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MathGroup Archive 2007

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Re: NDSolve with a constraint : how ?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72597] Re: NDSolve with a constraint : how ?
  • From: dh <dh at metrohm.ch>
  • Date: Wed, 10 Jan 2007 04:15:22 -0500 (EST)
  • References: <en82l6$6pp$1@smc.vnet.net>


Hi Cham,

here is a 2 dim. example for your problem. Imagine that you throw a 

stone with a angle p and given velocity upward. What is the angle to 

come as near as possible to a given point (the stone has not enough 

energy to reach the point)?

Method: we define a function that calculates the distance of closest 

approach and use this function in FindMinimum:



v=5; (*velocity*)

g=10; (*acceleration*)

reference={2,1}; (*point to approach*)

f:={fx[#],fy[#]}&; (*general 2 dim function*)

dist[p_Real]:=(

     eq=Sequence@@Thread[#]&/@{

            f''[t]=={0,-g}, (*these are Newtons equations *)

           f[0]=={0,0},(*initial conditions*)

           f'[0]==v{Sin[p],Cos[p]}

           };

     res=f[t] /. NDSolve[eq,f[t],{t,0,1}][[1]]; (*the trajectory*)

     t0=FindRoot[

         D[Plus@@((res-reference)^2),

             t]==0,{t,.5}];(*parameter t for closets approach*)

     di=Plus@@((res/.t0)- reference)^2;  (*calculate the distance of the 

closest approach*)

     Print["p=",p,", dist= ",di]; ParametricPlot[res,{t,0,1}];   (*make 

a picture*)

     di

     );

FindMinimum[dist[p],{p,0.5}]



Daniel



Cham wrote:

> 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 ?

> 



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