Flow - Maple Help

DifferentialGeometry

 Flow
 calculate the one parameter group of differeomorphisms (flow) of a vector field

 Calling Sequence Flow(X, var, options)

Parameters

 X - a vector field var - an unassigned Maple name, the flow parameter options - optional arguments to pass to the Maple command dsolve for solving the ordinary differential equations for the flow

Description

 • The flow of a vector field X on a manifold M is a one parameter group of transformations Phi_t: M -> M such that for all p in M, diff(Phi_t(p), t) = X(Phi_t(p)) and Phi_0(p) = p. For each fixed t, Phi_t is a local diffeomorphism of M and Phi_t o Phi_s = Phi_(t + s).
 • The flow of X is calculated by solving a first order system of ordinary differential equations with the Maple dsolve command.
 • If dsolve fails to solve these odes, the Flow command returns NULL.
 • The command Flow returns a transformation whose domain and range coincide with the manifold on which X is defined.
 • With the option ode = true, the system of odes (with initial conditions) defining the flow is returned.
 • With the option initialpoint = [x1 = a, x2 = b, ...], the flow though the specific point [a, b, ...] is calculated.
 • With the option dsolvehints = [hints], the list of optional arguments hints is passed to dsolve.
 • A customized ode solver can be used in place of dsolve though the use of the Preference command.
 • This command is part of the DifferentialGeometry package, and so can be used in the form Flow(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-Flow.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$
 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$

Example 1.

Calculate the flow Phi_t for the vector field X.

 > $X≔\mathrm{evalDG}\left(-y\mathrm{D_x}+x\mathrm{D_y}+\frac{1}{4}z\mathrm{D_z}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}\right]{,}{x}\right]{,}\left[\left[{3}\right]{,}\frac{{1}}{{4}}{}{z}\right]\right]\right]\right)$ (1)
 > $\mathrm{Phi_t}≔\mathrm{Flow}\left(X,t\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{ccc}\mathrm{cos}{}\left(t\right)& -\mathrm{sin}{}\left(t\right)& 0\\ \mathrm{sin}{}\left(t\right)& \mathrm{cos}{}\left(t\right)& 0\\ 0& 0& {ⅇ}^{\frac{1}{4}{}t}\end{array}\right]\right]\right]{,}\left[\left[{-}{y}{}{\mathrm{sin}}{}\left({t}\right){+}{x}{}{\mathrm{cos}}{}\left({t}\right){,}{x}\right]{,}\left[{y}{}{\mathrm{cos}}{}\left({t}\right){+}{x}{}{\mathrm{sin}}{}\left({t}\right){,}{y}\right]{,}\left[{z}{}{\mathrm{exp}}{}\left(\frac{{1}}{{4}}{}{t}\right){,}{z}\right]\right]\right]\right)$ (2)

Flow plots

Plot the flows for various initial conditions:

 > $C≔\mathrm{map2}\left(\mathrm{ApplyTransformation},\mathrm{Phi_t},\left\{\left[0,1,\frac{1}{2}\right],\left[1,0,0\right],\left[1,0,\frac{1}{2}\right]\right\}\right)$
 ${C}{≔}\left\{\left[{-}{\mathrm{sin}}{}\left({t}\right){,}{\mathrm{cos}}{}\left({t}\right){,}\frac{{{ⅇ}}^{\frac{{t}}{{4}}}}{{2}}\right]{,}\left[{\mathrm{cos}}{}\left({t}\right){,}{\mathrm{sin}}{}\left({t}\right){,}{0}\right]{,}\left[{\mathrm{cos}}{}\left({t}\right){,}{\mathrm{sin}}{}\left({t}\right){,}\frac{{{ⅇ}}^{\frac{{t}}{{4}}}}{{2}}\right]\right\}$ (3)
 > $\mathrm{plots}\left[\mathrm{spacecurve}\right]\left(C,t=-\mathrm{\pi }..\mathrm{\pi },\mathrm{axes}=\mathrm{normal}\right)$

We check that Phi_s o Phi_t is Phi_(t + s).

