form the evolutionary part of a vector field

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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 bundle E -> M

Description

•	The total part (TotalVector) of a generalized vector field X on the bundle E -> M is the generalized vector field Y on E -> M such that X - Y is a vertical vector and Hook(Y, omega) = 0 for any contact 1-form omega on J^1(E).

•	The vertical vector X - Y is called the evolutionary part of the vector field X.

•

The evolutionary part of a projectable vector field X has the following geometric interpretation. Let Phi _t: E -> E be the flow of X. Then Phi_t covers a map Psi_t: M -> M. If sigma: M -> E is a section of E, then the induced flow in the space of sections is defined to be sigma_t(x) = Phi_t (sigma(Psi_( - t)(x))). The derivative of sigma_t with respect to t, evaluated at t = 0, yields the components of the evolutionary part of X.

•

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 a space of 2 independent variables and 2 dependent variables.

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

J22 >

(2.1)

J22 >

(2.2)

J22 >

(2.3)

J22 >

(2.4)

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

J22 >

(2.5)

J22 >

(2.6)

J22 >

(2.7)

J22 >

(2.8)

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

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 E over a two dimensional base. Define the base space M separately.

J22 >

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 >

(2.13)

E >

(2.14)

E >

(2.15)

M >

(2.16)

Calculate the flow psi_t of Y4 and the flow Phi_t of X4.

M >

(2.17)

M >

(2.18)

Define a section sigma of E sending (x, y) to S(x, y).

E >

(2.19)

Calculate the induced flow on the space of sections.

M >

sigma[t] := [x = (x*cos(t)+y*sin(t))*cos(t)-(-x*sin(t)+y*cos(t))*sin(t), y = (x*cos(t)+y*sin(t))*sin(t)+(-x*sin(t)+y*cos(t))*cos(t), u[] = S(x*cos(t)+y*sin(t), -x*sin(t)+y*cos(t))*exp(t)]