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