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

Calling Sequences

     VerticalHomotopy(ω, options)

Parameters

     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

Details

Examples

Description

• 

Let π:EM 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, s1 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,s1J∞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 s1, then VerticalHomotopy(omega) returns a bi-form η of degree (r, s1) 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(...).

Details

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 Θi1i2ikα = duαui1i2ikℓα 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:JE JE 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  Ω1,2JE is given by ω = Aℓ α β   I  J(xi, uα, uiα, uijα, ...) dxℓΘIαΘJβ , then

   hVr,sω = 01 t A α β   I  Jxi, tuα, tuiα, tuijα, ... dxΘIαΘJβ ⅆt.

 

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 J3E for the bundle E =ℝ2×ℝℝ with coordinates x,y,ux,y.

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

 

Show that the form ω1 is dV closed.

E > 

omega1 := evalDG(Cu[] &w Cu[1] &w Cu[2]);

ω1CuCu1Cu2

(3.1)
E > 

VerticalExteriorDerivative(omega1);

0CuCu1Cu2Cu1,1

(3.2)

 

Apply the vertical homotopy operator to ω1.

E > 

eta1a := VerticalHomotopy(omega1);

eta1au23CuCu1u13CuCu2+u3Cu1Cu2

(3.3)

 

Check that the vertical exterior derivative of η1 gives ω1.

E > 

omega1 &minus VerticalExteriorDerivative(eta1a);

0CuCu1Cu2

(3.4)

 

Alternatives to η1 can be obtained using the path = "zigzag" option for the VerticalHomotopy command. See DeRhamHomotopy for more details.

E > 

eta1b := VerticalHomotopy(omega1, path = "zigzag");

eta1bu2CuCu1

(3.5)
E > 

omega1 &minus VerticalExteriorDerivative(eta1b);

0CuCu1Cu2

(3.6)
E > 

eta1c := VerticalHomotopy(omega1, path = "zigzag", variableorder = [u[1], u[2], u[], u[1, 1], u[1, 2], u[2, 2]]);

eta1cuCu1Cu2

(3.7)
E > 

omega1 &minus VerticalExteriorDerivative(eta1c);

0CuCu1Cu2

(3.8)

See Also

DifferentialGeometry

JetCalculus

HorizontalExteriorDerivative

HorizontalHomotopy

VerticalExteriorDerivative

ZigZag

 


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