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.

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

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

$X≔\mathrm{evalDG}\left(-y\mathrm{D\_x}+x\mathrm{D\_y}+\frac{1}{4}z\mathrm{D\_z}\right)$

$X≔-y\mathrm{D\_x}+x\mathrm{D\_y}+\frac{z\mathrm{D\_z}}{4}$

$\mathrm{Phi\_t}≔\mathrm{Flow}\left(X\,t\right)$

$\mathrm{Phi\_t}≔\left[x=-y\sin \left(t\right)+x\cos \left(t\right)\,y=y\cos \left(t\right)+x\sin \left(t\right)\,z=z{\ⅇ}^{\frac{t}{4}}\right]$

$\mathrm{combine}\left(\mathrm{ComposeTransformations}\left(\mathrm{eval}\left(\mathrm{Phi\_t}\,t=s\right)\,\mathrm{Phi\_t}\right)\right)$

$\left[x=-y\sin \left(t+s\right)+x\cos \left(t+s\right)\,y=y\cos \left(t+s\right)+x\sin \left(t+s\right)\,z=z{\ⅇ}^{\frac{t}{4}+\frac{s}{4}}\right]$

$\mathrm{eval}\left(\mathrm{Phi\_t}\,t=t+s\right)$

$\left[x=-y\sin \left(t+s\right)+x\cos \left(t+s\right)\,y=y\cos \left(t+s\right)+x\sin \left(t+s\right)\,z=z{\ⅇ}^{\frac{t}{4}+\frac{s}{4}}\right]$

$\mathrm{C1}≔\mathrm{ApplyTransformation}\left(\mathrm{Phi\_t}\,\left[a\,b\,c\right]\right)$

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

$\mathrm{Y1}≔\mathrm{DGzip}\left(\mathrm{diff}\left(\mathrm{C1}\,t\right)\,\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]\,''plus''\right)$

$\mathrm{Y1}≔-\left(b\cos \left(t\right)+a\sin \left(t\right)\right)\mathrm{D\_x}+\left(-b\sin \left(t\right)+a\cos \left(t\right)\right)\mathrm{D\_y}+\frac{c{\ⅇ}^{\frac{t}{4}}\mathrm{D\_z}}{4}$

$\mathrm{C2}≔\mathrm{ApplyTransformation}\left(\mathrm{Phi\_t}\,\left[x=a\,y=b\,z=c\right]\right)$

$\mathrm{C2}≔\left[x=-b\sin \left(t\right)+a\cos \left(t\right)\,y=b\cos \left(t\right)+a\sin \left(t\right)\,z=c{\ⅇ}^{\frac{t}{4}}\right]$

$\mathrm{Y2}≔\mathrm{eval}\left(X\,\mathrm{C2}\right)$

$\mathrm{Y2}≔-\left(b\cos \left(t\right)+a\sin \left(t\right)\right)\mathrm{D\_x}+\left(-b\sin \left(t\right)+a\cos \left(t\right)\right)\mathrm{D\_y}+\frac{c{\ⅇ}^{\frac{t}{4}}\mathrm{D\_z}}{4}$

$\mathrm{Y1}\&minus\mathrm{Y2}$

$0\mathrm{D\_x}$

$\mathrm{Flow}\left(X\,t\,\mathrm{initialpoint}=\left[x=1\,y=0\,z=0\right]\right)$

$\left[x=\cos \left(t\right)\,y=\sin \left(t\right)\,z=0\right]$

$\mathrm{Flow}\left(X\,t\,\mathrm{ode}=\mathrm{true}\right)$

$\left\{-\mathrm{\_z1}\left(t\right)+\frac{\ⅆ}{\ⅆt}\mathrm{\_z2}\left(t\right)\,\mathrm{\_z2}\left(t\right)+\frac{\ⅆ}{\ⅆt}\mathrm{\_z1}\left(t\right)\,-\frac{\mathrm{\_z3}\left(t\right)}{4}+\frac{\ⅆ}{\ⅆt}\mathrm{\_z3}\left(t\right)\right\},\left\{\mathrm{\_z1}\left(0\right)=x\,\mathrm{\_z2}\left(0\right)=y\,\mathrm{\_z3}\left(0\right)=z\right\},\left\{\mathrm{\_z1}\left(t\right)\,\mathrm{\_z2}\left(t\right)\,\mathrm{\_z3}\left(t\right)\right\}$

$\mathrm{Flow}\left(X\,t\,\mathrm{dsolvehints}=\left[\mathrm{method}=\mathrm{laplace}\right]\right)$

$\left[x=-y\sin \left(t\right)+x\cos \left(t\right)\,y=y\cos \left(t\right)+x\sin \left(t\right)\,z=z{\ⅇ}^{\frac{t}{4}}\right]$