NDSolve with a constraint : how ?

*To*: mathgroup at smc.vnet.net*Subject*: [mg72477] NDSolve with a constraint : how ?*From*: Cham <martin465 at sympatico.ca>*Date*: Sun, 31 Dec 2006 05:19:03 -0500 (EST)

I need to find a proper way to solve an equation with the NDSolve operation. I'm looking for a solution { x[ t ], y[ t ], z[ t ] } which should obey some constraint (the initial conditions { x[0], y[0], z[0] } are not well known and I only have some very approximate values). How should I do this ? More specifically, I'm using a simple code like this : NDSolve[ { x'[t] == Fx[ x[t], y[t], z[t] ], y'[t] == Fy[ x[t], y[t], z[t] ], z'[t] == Fz[ x[t], y[t], z[t] ], x[0] == x0, y[0] == y0, z[0] == z0 }, {x, y, z}, {t, 0, 100} ] Mathematica then finds easily a solution. But the solution I'm looking for must obey a constraint, and the inital conditions {x0, y0, z0} aren't well known. I need to find the initial values {x0, y0, z0} for which the horizontal distance is the closest to some constant, for an unknown "t" : rho = Sqrt[x[t]^2 + y[t]^2] = cste, for some unknown "t". How can I program Mathematica so it could find the right set of numbers {x0, y0, z0} ? For the moment, all I can do is find a solution from some approximate values {x0, y0, z0}, then check by trial and errors if there's a "t" which gives rho = ctse (approximately). If not, I have to use the NDSolve again, and again (changing a bit the inital values after each trial), until it works. This can be very long, especially since I don't know what value of "t" will satisfy the constraint. Any better idea ?