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JetCalculus[EvolutionaryVector] - form the evolutionary part of a vector field

Calling Sequences

     EvolutionaryVector(X)

Parameters

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

 

Description

Examples

Description

• 

Let π:E  M be a fiber bundle and let πk:JkE M be the associated jet bundle. Let (xi, uα, uiα, uijα, ...,uij  mα) be the local coordinates on JkE and let X = Aj   xi +Bβ    uβ (*) be a generalized vector field on E. The coefficients Ai and Bβ are functions on jet space. Then the evolutionary part of X is the generalized vertical vector field Xev = Bβ Aℓuℓβ    uβ.  Every vector field decomposes as a sum of its evolutionary and total parts X = Xtot + Xev .

• 

The evolutionary part of a projectable vector field X has the following geometric interpretation (The vector (*) is projectable if Ai=Aixj and Bβ = Bβ(xi, uα)). Let φt:E  E be the flow of X. Then φt covers a map ψt:MM. If σ:ME is a section of E, then the induced flow in the space of sections is defined by the section σtx=φtσψtx. The derivative of σt, evaluated at t = 0, yields Xev .

• 

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

Examples

withDifferentialGeometry:withJetCalculus:

 

Example 1.

Create the 1st order jet space of 2 independent variables x,y and 2 dependent variables u, v.

DGsetupx,y,u,v,J22,1:

 

Define a vector X1 and compute its total and evolutionary parts totX1and evolX1. Check that X1 = totX1+evolX1.

J22 > 

X1D_x

X1D_x

(2.1)
J22 > 

totX1TotalVectorX1

totX1D_x+u1D_u+v1D_v

(2.2)
J22 > 

evolX1EvolutionaryVectorX1

evolX1u1D_uv1D_v

(2.3)
J22 > 

totX1 &plus evolX1

D_x

(2.4)

 

Define a vector X2 and compute its total and evolutionary parts totX2 and evolX2. Check that X2 = totX2+evolX2.

J22 > 

X2D_u[]

X2D_u

(2.5)
J22 > 

totX2TotalVectorX2

totX20D_x

(2.6)
J22 > 

evolX2EvolutionaryVectorX2

evolX2D_u

(2.7)
J22 > 

totX2 &plus evolX2

D_u

(2.8)

 

Define a vector X3 and compute its total and evolutionary parts totX3 and evolX3. Check that X3 = totX3+ evolX3.

J22 > 

X3evalDGaD_x+bD_y+cD_u[]+dD_v[]

X3aD_x+bD_y+cD_u+dD_v

(2.9)
J22 > 

totX3TotalVectorX3

totX3aD_x+bD_y+u1a+u2bD_u+v1a+v2bD_v

(2.10)
J22 > 

evolX3EvolutionaryVectorX3

evolX3u1a+u2bcD_uv1a+v2bdD_v

(2.11)
J22 > 

totX3 &plus evolX3

aD_x+bD_y+cD_u+dD_v

(2.12)

 

Example 2.

In this example we illustrate the geometric interpretation of the evolutionary part of a projectable vector field. First define a 3-dimensional bundle E over a two dimensional base. Define the base space M separately.

J22 > 

DGsetupx,y,M:DGsetupx,y,u,E,2:

 

Define a vector field X4 and compute its evolutionary part evolX4. Define the projection Y4 of the vector field X4 onto the base manifold M.

E > 

X4evalDGyD_x+xD_y+u[]D_u[]

X4yD_x+xD_y+uD_u

(2.13)
E > 

evolX4EvolutionaryVectorX4

evolX4u2xu1yuD_u

(2.14)
E > 

ChangeFrameM

E

(2.15)
M > 

Y4evalDGyD_x+xD_y

Y4yD_x+xD_y

(2.16)

 

Calculate the flow ψt of Y4 and the flow φt of X4.

M > 

ψFlowY4,tt=t|FlowY4,tt=t

ψx=ysint+xcost,y=ycostxsint

(2.17)
M > 

ΦFlowX4,t

Φx=ysint+xcost,y=ycost+xsint,u=uⅇt

(2.18)

 

Define a section σ of E sending x,y  Ux,y.

E > 

σTransformationM,E,x=x,y=y,u[]=Ux,y

σx=x,y=y,u=Ux,y

(2.19)

 

Calculate the induced flow on the space of sections.

M > 

sigma_tComposeTransformationsΦ,σ,ψ

sigma_tx=ycostxsintsint+ysint+xcostcost,y=ycostxsintcost+ysint+xcostsint,u=Uysint+xcost,ycostxsintⅇt

(2.20)
M > 

ΣApplyTransformationsigma_t,x,y

Σycostxsintsint+ysint+xcostcost,ycostxsintcost+ysint+xcostsint,Uysint+xcost,ycostxsintⅇt

(2.21)
E > 

tΣt=0|tΣt=0

0,0,D1Ux,yyD2Ux,yx+Ux,y

(2.22)

 

Compare with the components of evolX4.

E > 

GetComponentsevolX4,D_x,D_y,D_u[]

0,0,xu2+yu1+u

(2.23)

See Also

DifferentialGeometry

JetCalculus

ApplyTransformation

ComposeTransformations

GetComponents

Prolong

TotalVector

Transformation

 


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