(omega3));
 ${0}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}$ (2.12)

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

 E > theta3 := ZigZag(omega3);
 ${\mathrm{θ3}}{≔}{-}{2}{}{{u}}_{{2}}^{{2}}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{}{\bigwedge }{}{{\mathrm{du}}}_{{1}}{-}{4}{}{{u}}_{{1}}{}{{u}}_{{2}}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{}{\bigwedge }{}{{\mathrm{du}}}_{{2}}{+}{2}{}{{u}}_{{2}}{}{\mathrm{dx}}{}{\bigwedge }{}{{\mathrm{du}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{du}}}_{{1}}{+}{2}{}{{u}}_{{1}}{}{\mathrm{dx}}{}{\bigwedge }{}{{\mathrm{du}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{du}}}_{{2}}{-}{2}{}{{u}}_{{2}}{}{\mathrm{dy}}{}{\bigwedge }{}{{\mathrm{du}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{du}}}_{{2}}$ (2.13)

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

 E > ExteriorDerivative(theta3);
 ${0}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{}{\bigwedge }{}{{\mathrm{du}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{du}}}_{{1}}$ (2.14)
 E > convert(theta3, DGbiform, [2, 1]);
 ${-}\left({2}{}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}}{+}{4}{}{{u}}_{{2}}{}{{u}}_{{1}{,}{2}}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}$ (2.15)