 EvolutionaryVector - Maple Help

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

 • 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 $E$. 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 ${A}^{i}={A}^{i}\left({x}^{j}\right)$ and = ${B}^{\mathrm{β}}($). Let  be the flow of $X$. Then covers a map ${\mathrm{ψ}}_{t}:M\to M$. If $\mathrm{σ}:M\to E$ is a section of $E$, then the induced flow in the space of sections is defined by the section. The derivative of ${\mathrm{\sigma }}_{t}$, evaluated at , yields ${X}_{\mathrm{ev}}$ .
 • 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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{JetCalculus}\right):$

Example 1.

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

 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u,v\right],\mathrm{J22},1\right):$

Define a vector ${X}_{1}$ and compute its total and evolutionary parts tot$\left({X}_{1}\right)$and evol$\left({X}_{1}\right)$. Check that ${X}_{1}$ = tot$\left({X}_{1}\right)$$+$evol$\left({X}_{1}\right).$

 J22 > $\mathrm{X1}≔\mathrm{D_x}$
 ${\mathrm{X1}}{≔}{\mathrm{D_x}}$ (2.1)
 J22 > $\mathrm{totX1}≔\mathrm{TotalVector}\left(\mathrm{X1}\right)$
 ${\mathrm{totX1}}{≔}{\mathrm{D_x}}{+}{{u}}_{{1}}{}{{\mathrm{D_u}}}_{\left[\right]}{+}{{v}}_{{1}}{}{{\mathrm{D_v}}}_{\left[\right]}$ (2.2)
 J22 > $\mathrm{evolX1}≔\mathrm{EvolutionaryVector}\left(\mathrm{X1}\right)$
 ${\mathrm{evolX1}}{≔}{-}{{u}}_{{1}}{}{{\mathrm{D_u}}}_{\left[\right]}{-}{{v}}_{{1}}{}{{\mathrm{D_v}}}_{\left[\right]}$ (2.3)
 J22 > $\mathrm{totX1}&plus\mathrm{evolX1}$
 ${\mathrm{D_x}}$ (2.4)

Define a vector and compute its total and evolutionary parts tot$\left({X}_{2}\right)$ and evol$\left({X}_{2}\right)$. Check that ${X}_{2}$ = tot$\left({X}_{2}\right)$$+$evol$\left({X}_{2}\right).$

 J22 > $\mathrm{X2}≔{\mathrm{D_u}}_{[]}$
 ${\mathrm{X2}}{≔}{{\mathrm{D_u}}}_{\left[\right]}$ (2.5)
 J22 > $\mathrm{totX2}≔\mathrm{TotalVector}\left(\mathrm{X2}\right)$
 ${\mathrm{totX2}}{≔}{0}{}{\mathrm{D_x}}$ (2.6)
 J22 > $\mathrm{evolX2}≔\mathrm{EvolutionaryVector}\left(\mathrm{X2}\right)$
 ${\mathrm{evolX2}}{≔}{{\mathrm{D_u}}}_{\left[\right]}$ (2.7)
 J22 > $\mathrm{totX2}&plus\mathrm{evolX2}$
 ${{\mathrm{D_u}}}_{\left[\right]}$ (2.8)

Define a vector ${X}_{3}$ and compute its total and evolutionary parts tot$\left({X}_{3}\right)$ and evol$\left({X}_{3}\right)$. Check that ${X}_{3}$ = tot$\left({X}_{3}\right)$$+$ evol$\left({X}_{3}\right).$

