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The solution to this system of differential equations:

NDSolve[{
    x''[t] == k[t] y'[t], y''[t] == -k[t] x'[t], 
    x'[0] == A, y'[0] == 0, x[0] == 0, y[0] == 0}, 
  {x, y}, {t, 0, 10}]

satisfies this equation:

x'[t]^2 + y'[t]^2 == A

But when NDSolve solves the system, the speed will eventually move away from it's initial value A.

Is there anyway to make NDSolve keep the speed normalized?

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    $\begingroup$ @ Coolwater: This can be easily done: eliminate y'[t] from x'[t]^2+y'[t]^2==A^2, i.e. y'[t] = +-Sqrt[A^2-x'[t]^2]. Then your only remaining equation will be {x''[t]==+-k[t] Sqrt[A^2-x'[t]^2],x'[0] == A, x[0] == 0}. After specifying k(t), A and the sign of the Sqrt you can NSolve it. $\endgroup$ Aug 21, 2014 at 7:14
  • $\begingroup$ I don't understand how that is useful. When the order of the equation system is reduced to 3, how would I know what the sign of the Sqrt is? It's needed for both y'[t] and x''[t], so only x'[t]and k[t] is available to determine it, which I believe is impossible?! $\endgroup$
    – Coolwater
    Aug 21, 2014 at 8:16
  • $\begingroup$ you should just NDSolve the equation for both signs and then look how they fit together. $\endgroup$ Aug 21, 2014 at 8:22
  • $\begingroup$ BTW the system describes the motion of a charged particle in the x-y-plane in a time-dependent magnetic field along the z-axis given by k(t). It leads to interesting trajectories. Take k(t) = t or k(t) = Sin(t/5) for instance. $\endgroup$ Aug 21, 2014 at 8:25

1 Answer 1

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As it is (unspecified k[t], A) NDSolve will not work. However the equations can be handled analytically. After a simple manipulation you can decouple them and get :

rawx[t_] = x[t] /. DSolve[{k[t] x'''[t] - k'[t] x''[t] == -k[t]^3 x'[t]}, x[t], t]

rawy[t_] = y[t] /. First@DSolve[{D[#, {t, 2}]/k[t] == y'[t]}, y[t], t] & /@ rawx[t]

Now you can check that indeed :

rawx'[t]^2 + rawy'[t]^2 // Simplify
(* {-2 C[1], -2 C[1]} *)

A quicker way is to rename x'[t]->m[t], y'[t]->y[t] and then :

raw = DSolve[{m'[t] == k[t] n[t], n'[t] == -k[t] m[t]}, {m[t], n[t]}, t]
sol[t_] = {m[t], n[t]} /. First@raw

sol[t]^2 // Total // Simplify
(* C[1]^2 + C[2]^2 *)
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