HorizontalHomotopy - Maple Help

JetCalculus[HorizontalHomotopy] - apply the horizontal homotopy operator to a bi-form on a jet space

Calling Sequences

HorizontalHomotopy(${\mathrm{ω}}$, options)

Parameters

$\mathrm{\omega }$        - a differential bi-form on the jet space

options - any of  the optional arguments used in the commands DeRhamHomotopy

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 ${\mathrm{Ω}}^{p}\left({J}^{\mathrm{∞}}\left(E\right)\right)$ decomposes into a direct sum , where  is the space of bi-forms of horizontal degree $r$ and vertical degree The horizontal exterior derivative  is a mapping  with the following properties. A form is called closed if and exact if there is a bi-form such that . Since every exact bi-form is closed.

[i] If $r, then every closed bi-form is exact,

[ii] If and and where $E$ is the Euler-Lagrange operator, then .

[iii] If and and where is the integration by parts operator, then .

There are a number of algorithms for finding the bi-form One approach is to use the horizontal homotopy operators . Similar to the DeRham homotopy operator, these homotopy operators satisfy the identities

[i]  if

[ii]  if  and and

[iii]  if  and and

 • If  is a bi-form of degree with  then HorizontalHomotopy(${\mathrm{ω}}$) returns a bi-form of degree (.
 • For  the operators ${h}_{{}^{}H}^{r,s}$ are total differential operators and therefore, unlike the usual homotopy operators for the de Rham complex or the vertical homotopy operators for bi-forms on jet spaces, do not involve any quadratures. For the horizontal homotopy does involve quadratures and the optional arguments used in the commands DeRhamHomotopy or VerticalHomotopy can be invoked.
 • The command HorizontalHomotopy is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form HorizontalHomotopy(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-HorizontalHomotopy(...).
 Details Here are the explicit formulas for the horizontal homotopy operators. Let , ..., be a local system of jet coordinates and let . Let  be a $k$-th order bi-form with and let ${E}_{\mathrm{α}}^{I}\left(\mathrm{ω}\right)$be the higher (interior product) Euler operators. Let  be the multi-total derivative operator and let  . Then    .   For the horizontal homotopy operator is defined in terms of the vertical exterior derivative ${d}_{V}$  and the vertical homotopy operator  ${h}_{V}^{r,s}$ by  For further information, see: [i] Ian M. Anderson, Notes on the Variational Bicomplex. [ii] Niky. Kamran, Selected Topics in the Geometrical Study of Differential Equations, CBMS Lecture Series, 2002. [iii] Peter J. Olver, Applications of Lie Groups to Differential Equations, Chapter 5.

Examples

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

Example 1.

Create the jet space for the bundle with coordinates

 > $\mathrm{DGsetup}\left(\left[x\right],\left[u\right],E,3\right):$

Show that the EulerLagrange form for is 0 so that ${\mathrm{ω}}_{}{}_{1}$ is ${d}_{H}$ exact.

 E > $\mathrm{ω1}≔\mathrm{evalDG}\left(\left(u\left[1,1,1\right]u\left[1\right]+xu\left[1,1,1\right]u\left[1,1\right]+2u\left[1,1\right]u\left[1,1,1\right]+xu\left[1\right]u\left[1,1,1,1\right]\right)\mathrm{Dx}\right)$
 ${\mathrm{ω1}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}{,}{1}{,}{1}}{+}{x}{}{{u}}_{{1}{,}{1}{,}{1}}{}{{u}}_{{1}{,}{1}}{+}{{u}}_{{1}{,}{1}{,}{1}}{}{{u}}_{{1}}{+}{2}{}{{u}}_{{1}{,}{1}}{}{{u}}_{{1}{,}{1}{,}{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}{,}{1}{,}{1}}{+}{x}{}{{u}}_{{1}{,}{1}{,}{1}}{}{{u}}_{{1}{,}{1}}{+}{{u}}_{{1}{,}{1}{,}{1}}{}{{u}}_{{1}}{+}{2}{}{{u}}_{{1}{,}{1}}{}{{u}}_{{1}{,}{1}{,}{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}{,}{1}{,}{1}}{+}{x}{}{{u}}_{{1}{,}{1}{,}{1}}{}{{u}}_{{1}{,}{1}}{+}{}_{}\right]\right]\right]\right)$