# State transition matrix to change initial conditions.

1. Jul 8, 2011

Hey folks, I have an orbit in the circular restricted three body problem with initial conditions

[x(0), 0, z(0), 0, y'(0), 0]

I'm following this paper

on how to correct these initial conditions given the state transition matrix at a certain time (in this case, the halfway point of the orbit which I'll call t/2).

The orbit is integrated until it crosses the y axis again and at that point the position and velocities are given by

[x(t/2), 0, z(t/2), x'(t/2), y'(t/2), z'(t/2)].

I want to use the state transition matrix to drive x'(t/2) and z'(t/2) to zero by changing the initial condition but I cant get this to work. I have the state transition matrix at t/2 but following the methods in the above paper does not send them to zero. They both decrease for a couple of iterations but then one start to increase.

I can send either x'(t/2), or z'(t/2) to zero (by a little trial and error) but as you decrease one the other increases, it just seems impossible to send them both to zero. I'm following the exact method described in the above paper and it should only take 3-4 iterations apparently. My state transition matrix is correct as it been confirmed by a seperate code I found online.

So, I have...

State transition matrix at t/2

Initial conditions

position and velocity at t/2

The changes desired in the position and velocity at t/2 (i.e. -x'(t/2) and -z'(t/2)).

What do I need to do to find the changes in initial conditions. I've now spent 2 weeks on this trying every method I can find and so far nothing has worked.

If anyone needs any more info (numerical values of state trans matrix, initial conditions, etc) then just ask.

2. Jul 19, 2011