 > $\mathrm{combine}\left(\mathrm{ComposeTransformations}\left(\mathrm{eval}\left(\mathrm{Phi_t},t=s\right),\mathrm{Phi_t}\right)\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{ccc}\mathrm{cos}{}\left(t+s\right)& -\mathrm{sin}{}\left(t+s\right)& 0\\ \mathrm{sin}{}\left(t+s\right)& \mathrm{cos}{}\left(t+s\right)& 0\\ 0& 0& {ⅇ}^{\frac{1}{4}{}t+\frac{1}{4}{}s}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{\mathrm{cos}}{}\left({t}{+}{s}\right){-}{y}{}{\mathrm{sin}}{}\left({t}{+}{s}\right){,}{x}\right]{,}\left[{x}{}{\mathrm{sin}}{}\left({t}{+}{s}\right){+}{y}{}{\mathrm{cos}}{}\left({t}{+}{s}\right){,}{y}\right]{,}\left[{z}{}{\mathrm{exp}}{}\left(\frac{{1}}{{4}}{}{t}{+}\frac{{1}}{{4}}{}{s}\right){,}{z}\right]\right]\right]\right)$ (4)
 > $\mathrm{eval}\left(\mathrm{Phi_t},t=t+s\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{ccc}\mathrm{cos}{}\left(t+s\right)& -\mathrm{sin}{}\left(t+s\right)& 0\\ \mathrm{sin}{}\left(t+s\right)& \mathrm{cos}{}\left(t+s\right)& 0\\ 0& 0& {ⅇ}^{\frac{1}{4}{}t+\frac{1}{4}{}s}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{\mathrm{cos}}{}\left({t}{+}{s}\right){-}{y}{}{\mathrm{sin}}{}\left({t}{+}{s}\right){,}{x}\right]{,}\left[{x}{}{\mathrm{sin}}{}\left({t}{+}{s}\right){+}{y}{}{\mathrm{cos}}{}\left({t}{+}{s}\right){,}{y}\right]{,}\left[{z}{}{\mathrm{exp}}{}\left(\frac{{1}}{{4}}{}{t}{+}\frac{{1}}{{4}}{}{s}\right){,}{z}\right]\right]\right]\right)$ (5)

We check that the derivative of the flow with respect to t coincides with the vector field evaluated along the flow:

 > $\mathrm{C1}≔\mathrm{ApplyTransformation}\left(\mathrm{Phi_t},\left[a,b,c\right]\right)$
 ${\mathrm{C1}}{≔}\left[{-}{b}{}{\mathrm{sin}}{}\left({t}\right){+}{a}{}{\mathrm{cos}}{}\left({t}\right){,}{b}{}{\mathrm{cos}}{}\left({t}\right){+}{a}{}{\mathrm{sin}}{}\left({t}\right){,}{c}{}{{ⅇ}}^{\frac{{t}}{{4}}}\right]$ (6)

Differentiate the components of this curve with respect to t.

