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DifferentialGeometry

  

Flow

  

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

 

Calling Sequence

Parameters

Description

Examples

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

withDifferentialGeometry:

DGsetupx,y,z,M:

 

 Example 1.

Calculate the flow Phi_t for the vector field X.

XevalDGyD_x+xD_y+1zD_z4

XyD_x+xD_y+zD_z4

(1)

Phi_tFlowX,t

Phi_tx=ysint+xcost,y=ycost+xsint,z=zⅇt4

(2)

Flow plots

Plot the flows for various initial conditions:

Cmap2ApplyTransformation,Phi_t,1,0,0,1,0,12,0,1,12

Csint,cost,12ⅇ14t,cost,sint,0,cost,sint,12ⅇ14t

(3)

plots[spacecurve]C,t=π..π,axes=normal

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

combineComposeTransformationsPhi_tt=s|Phi_tt=s,Phi_t

x=xcost+sysint+s,y=xsint+s+ycost+s,z=zⅇt4+s4

(4)

Phi_tt=t+s|Phi_tt=t+s

x=xcost+sysint+s,y=xsint+s+ycost+s,z=zⅇt4+s4

(5)

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

C1ApplyTransformationPhi_t,a,b,c

C1bsint+acost,bcost+asint,cⅇ14t

(6)

Differentiate the components of this curve with respect to t.

Y1DGziptC1,D_x,D_y,D_z,plus

Y1bcost+asintD_x+bsint+acostD_y+cⅇt4D_z4

(7)

C2ApplyTransformationPhi_t,x=a,y=b,z=c

C2x=bsint+acost,y=bcost+asint,z=cⅇ14t

(8)

Y2evalX,C2

Y2bcost+asintD_x+bsint+acostD_y+cⅇt4D_z4

(9)

Y1 &minus Y2

0D_x

(10)

 

Example 2.

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

FlowX,t,initialpoint=x=1,y=0,z=0

x=cost,y=sint,z=0

(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.

FlowX,t,ode=true

_z1t+ⅆⅆt_z2t,_z2t+ⅆⅆt_z1t,14_z3t+ⅆⅆt_z3t,_z10=x,_z20=y,_z30=z,_z1t,_z2t,_z3t

(12)

 

Example 4.

FlowX,t,dsolvehints=method=laplace

x=ysint+xcost,y=ycost+xsint,z=zⅇt4

(13)
M > 

See Also

DifferentialGeometry

ApplyTransformation

ComposeTransformations

InfinitesimalTransformation

Preferences

Transformation

 


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