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JetCalculus[TotalVector] - form the total part of a vector field

Calling Sequences

     TotalVector(X)

Parameters

     X    - a vector field or a generalized vector field on a fiber bundle

Description

• 

Let π:EM be a fiber bundle, with base dimension n and fiber dimension m and let πk:JkE  M  be the k-th jet bundle with jet coordinates (xi, uα, uiα, uijα, ..., uij  kα). A total vector field on jet space is a vector field Y of the form Y= AℓDℓ , where the coefficients Aℓ are functions on the jet space JkE and Dℓ is the total vector field for the coordinate xℓ , that is,

Dℓ = xℓ + uℓαuα + uiℓα uiα + uijℓα uijα  + 

Total vector fields may be characterized intrinsically as generalized vector fields which annihilate all contact 1-forms. If X = Aℓ    xℓ +Bα     uα is a generalized vector field on E, then the total part is

 Xtot = Aℓ x + uαuα and the evolutionary part is Xev = Bα Aℓuℓα    uα

The prolongation of Xtot is the total vector field pr(Xtot) = ADℓ.

• 

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

Examples

withDifferentialGeometry:withJetCalculus:

 

Example 1.

Create the jet space J2E for the bundle with local coordinates x, y, u, v x, y. We calculate the total part of some vector fields.

DGsetupx,y,u,v,E,2:

 

Define a vector X1 and compute its total part.

E > 

X1:=evalDGD_x

X1:=D_x

(2.1)
E > 

totX1:=TotalVectorX1

totX1:=D_x+u1D_u[]+v1D_v[]

(2.2)

 

The prolongation of tot(X1) is the total derivativewith respect to x.

E > 

ProlongtotX1,2

D_x+u1D_u[]+v1D_v[]+u1,1D_u1+u1,2D_u2+v1,1D_v1+v1,2D_v2+u1,1,1D_u1,1+u1,1,2D_u1,2+u1,2,2D_u2,2+v1,1,1D_v1,1+v1,1,2D_v1,2+v1,2,2D_v2,2

(2.3)

 

Define a vector X2 and compute its total part.

E > 

X2:=evalDGD_u[]

X2:=D_u[]

(2.4)
E > 

TotalVectorX2

0D_x

(2.5)

 

Define a vector X3 and compute its total part.

E > 

X3:=evalDGaD_x+bD_y+cD_u[]+dD_v[]

X3:=aD_x+bD_y+cD_u[]+dD_v[]

(2.6)
E > 

totX3:=TotalVectorX3

totX3:=aD_x+bD_y+bu2+au1D_u[]+bv2+av1D_v[]

(2.7)

 

Example 2.

We show that the total part of a vector field annihilates the 1st order contact forms.

E > 

DGsetupx,y,z,u,v,w,J33,3:

J33 > 

X4:=w1,2,3D_z

X4:=w1,2,3D_z

(2.8)
J33 > 

totX4:=TotalVectorX4

totX4:=w1,2,3D_z+w1,2,3u3D_u[]+w1,2,3v3D_v[]+w1,2,3w3D_w[]

(2.9)

 

A total vector field always annihilates the first order contact 1-forms.

J33 > 

ω1:=convertCu[],DGform;ω2:=convertCv[],DGform;ω3:=convertCw[],DGform

ω1:=u1dxu2dyu3dz+du[]

ω2:=v1dxv2dyv3dz+dv[]

ω3:=w1dxw2dyw3dz+dw[]

(2.10)
J33 > 

HooktotX4,ω1,HooktotX4,ω2,HooktotX4,ω3

0,0,0

(2.11)

 

A vector field is always the sum of its total and evolutionary parts.

J33 > 

evolX4:=EvolutionaryVectorX4

evolX4:=w1,2,3u3D_u[]w1,2,3v3D_v[]w1,2,3w3D_w[]

(2.12)
J33 > 

totX4 &plus evolX4

w1,2,3D_z

(2.13)

See Also

DifferentialGeometry, JetCalculus, EvolutionaryVector, Hook, Prolong


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