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JetCalculus[ZigZag] - lift a dH-closed form on a jet space to a d-closed form

Calling Sequences

     ZigZag(ω)

Parameters

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

 

Description

Examples

Description

• 

Let π:EM 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 on J∞E decomposes as a direct sum of bi-forms

ΩpJ∞ = r+s =p Ωr,sJ∞E.

Let &omega;  &Omega;r&comma;sJE be a bi-form of degree r&comma;s. I f r <n &comma; suppose that dH &omega;&equals;0 or, if r&equals; n, that IdV &omega; &equals; 0. See HorizontalExteriorDerivative, VerticalExteriorDerivative, and IntegrationByParts for the definitions of the space Ωr&comma;sJE, the horizontal exterior derivative dH, the vertical exterior derivative dV&comma; and the integration by parts operator I&period; 

Given that dH &omega;&equals;0 or IdV &omega; &equals; 0&comma; define a degree p&equals;r&plus;s form &theta;  ΩpJ  by

&theta; &equals; &theta;0  &theta;1 &plus;&theta;2 &plus;&theta;r where &theta;0 &equals; &omega; and dV&theta;i&equals;dH&theta;i&plus;1&period;

The forms &theta;i are of bi-degree ri&comma; s&plus;i&period; The forms &theta;i can be calculated inductively using the horizontal homotopy operators &theta;i&plus;1&equals; hHri&comma; s&plus;i&plus;1dV&theta;i. The fundamental property of this construction is that the form &theta; is always closed with respect to the standard exterior derivative, that is, d&theta; &equals; 0.

• 

If &omega; is a bi-form of degree r&comma;s&comma;then ZigZag(ω) returns the differential form &theta; of degree r&plus;s.

• 

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 J3Efor the bundle E&equals; &reals;2 ×&reals; with coordinates x&comma;y&comma;ux&comma;y.

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

 

Define a type (1, 0) form &omega;1 and show that it is dH -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);

&omega;1u1,1u1,1,2+u1,2u1,1,1Dx+u1,1u1,2,2+u1,2u1,1,2Dy

(2.1)
E > 

HorizontalExteriorDerivative(omega1);

0DxDy

(2.2)

 

Apply the ZigZag command to &omega;1 to obtain a form &theta;1.

E > 

theta1 := ZigZag(omega1);

&theta;1u1,2du1,1+u1,1du1,2

(2.3)

 

Check that &theta;1 is d-closed and that its [1, 0] component matches &omega;1.

E > 

ExteriorDerivative(theta1);

0dxdy

(2.4)
E > 

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

u1,1u1,1,2+u1,2u1,1,1Dx+u1,1u1,2,2+u1,2u1,1,2Dy

(2.5)

 

Example 2.

Define a type (2, 0) form &omega;2 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);

&omega;2u1u1,2,2+u2u1,2,2+u1,22+u2,2u1,2DxDy

(2.6)
E > 

EulerLagrange(omega2);

0DxDyCu

(2.7)

 

Apply the ZigZag command to &omega;2 to obtain a 2-form &theta;2. 

E > 

theta2 := ZigZag(omega2);

&theta;2u1,23+u2,26dxdu12u1,23dxdu2u13+2u23dxdu1,2u16dxdu2,2+u2,23dydu1+u1,23+u2,26dydu2+u23dydu1,2+u26+u13dydu2,213dudu1,2+16dudu2,2+13du1du2

(2.8)

 

Check that &theta;2 is dclosed and that its [2, 0] component matches &omega;2.

E > 

ExteriorDerivative(theta2);

0dxdydu

(2.9)
E > 

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

u1u1,2,2+u2u1,2,2+u1,22+u2,2u1,2DxDy

(2.10)

 

Example 3.

Define a type (2, 1) form &omega;3 and show that IdV&omega;3 &equals;0.

E > 

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

&omega;32u1u2,2+4u2u1,2DxDyCu

(2.11)
E > 

IntegrationByParts(VerticalExteriorDerivative(omega3));

0DxDyCuCu1

(2.12)

 

Apply the ZigZag command to &omega;3 to obtain a form &theta;3.

E > 

theta3 := ZigZag(omega3);

&theta;32u22dxdydu14u1u2dxdydu2+2u2dxdudu1+2u1dxdudu22u2dydudu2

(2.13)

 

Check that &theta;3 is dclosed and that its [2, 1] component matches &omega;3&period;

E > 

ExteriorDerivative(theta3);

0dxdydudu1

(2.14)
E > 

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

2u1u2,2+4u2u1,2DxDyCu

(2.15)

See Also

DifferentialGeometry

JetCalculus

DeRhamHomotopy

EulerLagrange

ExteriorDerivative

HorizontalExteriorDerivative

HorizontalHomotopy

IntegrationByParts

VerticalExteriorDerivative

VerticalHomotopy

 


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