# Understanding the (restricted) three body problem

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1. Feb 16, 2016

### throneoo

Hello. Recently I've been trying to understand this problem but I'm not sure if I'm doing it correctly. So I'd like to show you some statements formed from my understanding on the subject and you guys could correct me if I'm wrong.

General:
1)The problem has 3*3=9 degrees of freedom and n independent first integrals is needed to solve it using quadrature (Integrable).

2)The independent first integrals here is the center of mass/ total momentum/ total angular momentum (3) and total energy (1) =4 which is not enough to be qualified as integrable.

Circularly restricted (3D)
3) In the inertial frame, the lagrangian is explicitly time dependent as the distances from the two massive bodies depend explicitly on time (constant rotation), hence both the lagrangian and hamiltonian are not conserved.

4)Transforming to the synodic (rotating) frame removes the explicit time dependence so those quantities are conserved, but the hamiltonian is not the total mechanical energy (due to the centrifugal and coriolis force)

5)Total energy is not conserved in the synodic frame due to the inertial forces. Presumably it is conserved in the inertial frame despite the fact that the gravitational influence from the smallest body is neglected.

6) The Lagrangian points are the only equilibrium points in the synodic frame.

Thanks for your help! I might add more to the thread.

2. Feb 18, 2016

### dextercioby

This is an interesting problem for me, too. What books are you using?

3. Feb 18, 2016

### throneoo

Celestial Dynamics by Dvorak and Lhotka is the main text. I also read a lot of articles on this subject