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Re: Solving ODE for rotational, irrotational vector field

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104901] Re: Solving ODE for rotational, irrotational vector field
  • From: "Sjoerd C. de Vries" <sjoerd.c.devries at gmail.com>
  • Date: Fri, 13 Nov 2009 05:51:37 -0500 (EST)
  • References: <hdgqbm$ilg$1@smc.vnet.net>

If you use the intial condition {x[0], y[0]} == {1, -1}} you get the
other half of the circles.

If t passes Pi/2, x jumps from + to - Sqrt[2] while y remains at 0.
Perhaps Mathematica should have drawn it as an exclusion.

Cheers -- Sjoerd

On Nov 12, 1:08 pm, Murray Eisenberg <mur... at math.umass.edu> wrote:
> 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                     mur... 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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