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Let's consider the three body problem.

The motion of all bodies is a manifold of dim 18. But I will consider that the mass of the third body is neglictible and I am interested in the motion of the third body (in the case, this is the restricted three body problem (non necessary planar nor circular)).

0) What is the new manifold that we have to consider ? Does the set of all the movement possible by the third body is a manifold = M'?

1) When we talk about the symplectic manifold, what is it in this case ? Is it the cotangent bundle of M or M' or an other manifold?

What is the exterior 2-form of the structure ?

2) We considered M as the set of position and velocities, does it change something (in particular with the symplectic form) if we consider just the orbital elements of the third body ?

3) It is proved that Lagrange equation is a Hamiltonian system with a,e,i,RA (right ascension), w (argument of periapsis), M (mean anomalie). Is the the action-angles formulation proposed by Liouville's theorem ?

Thanks for your help :)

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# Restricted three body problem

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