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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.
Define a vector X2 and compute its total and evolutionary parts totX2 and evolX2. Check that X2 = totX2 + evolX2.
Define a vector X3 and compute its total and evolutionary parts totX3 and evolX3. Check that X3 = totX3 + evolX3.
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.
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.
Calculate the flow psi_t of Y4 and the flow Phi_t of X4.
Define a section sigma of E sending (x, y) to S(x, y).
Calculate the induced flow on the space of sections.
Compare with the components of evolX4.
See Also
DifferentialGeometry, JetCalculus, ApplyTransformation, ComposeTransformations, GetComponents, Prolong, TotalVector, Transformation
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