VerticalHomotopy - Maple Help

JetCalculus[VerticalHomotopy] - apply the vertical homotopy operator to a bi-form on a jet space

Calling Sequences

VerticalHomotopy(${\mathbf{ω}}$, options)

Parameters

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

options   - various keyword arguments for specifying the integration path used by the vertical homotopy operator.

Description

 • Let be a fiber bundle and let  be the associated infinite jet bundle. Let be a bi-form of degree $\left(r,s\right)$on ${J}^{\mathrm{∞}}\left(E\right)$. Then is called ${d}_{V}$ closed if where ${d}_{V}$denotes the vertical exterior derivative and is called ${d}_{V}$ exact if there exists a bi-form of degree such that . Every closed bi-form is ${d}_{V}$ exact in some neighborhood about each point in jet space. If then there are numerous algorithms for finding a bi-form such that  One approach is to use the vertical homotopy operators

.

These operators satisfyso that if then where .

 • If is a bi-form of degree with $s\ge 1$, then VerticalHomotopy(omega) returns a bi-form $\mathrm{η}$ of degree (such that .
 • The optional arguments available to DeRhamHomotopy can also be invoked with VerticalHomotopy.
 • The command VerticalHomotopy is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form VerticalHomotopy(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-VerticalHomotopy(...).
 Details Here are the explicit formulas for the vertical homotopy operators. Let , ..., be a local system of jet coordinates and let be the contact forms. The vertical radial vector field on is  and its prolongation to jet space is pr The flow of the vector field pr $R$ is the transformation  given by = . The vertical homotopy operators are then defined in terms of pr and ${\mathrm{\Phi }}_{t}$  and the interior product operator (see Hook) by . As a concrete example, if is given by (, ...), then    .   Thus the formulas for the vertical homotopy operators are essentially the same as that for the standard de Rham homotopy operators.

Examples

 > with(DifferentialGeometry): with(JetCalculus):

Example 1.

Create the jet space ${J}^{3}\left(E\right)$ for the bundle ${\mathrm{ℝ}}^{2}×\mathrm{ℝ}\mathit{\to }\mathrm{ℝ}$ with coordinates $\left(x,y,u\right)\to \left(x,y\right)$.

 > DGsetup([x, y], [u], E, 1):

Show that the form is ${d}_{V}$ closed.

 E > omega1 := evalDG(Cu[] &w Cu[1] &w Cu[2]);
 ${\mathrm{ω1}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{0}{,}{3}\right]\right]{,}\left[\left[\left[{3}{,}{4}{,}{5}\right]{,}{1}\right]\right]\right]\right)$ (3.1)
 E > VerticalExteriorDerivative(omega1);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{0}{,}{4}\right]\right]{,}\left[\left[\left[{3}{,}{4}{,}{5}{,}{6}\right]{,}{0}\right]\right]\right]\right)$ (3.2)

Apply the vertical homotopy operator to ${\mathrm{ω}}_{1}.$

 E > eta1a := VerticalHomotopy(omega1);
 ${\mathrm{eta1a}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{0}{,}{2}\right]\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}\frac{{{u}}_{{2}}}{{3}}\right]{,}\left[\left[{3}{,}{5}\right]{,}{-}\frac{{{u}}_{{1}}}{{3}}\right]{,}\left[\left[{4}{,}{5}\right]{,}\frac{{{u}}_{\left[\right]}}{{3}}\right]\right]\right]\right)$ (3.3)

Check that the vertical exterior derivative of gives ${\mathrm{ω}}_{1}.$

 E > omega1 &minus VerticalExteriorDerivative(eta1a);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{0}{,}{3}\right]\right]{,}\left[\left[\left[{3}{,}{4}{,}{5}\right]{,}{0}\right]\right]\right]\right)$ (3.4)

Alternatives to can be obtained using the path = "zigzag" option for the VerticalHomotopy command. See DeRhamHomotopy for more details.

 E > eta1b := VerticalHomotopy(omega1, path = "zigzag");
 ${\mathrm{eta1b}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{0}{,}{2}\right]\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{{u}}_{{2}}\right]\right]\right]\right)$ (3.5)
 E > omega1 &minus VerticalExteriorDerivative(eta1b);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{0}{,}{3}\right]\right]{,}\left[\left[\left[{3}{,}{4}{,}{5}\right]{,}{0}\right]\right]\right]\right)$ (3.6)
 E > eta1c := VerticalHomotopy(omega1, path = "zigzag", variableorder = [u[1], u[2], u[], u[1, 1], u[1, 2], u[2, 2]]);
 ${\mathrm{eta1c}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{0}{,}{2}\right]\right]{,}\left[\left[\left[{4}{,}{5}\right]{,}{{u}}_{\left[\right]}\right]\right]\right]\right)$ (3.7)
 E > omega1 &minus VerticalExteriorDerivative(eta1c);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{0}{,}{3}\right]\right]{,}\left[\left[\left[{3}{,}{4}{,}{5}\right]{,}{0}\right]\right]\right]\right)$ (3.8)