ZigZag - Maple Help
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JetCalculus[ZigZag] - lift a -closed form on a jet space to a -closed form

Calling Sequences

     ZigZag()

Parameters

           - a differential bi-form on the jet space of a fiber bundle

 

Description

Examples

Description

• 

Let be a fiber bundle, with base dimension  and fiber dimension  and let be the infinite jet bundle of . The space of -forms on decomposes as a direct sum of bi-forms

Let be a bi-form of degree . I f suppose that  or, if , that . See HorizontalExteriorDerivative, VerticalExteriorDerivative, and IntegrationByParts for the definitions of the space , the horizontal exterior derivative , the vertical exterior derivative and the integration by parts operator

Given that  or define a degree form  by

 where and

The forms  are of bi-degree The forms can be calculated inductively using the horizontal homotopy operators . The fundamental property of this construction is that the form is always closed with respect to the standard exterior derivative, that is,

• 

If  is a bi-form of degree then ZigZag() returns the differential form of degree .

• 

The command ZigZag is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form ZigZag(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-ZigZag(...).

Examples

with(DifferentialGeometry): with(JetCalculus):

 

Example 1.

Create the jet space for the bundle with coordinates .

DGsetup([x, y], [u], E, 3):

 

Define a type (1, 0) form and show that it is  -closed.

E > 

omega1 := evalDG((u[1, 2]*u[1, 1, 1] + u[1 ,1]*u[1, 1, 2])*Dx + (u[1, 2]*u[1 ,1, 2] + u[1, 1]*u[1, 2, 2])*Dy);

(2.1)
E > 

HorizontalExteriorDerivative(omega1);

(2.2)

 

Apply the ZigZag command to  to obtain a form .

E > 

theta1 := ZigZag(omega1);

(2.3)

 

Check that  is -closed and that its [1, 0] component matches .

E > 

ExteriorDerivative(theta1);

(2.4)
E > 

convert(theta1, DGbiform, [1, 0]);

(2.5)

 

Example 2.

Define a type (2, 0) form  and show that its Euler-Lagrange form is 0.

E > 

omega2 := evalDG((- u[2, 2]*u[1, 2] - u[2]*u[1, 2, 2] - u[1]*u[1, 2, 2] - u[1, 2]^2)*Dx &w Dy);

(2.6)
E > 

EulerLagrange(omega2);

(2.7)

 

Apply the ZigZag command to  to obtain a 2-form . 

E > 

theta2 := ZigZag(omega2);

(2.8)

 

Check that  is closed and that its [2, 0] component matches .

E > 

ExteriorDerivative(theta2);

(2.9)
E > 

convert(theta2, DGbiform, [2, 0]);

(2.10)

 

Example 3.

Define a type (2, 1) form  and show that .

E > 

omega3 := EulerLagrange(u[1]*u[2]^2*Dx &w Dy);

(2.11)
E > 

IntegrationByParts(VerticalExteriorDerivative(omega3));

(2.12)

 

Apply the ZigZag command to  to obtain a form .

E > 

theta3 := ZigZag(omega3);

(2.13)

 

Check that  is closed and that its [2, 1] component matches

E > 

ExteriorDerivative(theta3);

(2.14)
E > 

convert(theta3, DGbiform, [2, 1]);

(2.15)

See Also

DifferentialGeometry

JetCalculus

DeRhamHomotopy

EulerLagrange

ExteriorDerivative

HorizontalExteriorDerivative

HorizontalHomotopy

IntegrationByParts

VerticalExteriorDerivative

VerticalHomotopy

 


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