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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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