Invariant manifolds are used to calculate low-energy trajectories for spacecraft transiting between Lagrange points. I understand that an invariant manifold is a topological manifold that is invariant under the action of the dynamical system. I assume the “action of the dynamical system” refers to the laws of motion and gravity. But which quantity is “invariant” referring to? Total energy (kinetic plus gravitational potential)?
In a 2-body system, a conic section trajectory is “invariant” in that the sum of the kinetic and gravitational energy is constant. Is this a useful analogy for the invariant manifolds in a 3-body system?
I have a very limited math background. I'm trying to develop an intuitive understanding.