 J22 > $\mathrm{X3}≔\mathrm{evalDG}\left(a\mathrm{D_x}+b\mathrm{D_y}+c{\mathrm{D_u}}_{[]}+d{\mathrm{D_v}}_{[]}\right)$
 ${\mathrm{X3}}{≔}{a}{}{\mathrm{D_x}}{+}{b}{}{\mathrm{D_y}}{+}{c}{}{{\mathrm{D_u}}}_{\left[\right]}{+}{d}{}{{\mathrm{D_v}}}_{\left[\right]}$ (2.9)
 J22 > $\mathrm{totX3}≔\mathrm{TotalVector}\left(\mathrm{X3}\right)$
 ${\mathrm{totX3}}{≔}{a}{}{\mathrm{D_x}}{+}{b}{}{\mathrm{D_y}}{+}\left({{u}}_{{1}}{}{a}{+}{{u}}_{{2}}{}{b}\right){}{{\mathrm{D_u}}}_{\left[\right]}{+}\left({{v}}_{{1}}{}{a}{+}{{v}}_{{2}}{}{b}\right){}{{\mathrm{D_v}}}_{\left[\right]}$ (2.10)
 J22 > $\mathrm{evolX3}≔\mathrm{EvolutionaryVector}\left(\mathrm{X3}\right)$
 ${\mathrm{evolX3}}{≔}{-}\left({{u}}_{{1}}{}{a}{+}{{u}}_{{2}}{}{b}{-}{c}\right){}{{\mathrm{D_u}}}_{\left[\right]}{-}\left({{v}}_{{1}}{}{a}{+}{{v}}_{{2}}{}{b}{-}{d}\right){}{{\mathrm{D_v}}}_{\left[\right]}$ (2.11)
 J22 > $\mathrm{totX3}&plus\mathrm{evolX3}$
 ${a}{}{\mathrm{D_x}}{+}{b}{}{\mathrm{D_y}}{+}{c}{}{{\mathrm{D_u}}}_{\left[\right]}{+}{d}{}{{\mathrm{D_v}}}_{\left[\right]}$ (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 separately.

 J22 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right):$$\mathrm{DGsetup}\left(\left[x,y\right],\left[u\right],E,2\right):$

Define a vector field and compute its evolutionary part evolDefine the projection of the vector field ${X}_{4}$ onto the base manifold $M.$

 E > $\mathrm{X4}≔\mathrm{evalDG}\left(-y\mathrm{D_x}+x\mathrm{D_y}+{u}_{[]}{\mathrm{D_u}}_{[]}\right)$
 ${\mathrm{X4}}{≔}{-}{y}{}{\mathrm{D_x}}{+}{x}{}{\mathrm{D_y}}{+}{{u}}_{\left[\right]}{}{{\mathrm{D_u}}}_{\left[\right]}$ (2.13)
 E > $\mathrm{evolX4}≔\mathrm{EvolutionaryVector}\left(\mathrm{X4}\right)$
 ${\mathrm{evolX4}}{≔}{-}\left({{u}}_{{2}}{}{x}{-}{{u}}_{{1}}{}{y}{-}{{u}}_{\left[\right]}\right){}{{\mathrm{D_u}}}_{\left[\right]}$ (2.14)
 E > $\mathrm{ChangeFrame}\left(M\right)$
 ${E}$ (2.15)
 M > $\mathrm{Y4}≔\mathrm{evalDG}\left(-y\mathrm{D_x}+x\mathrm{D_y}\right)$
 ${\mathrm{Y4}}{≔}{-}{y}{}{\mathrm{D_x}}{+}{x}{}{\mathrm{D_y}}$ (2.16)

Calculate the flow ${\mathrm{ψ}}_{-t}$ of and the flow ${\mathrm{φ}}_{t}$ of ${X}_{4}$.

 M > $\mathrm{ψ}≔\genfrac{}{}{0}{}{\mathrm{Flow}\left(\mathrm{Y4},t\right)}{\phantom{t=-t}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{Flow}\left(\mathrm{Y4},t\right)}}{t=-t}$
 ${\mathrm{\psi }}{≔}\left[{x}{=}{y}{}{\mathrm{sin}}{}\left({t}\right){+}{x}{}{\mathrm{cos}}{}\left({t}\right){,}{y}{=}{y}{}{\mathrm{cos}}{}\left({t}\right){-}{x}{}{\mathrm{sin}}{}\left({t}\right)\right]$ (2.17)
 M > $\mathrm{Φ}≔\mathrm{Flow}\left(\mathrm{X4},t\right)$
 ${\mathrm{\Phi }}{≔}\left[{x}{=}{-}{y}{}{\mathrm{sin}}{}\left({t}\right){+}{x}{}{\mathrm{cos}}{}\left({t}\right){,}{y}{=}{y}{}{\mathrm{cos}}{}\left({t}\right){+}{x}{}{\mathrm{sin}}{}\left({t}\right){,}{{u}}_{\left[\right]}{=}{{u}}_{\left[\right]}{}{{ⅇ}}^{{t}}\right]$ (2.18)

Define a section of $E$ sending .

