JetCalculus[HorizontalHomotopy] - apply the horizontal homotopy operator to a bi-form on a jet space
ω - a differential bi-form on the jet space
options - any of the optional arguments used in the commands DeRhamHomotopy
Let π:E→M be a fiber bundle, with base dimension n and fiber dimension m and let π∞ :J∞E →M be the infinite jet bundle of E. The space of p-forms ΩpJ∞E decomposes into a direct sum ΩpJ∞ = ⨁r+s =p Ωr,sJ∞E, where Ωr,s J∞E is the space of bi-forms of horizontal degree r and vertical degree s. The horizontal exterior derivative is a mapping dH :Ωr,sJ∞E→ Ωr+1,sJ∞E with the following properties. A form ω ∈ Ωr,s J∞E is called dH closed if dH ω = 0 and dH exact if there is a bi-form η∈ Ωr−1,s J∞E such that ω = dH η. Since dH∘dH =0, every dH exact bi-form is dH closed.
[i] If r<n, then every dH closed bi-form ω is dH exact, ω = dH η.
[ii] If r =n and s=0 and Eω =0, where E is the Euler-Lagrange operator, then ω = dH η.
[iii] If r =n and s>0 and Iω =0, where I is the integration by parts operator, then ω = dH η.
There are a number of algorithms for finding the bi-form η. One approach is to use the horizontal homotopy operators hHr,s : Ωr,sJ∞E → Ωr−1,sJ∞E. Similar to the DeRham homotopy operator, these homotopy operators satisfy the identities
[i] hHr+1, s dH ω + dHhHr,s ω = ω if r<n ;
[ii] dHhHr,s ω = ω if r =n and s =0 and Eω =0.
[iii] dHhHr,s ω = ω if r =n and s >0 and Iω =0.
If ω is a bi-form of degree r,s with r>0 then HorizontalHomotopy(ω) returns a bi-form of degree (r−1, s).
For s >0 the operators hHr,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 s= 0 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(...).
Here are the explicit formulas for the horizontal homotopy operators. Let (xi, uα, uiα, uijα, ..., uij ⋅⋅⋅ kα, ....) be a local system of jet coordinates and let Θα = duα−uℓαdxℓ. Let ω ∈ Ωr,sJ∞ E be a k-th order bi-form with s≥1 and let EαIω∈ Ωr,s−1J∞ Ebe the higher (interior product) Euler operators. Let DI = Di1i2⋅⋅⋅iℓ be the multi-total derivative operator and let ωj = ιDjω. Then
hHr,sω = 1s∑I =0k−1 |I|+1n−r+I +1DI Θα ∧EαIjωj.
For s=0, the horizontal homotopy operator is defined in terms of the vertical exterior derivative dV and the vertical homotopy operator hVr,s by
hHr,sω = hVr−1, 1hHr,1dVω.
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.
Create the jet space J3E for the bundle E with coordinates x, u→ x.
Show that the EulerLagrange form for ω1 is 0 so that ω1 is dH exact.