Hamiltonian, synodic

Let us consider a rotating or synodic reference frame centered at the barycenter of P and P2 assume that the units of measure are such that the relative angular velocity of P and P2 is unity. Then, the coordinates of Pi and P2 are, respectively, Pi( 2>0) and P2(—Mi,0). Let (qi,q2) be the coordinates of S in the synodic frame (see Figure 4). In order to derive the synodic Hamiltonian, we consider the generating function... 

It is trivial to check that the synodic Hamiltonian takes the form... 

In a synodic reference frame, the Hamiltonian of the planar, circular, restricted three-body problem is given by... 

In the framework of the circular, restricted three-body problem, let us consider the motion in the 3-dimensional space of the three bodies S, Pi and P2. The primaries move in the c/ic/2 plane around their common center of mass, while in the synodic frame their coordinates become Pi(p2,0,0), P2(—Pu 0) 0)- Assume that the (/lfjg-plane rotates with unit angular velocity about the vertical axis. Then, the Hamiltonian function is given by... 

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