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Solving ODE for rotational, irrotational vector field

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104888] Solving ODE for rotational, irrotational vector field
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Thu, 12 Nov 2009 06:07:19 -0500 (EST)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • Reply-to: murray at math.umass.edu

The vector field

   F[{x_,y_}] := {y/(x^2 + y^2), -(x/(x^2 + y^2))}

is smooth (except at the origin, where it's undefined). And, as is 
well-known, the trajectories of the vector differential equation

   {x'[t],y'[t]} == F[{x[t],y[t]}]

are circular about the origin.  And yet this vector field is 
"irrotational", i.e., its curl is {0,0,0} everywhere the field is defined.

I'd like to show that the trajectories really are circular by explicitly 
finding them.  So I tried finding the solutions of the differential 
equation like this:

   DSolve[Thread[D[{x[t],y[t]},t]==F[{x[t],y[t]}]],{x[t],y[t]},t

One gets two solutions.  In fact, if you include initial conditions, e.g.,

   DSolve[{Thread[
    D[{x[t], y[t]}, t] == F[{x[t], y[t]}]], {x[0], y[0]} == {1,
     1}}, {x[t], y[t]}, t]

... you still get two solutions. The components of each solution involve 
Tan and ArcTan, so I assume that's why there are two pieces. But when I 
piece them together by doing ParametricPlot of both on the same axes, I 
don't get circles: I get semi-circles along with the x-axis, which 
clearly seems to be wrong.

Can anybody shed light on this mathematically or Mathematicaly?  In 
particular, are the domains of solutions not {-Infinity,Infinity}?




-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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