# State transition matrix to change initial conditions.

In summary, the individual is seeking help with correcting the initial conditions for an orbit in the circular restricted three body problem by using the state transition matrix at a certain time. They have followed the methods described in a paper but have been unable to send both x'(t/2) and z'(t/2) to zero simultaneously. They have spent two weeks trying various methods without success.
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 can't 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 separate 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.

Gonna have to bump this.

2 weeks later, progress = 0.

## 1. What is a state transition matrix?

A state transition matrix is a mathematical tool used in systems analysis and control theory to describe the evolution of a system over time. It is a square matrix that represents the relationship between the current state of a system and its next state.

## 2. How is a state transition matrix used to change initial conditions?

A state transition matrix can be used to change initial conditions by multiplying it with the initial state vector. This will result in a new state vector that represents the system at a later time. By changing the initial state vector, the initial conditions of the system can be altered.

## 3. What information is needed to create a state transition matrix?

To create a state transition matrix, you will need the system's state variables, which represent the system's current state, as well as the system's dynamics, which describe how the state variables change over time. These dynamics are usually represented by a set of differential equations.

## 4. What are the advantages of using a state transition matrix?

The use of a state transition matrix allows for a systematic and efficient way of analyzing the behavior of a system over time. It also enables the prediction of future states based on the current state and system dynamics. Additionally, it can be used to design control systems that can manipulate the state of a system to achieve a desired outcome.

## 5. Are there any limitations to using a state transition matrix?

While state transition matrices are a powerful tool, they do have limitations. They can only be used for systems that can be described by linear dynamics, which means that the relationships between the state variables must be linear. Additionally, state transition matrices assume that the system is time-invariant, meaning that its dynamics do not change over time.

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