[i] If $r<n$, then every $d_H$closed bi-form $\mathrm{\&omega;}$is $d_H$exact, $\mathrm{\omega } \&equals; d_{H }\mathrm{\&eta;}\&period;$

[ii] If $r \&equals;n$and $s\&equals;0$and $E\left(\mathrm{\&omega;}\right) \&equals;0\&comma;$where $E$ is the Euler-Lagrange operator, then $\mathrm{\omega } \&equals; d_{H }\mathrm{\&eta;}$.

[iii] If $r \&equals;n$and $s\&gt;0$and $I\left(\mathrm{\&omega;}\right) \&equals;0\&comma;$where $I$is the integration by parts operator, then $\mathrm{\omega } \&equals; d_{H }\mathrm{\&eta;}$.

There are a number of algorithms for finding the bi-form $\mathrm{\&eta;}\&period;$One approach is to use the horizontal homotopy operators $h_{{}^{}H}^{r\&comma;s} \&colon; {\mathrm{\Omega }}^{\left(r\&comma;s\right)}\left(J^{\infty }\left(E\right)\right) \to {\mathrm{\Omega }}^{\left(r-1\&comma;s\right)}\left(J^{\infty }\left(E\right)\right)$. Similar to the DeRham homotopy operator, these homotopy operators satisfy the identities

[i] $h_{{}^{}H}^{r\&plus;1\&comma; s} \left(d_H \mathrm{\&omega;}\right) \&plus; d_H\left(h_{{}^{}H}^{r\&comma;s} \left(\mathrm{\&omega;}\right)\right) \&equals; \mathrm{\&omega;}$ if  $r<n \&semi;$

[ii] $d_H\left(h_{{}^{}H}^{r\&comma;s} \left(\mathrm{\&omega;}\right)\right) \&equals; \mathrm{\&omega;}$ if  $r \&equals;n$and $s \&equals;0$and $E\left(\mathrm{\&omega;}\right) \&equals;0.$

[iii] $d_H\left(h_{{}^{}H}^{r\&comma;s} \left(\mathrm{\&omega;}\right)\right) \&equals; \mathrm{\&omega;}$ if  $r \&equals;n$and $s \&gt;0$and $I\left(\mathrm{\&omega;}\right) \&equals;0.$

Here are the explicit formulas for the horizontal homotopy operators. Let $\&lpar;x^i\&comma; u^{\mathrm{\&alpha;}}\&comma; u_{i_{}}^{\mathrm{\&alpha;}}\&comma; u_{i_{}j}^{\mathrm{\&alpha;}}$, ..., $u_{\mathrm{ij} \cdot \cdot \cdot k}^{\mathrm{\&alpha;}}\&comma; ....\&rpar;$be a local system of jet coordinates and let ${\mathrm{\&Theta;}}^{\mathrm{\&alpha;}} \&equals; {\mathrm{du}}^{\mathrm{\&alpha;}}-u_{\mathrm{\&ell;}}^{\mathrm{\&alpha;}}{\mathrm{dx}}^{\mathrm{\&ell;}}$. Let $\mathrm{\omega } \in {\mathrm{\Omega }}^{\left(r\&comma;s\right)}\left(J^{\mathit{\infty }\mathit{ }}\left(E\right)\right)$ be a $k$-th order bi-form with $s\ge 1$and let $E_{\mathrm{\&alpha;}}^I\left(\mathrm{\&omega;}\right)$$\in {\mathrm{\&Omega;}}^{\left(r\&comma;s-1\right)}\left(J^{\mathit{\infty }\mathit{ }}\left(E\right)\right)$be the higher (interior product) Euler operators. Let $$${\mathrm{D}}_I \&equals; {\mathrm{D}}_{i_1{i}_2\cdot \cdot \cdot i_{\mathrm{\&ell;}}}$ be the multi-total derivative operator and let  ${\mathrm{\&omega;}}_j \&equals; {\mathrm{\&iota;}}_{{\mathrm{D}}_j}\left(\mathrm{\&omega;}\right)$. Then

$$$h_H^{r\&comma;s}\left(\mathrm{\&omega;}\right) \&equals; \frac{1}{s}\sum _{\left|I\right| \&equals;0}^{k-1}$ $\frac{\&verbar;I\&verbar;\&plus;1}{n-r\&plus;\left|I\right| \&plus;1}{\mathrm{D}}_I \left({\mathrm{\&Theta;}}^{\mathrm{\&alpha;}} \wedge E_{\mathrm{\&alpha;}}^{\mathrm{Ij}}\left({\mathrm{\&omega;}}_j\right)\right)$.

For $s\&equals;0\&comma;$the horizontal homotopy operator is defined in terms of the vertical exterior derivative $d_V$  and the vertical homotopy operator  $h_V^{r\&comma;s}$ by

$$$$$h_H^{r\&comma;s}\left(\mathrm{\&omega;}\right) \&equals; h_V^{r-1\&comma; 1}\left(h_H^{r\&comma;1}\left(d_V\mathrm{\&omega;}\right)\right)\&period;$

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.

Example 1.

Create the jet space $J^3\left(E\right)$for the bundle $E$with coordinates $\left(x\&comma; u\right)\to x\&period;$

Show that the EulerLagrange form for ${\mathrm{\&omega;}}_{1 }$is 0 so that ${\mathrm{\&omega;}}_{}{}_1$ is $d_H$ exact.