JetCalculus[VerticalHomotopy] - apply the vertical homotopy operator to a bi-form on a jet space
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.
Let π:E→M be a fiber bundle and let π∞:J∞E → M be the associated infinite jet bundle. Let ω ∈Ωr,sJ∞E be a bi-form of degree r,son J∞E. Then ω is called dV closed if dV ω=0, where dVdenotes the vertical exterior derivative and ω is called dV exact if there exists a bi-form of degree r, s−1 such that ω = dV η. Every dV closed bi-form is dV exact in some neighborhood about each point in jet space. If dV ω=0,then there are numerous algorithms for finding a bi-form η such that ω = dV η. One approach is to use the vertical homotopy operators
hVr,s : Ωr,sJ∞E → Ωr,s−1J∞E.
These operators satisfy hvr, s+1 dV ω + dVhVr,s ω = ω so that if dV ω=0, then ω = dV η where η = hVr,s ω.
If ω is a bi-form of degree r, s with s≥1, then VerticalHomotopy(omega) returns a bi-form η of degree (r, s−1) such that ω = dVη.
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(...).
Here are the explicit formulas for the vertical homotopy operators. Let (xi, uα, uiα, uijα, ..., uij ⋅⋅⋅ kα, ....) be a local system of jet coordinates and let Θi1i2⋅⋅⋅ikα = duα−ui1i2⋅⋅⋅ikℓα dxℓ be the contact forms. The vertical radial vector field on E is R = uα ∂ ∂uα and its prolongation to jet space is
pr R = uα ∂ ∂uα + uiα ∂ ∂uiα + uijα ∂ ∂uij α + ⋅⋅⋅⋅
The flow of the vector field pr R is the transformation Φt:J∞E→ J∞E given by Φtxi, uα, uiα, uijα, ...= xi, etuα, etuiα, etuijα, .... The vertical homotopy operators are then defined in terms of pr R and Φt and the interior product operator ι (see Hook) by
hVr,sω = ∫011t Φlogt* ιpr Rω dt .
As a concrete example, if ω ∈ Ω1,2J∞E is given by ω = Aℓ α β I J(xi, uα, uiα, uijα, ...) dxℓ∧ΘIα∧ΘJβ , then
hVr,sω = ∫01 t Aℓ α β I Jxi, tuα, tuiα, tuijα, ... dxℓ∧ΘIα∧ΘJβ ⅆt.
Thus the formulas for the vertical homotopy operators are essentially the same as that for the standard de Rham homotopy operators.
Create the jet space J3E for the bundle E =ℝ2×ℝ→ℝ with coordinates x,y,u→x,y.
DGsetup([x, y], [u], E, 1):
Show that the form ω1 is dV closed.
omega1 := evalDG(Cu &w Cu &w Cu);
Apply the vertical homotopy operator to ω1.
eta1a := VerticalHomotopy(omega1);
Check that the vertical exterior derivative of η1 gives ω1.
omega1 &minus VerticalExteriorDerivative(eta1a);
Alternatives to η1 can be obtained using the path = "zigzag" option for the VerticalHomotopy command. See DeRhamHomotopy for more details.
eta1b := VerticalHomotopy(omega1, path = "zigzag");
omega1 &minus VerticalExteriorDerivative(eta1b);
eta1c := VerticalHomotopy(omega1, path = "zigzag", variableorder = [u, u, u, u[1, 1], u[1, 2], u[2, 2]]);
omega1 &minus VerticalExteriorDerivative(eta1c);
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