How Does the Restricted Three Body Problem Define Motion Manifolds?

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The discussion centers on the Restricted Three Body Problem (R3BP) and its implications for motion manifolds, specifically focusing on the manifold dimension of 18 when considering two massive bodies and a negligible third body. The participants explore the nature of the symplectic manifold in this context, questioning whether it pertains to the cotangent bundle of the manifold M or a new manifold M'. Additionally, they examine the impact of using orbital elements instead of position and velocities on the symplectic form. The conversation also touches on the relationship between Lagrange's equations and Hamiltonian systems, particularly in relation to action-angle variables as proposed by Liouville's theorem.

PREREQUISITES
  • Understanding of the Restricted Three Body Problem (R3BP)
  • Familiarity with symplectic geometry and manifolds
  • Knowledge of Lagrangian and Hamiltonian mechanics
  • Basic concepts of orbital mechanics, including orbital elements
NEXT STEPS
  • Research the properties of symplectic manifolds in the context of R3BP
  • Study the cotangent bundle and its applications in mechanics
  • Explore the implications of Liouville's theorem on action-angle variables
  • Investigate the transition from position and velocities to orbital elements in dynamical systems
USEFUL FOR

This discussion is beneficial for physicists, mathematicians, and researchers specializing in celestial mechanics, dynamical systems, and symplectic geometry, particularly those focused on the complexities of the Restricted Three Body Problem.

baxter
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Hi

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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