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In the book called "Chaotic Worlds: from Order to Disorder in Gravitational N-Body Dynamical Systems", link here, the author states that if the dynamical system has an integral of motion which is not stationary along the periodic orbit, the monodromy matrix has a unit eigenvalue. Check here to see what a monodromy matrix is.

So, assuming we have a periodic orbit with initial conditions $x_0 = [r_0, v_0]^T$. Is the Jacobi constant which is the integral of motion of the CR3BP dynamical system constant?

My intuition says the the Jacobi constant is stationary as it is calculated from the initial conditions of the periodic orbit but why do we get unity eigenvalues.

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Yes it is constant along the trajectory as the Jacobi energy is computed using the initial state vector. From Wikipedia's Jacobi integral:

In celestial mechanics, Jacobi's integral (also known as the Jacobi integral or Jacobi constant) is the only known conserved quantity for the circular restricted three-body problem. Unlike in the two-body problem, the energy and momentum of the system are not conserved separately and a general analytical solution is not possible. The integral has been used to derive numerous solutions in special cases.

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  • $\begingroup$ I've added a quote of a bit of the link so that this isn't a link-only answer. If the link ever breaks, readers will now have at least some information to go on and the answer will still have value. $\endgroup$ – uhoh Aug 31 at 0:32

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