 > $\mathrm{Y1}≔\mathrm{DGzip}\left(\mathrm{diff}\left(\mathrm{C1},t\right),\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right],"plus"\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{b}{}{\mathrm{cos}}{}\left({t}\right){-}{a}{}{\mathrm{sin}}{}\left({t}\right)\right]{,}\left[\left[{2}\right]{,}{-}{b}{}{\mathrm{sin}}{}\left({t}\right){+}{a}{}{\mathrm{cos}}{}\left({t}\right)\right]{,}\left[\left[{3}\right]{,}\frac{{1}}{{4}}{}{c}{}{\mathrm{exp}}{}\left(\frac{{1}}{{4}}{}{t}\right)\right]\right]\right]\right)$ (7)
 > $\mathrm{C2}≔\mathrm{ApplyTransformation}\left(\mathrm{Phi_t},\left[x=a,y=b,z=c\right]\right)$
 ${\mathrm{C2}}{≔}\left[{x}{=}{-}{b}{}{\mathrm{sin}}{}\left({t}\right){+}{a}{}{\mathrm{cos}}{}\left({t}\right){,}{y}{=}{b}{}{\mathrm{cos}}{}\left({t}\right){+}{a}{}{\mathrm{sin}}{}\left({t}\right){,}{z}{=}{c}{}{{ⅇ}}^{\frac{{t}}{{4}}}\right]$ (8)
 > $\mathrm{Y2}≔\mathrm{eval}\left(X,\mathrm{C2}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{b}{}{\mathrm{cos}}{}\left({t}\right){-}{a}{}{\mathrm{sin}}{}\left({t}\right)\right]{,}\left[\left[{2}\right]{,}{-}{b}{}{\mathrm{sin}}{}\left({t}\right){+}{a}{}{\mathrm{cos}}{}\left({t}\right)\right]{,}\left[\left[{3}\right]{,}\frac{{1}}{{4}}{}{c}{}{\mathrm{exp}}{}\left(\frac{{1}}{{4}}{}{t}\right)\right]\right]\right]\right)$ (9)
 > $\mathrm{Y1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{Y2}$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right)$ (10)

Example 2.

We find the flow of the vector X through the point (1, 0, 0).

 > $\mathrm{Flow}\left(X,t,\mathrm{initialpoint}=\left[x=1,y=0,z=0\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{rrr}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\right]\right]{,}\left[\left[{\mathrm{cos}}{}\left({t}\right){,}{x}\right]{,}\left[{\mathrm{sin}}{}\left({t}\right){,}{y}\right]{,}\left[{0}{,}{z}\right]\right]\right]\right)$ (11)

Example 3.

We obtain the ode defining the flow for X.  The result consists of a sequence of 3 sets: the ode, the initial conditions, and the dependent variables.

 > $\mathrm{Flow}\left(X,t,\mathrm{ode}=\mathrm{true}\right)$
 $\left\{{-}{\mathrm{_z1}}{}\left({t}\right){+}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_z2}}{}\left({t}\right){,}{\mathrm{_z2}}{}\left({t}\right){+}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_z1}}{}\left({t}\right){,}{-}\frac{{\mathrm{_z3}}{}\left({t}\right)}{{4}}{+}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_z3}}{}\left({t}\right)\right\}{,}\left\{{\mathrm{_z1}}{}\left({0}\right){=}{x}{,}{\mathrm{_z2}}{}\left({0}\right){=}{y}{,}{\mathrm{_z3}}{}\left({0}\right){=}{z}\right\}{,}\left\{{\mathrm{_z1}}{}\left({t}\right){,}{\mathrm{_z2}}{}\left({t}\right){,}{\mathrm{_z3}}{}\left({t}\right)\right\}$ (12)

Example 4.

 > $\mathrm{Flow}\left(X,t,\mathrm{dsolvehints}=\left[\mathrm{method}=\mathrm{laplace}\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{ccc}\mathrm{cos}{}\left(t\right)& -\mathrm{sin}{}\left(t\right)& 0\\ \mathrm{sin}{}\left(t\right)& \mathrm{cos}{}\left(t\right)& 0\\ 0& 0& {ⅇ}^{\frac{1}{4}{}t}\end{array}\right]\right]\right]{,}\left[\left[{-}{y}{}{\mathrm{sin}}{}\left({t}\right){+}{x}{}{\mathrm{cos}}{}\left({t}\right){,}{x}\right]{,}\left[{y}{}{\mathrm{cos}}{}\left({t}\right){+}{x}{}{\mathrm{sin}}{}\left({t}\right){,}{y}\right]{,}\left[{z}{}{\mathrm{exp}}{}\left(\frac{{1}}{{4}}{}{t}\right){,}{z}\right]\right]\right]\right)$ (13)
 M >