EvolutionaryVector - Maple Help
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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 be a fiber bundle and let  be the associated jet bundle. Let , ..., be the local coordinates on and let  (*) be a generalized vector field on . The coefficients and are functions on jet space. Then the evolutionary part of is the generalized vertical vector field .  Every vector field decomposes as a sum of its evolutionary and total parts .

• 

The evolutionary part of a projectable vector field has the following geometric interpretation (The vector (*) is projectable if  and = ). Let  be the flow of . Then covers a map . If  is a section of , then the induced flow in the space of sections is defined by the section. The derivative of , evaluated at , yields  .

• 

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

 

Example 1.

Create the 1st order jet space of 2 independent variables and 2 dependent variables .

 

Define a vector  and compute its total and evolutionary parts totand evol. Check that  = totevol

J22 > 

(2.1)
J22 > 

(2.2)
J22 > 

(2.3)
J22 > 

(2.4)

 

Define a vector and compute its total and evolutionary parts tot and evol. Check that  = totevol

J22 > 

(2.5)
J22 > 

(2.6)
J22 > 

(2.7)
J22 > 

(2.8)

 

Define a vector  and compute its total and evolutionary parts tot and evol. Check that  = tot evol

J22 > 

(2.9)
J22 > 

(2.10)
J22 > 

(2.11)
J22 > 

(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  over a two dimensional base. Define the base space separately.

J22 > 

 

Define a vector field and compute its evolutionary part evolDefine the projection of the vector field  onto the base manifold

E > 

(2.13)
E > 

(2.14)
E > 

(2.15)
M > 

(2.16)

 

Calculate the flow  of and the flow  of .

M > 

(2.17)
M > 

(2.18)

 

Define a section of  sending .

E > 

(2.19)

 

Calculate the induced flow on the space of sections.

M > 

(2.20)
M > 

(2.21)
E > 

(2.22)

 

Compare with the components of evol

E > 

(2.23)

See Also

DifferentialGeometry

JetCalculus

ApplyTransformation

ComposeTransformations

GetComponents

Prolong

TotalVector

Transformation

 


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