 E > $\mathrm{σ}≔\mathrm{Transformation}\left(M,E,\left[x=x,y=y,{u}_{[]}=U\left(x,y\right)\right]\right)$
 ${\mathrm{\sigma }}{≔}\left[{x}{=}{x}{,}{y}{=}{y}{,}{{u}}_{\left[\right]}{=}{U}{}\left({x}{,}{y}\right)\right]$ (2.19)

Calculate the induced flow on the space of sections.

 M > $\mathrm{sigma_t}≔\mathrm{ComposeTransformations}\left(\mathrm{Φ},\mathrm{σ},\mathrm{ψ}\right)$
 ${\mathrm{sigma_t}}{≔}\left[{x}{=}{-}\left({y}{}{\mathrm{cos}}{}\left({t}\right){-}{x}{}{\mathrm{sin}}{}\left({t}\right)\right){}{\mathrm{sin}}{}\left({t}\right){+}\left({y}{}{\mathrm{sin}}{}\left({t}\right){+}{x}{}{\mathrm{cos}}{}\left({t}\right)\right){}{\mathrm{cos}}{}\left({t}\right){,}{y}{=}\left({y}{}{\mathrm{cos}}{}\left({t}\right){-}{x}{}{\mathrm{sin}}{}\left({t}\right)\right){}{\mathrm{cos}}{}\left({t}\right){+}\left({y}{}{\mathrm{sin}}{}\left({t}\right){+}{x}{}{\mathrm{cos}}{}\left({t}\right)\right){}{\mathrm{sin}}{}\left({t}\right){,}{{u}}_{\left[\right]}{=}{U}{}\left({y}{}{\mathrm{sin}}{}\left({t}\right){+}{x}{}{\mathrm{cos}}{}\left({t}\right){,}{y}{}{\mathrm{cos}}{}\left({t}\right){-}{x}{}{\mathrm{sin}}{}\left({t}\right)\right){}{{ⅇ}}^{{t}}\right]$ (2.20)
 M > $\mathrm{Σ}≔\mathrm{ApplyTransformation}\left(\mathrm{sigma_t},\left[x,y\right]\right)$
 ${\mathrm{\Sigma }}{≔}\left[{-}\left({y}{}{\mathrm{cos}}{}\left({t}\right){-}{x}{}{\mathrm{sin}}{}\left({t}\right)\right){}{\mathrm{sin}}{}\left({t}\right){+}\left({y}{}{\mathrm{sin}}{}\left({t}\right){+}{x}{}{\mathrm{cos}}{}\left({t}\right)\right){}{\mathrm{cos}}{}\left({t}\right){,}\left({y}{}{\mathrm{cos}}{}\left({t}\right){-}{x}{}{\mathrm{sin}}{}\left({t}\right)\right){}{\mathrm{cos}}{}\left({t}\right){+}\left({y}{}{\mathrm{sin}}{}\left({t}\right){+}{x}{}{\mathrm{cos}}{}\left({t}\right)\right){}{\mathrm{sin}}{}\left({t}\right){,}{U}{}\left({y}{}{\mathrm{sin}}{}\left({t}\right){+}{x}{}{\mathrm{cos}}{}\left({t}\right){,}{y}{}{\mathrm{cos}}{}\left({t}\right){-}{x}{}{\mathrm{sin}}{}\left({t}\right)\right){}{{ⅇ}}^{{t}}\right]$ (2.21)
 E > $\genfrac{}{}{0}{}{\frac{\partial }{\partial t}\mathrm{Σ}}{\phantom{t=0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{\partial }{\partial t}\mathrm{Σ}}}{t=0}$
 $\left[{0}{,}{0}{,}{{\mathrm{D}}}_{{1}}{}\left({U}\right){}\left({x}{,}{y}\right){}{y}{-}{{\mathrm{D}}}_{{2}}{}\left({U}\right){}\left({x}{,}{y}\right){}{x}{+}{U}{}\left({x}{,}{y}\right)\right]$ (2.22)

Compare with the components of evol$\left({X}_{4}\right).$

 E > $\mathrm{GetComponents}\left(\mathrm{evolX4},\left[\mathrm{D_x},\mathrm{D_y},{\mathrm{D_u}}_{[]}\right]\right)$
 $\left[{0}{,}{0}{,}{-}{x}{}{{u}}_{{2}}{+}{y}{}{{u}}_{{1}}{+}{{u}}_{\left[\right]}\right]$ (2.23)