Geometry of Motion near L1, L2, 3-Body Dynamical Systems Analysis, McGehee Representation | Topic 10

We analyze the phase space geometry of trajectories near the Lagrange points L1 and L2, the "gateways" on either side of m2 along the m1-m2 line. We focus on particles which have a 3-body energy just above that of the point L1 or L2, when an energy bottleneck has opened up, for which we can linearize the dynamics near the equilibrium point.
To better understand the orbit structure on the phase space, we make a linear change of coordinates with the eigenvectors, u1, u2, w1, w2, as the axes of the new system. Using the corresponding new coordinates (ξ, η, ζ1, ζ2), the differential equations assume a particularly simple form.
We can interpret that there are four types of orbits. On each energy surface, there's a single (unstable) periodic orbit, called a Lyapunov orbit, surrounding the Lagrange point. There are orbits asymptotic towards and away from the periodic orbit. There are also orbits going from one side of the bottleneck to the other, called transit orbits, and those which remain on their side, called non-transit orbits.
We also show the McGehee Representation of the Equilibrium Region. On each energy surface, the Equilibrium Region is topologically equivalent to a 2-sphere cross an interval, the boundaries being called the bounding spheres.
▶️ Next: Categories of Motion in the Equilibrium Regions of L1, L2, & L3
• Trajectory Types Near ...
▶️ Previous: Motion Near L1 and L2: Linearized Equations of Motion
• Motion Near L1 and L2:...
▶️ Three-Body Problem Introduction
• Three Body Problem Int...
▶️ Related: Applications to Dynamical Astronomy
• Interplanetary Transpo...
► Reference: Section 2.7, "Geometry of Solutions near the Equilibria" of my open-access PDF book:
Dynamical Systems, the Three-Body Problem and Space Mission Design.
Koon, Lo, Marsden, Ross (2022)
shaneross.com/b...
► PDF Lecture Notes (Lecture 6 for this video)
is.gd/3BodyNotes
The effective potential energy (also called the augmented potential) is a way to include both the effects of gravity and the centrifugal force of the rotating frame. The critical points of the effective potential energy function, Ū(x,y,z), are the Lagrange points, equilibrium points in the rotating frame (a.k.a., relative equilibria).
The circular restricted 3-body problem (CR3BP) describes the motion of a body moving in the gravitational field of two primaries that are orbiting in a circle about their common center of mass, with trajectories such as Lagrange points, halo orbits, Lyapunov planar orbits, quasi-periodic orbits, quasi-halos, low-energy trajectories, etc.
• The two primaries could be the Earth and Moon, the Sun and Earth, the Sun and Jupiter, etc.
• The equations have been non-dimensionalized
• The mass parameter μ = m2 / (m1 + m2) is the main factor determining the type of motion possible for the spacecraft. It is analogous to the Reynold's number Re in fluid mechanics, as it determines the onset of new types of behavior.
► Dr. Shane Ross is an Aerospace Engineering Professor at Virginia Tech. He has a Ph.D. from Caltech (California Institute of Technology) and worked at NASA/JPL and Boeing.
► Twitter: / rossdynamicslab
► Related Courses and Series Playlists by Dr. Ross
📚3-Body Problem Orbital Dynamics Course
is.gd/3BodyPro...
📚Space Manifolds
is.gd/SpaceMan...
📚Space Vehicle Dynamics
is.gd/SpaceVeh...
📚Lagrangian and 3D Rigid Body Dynamics
is.gd/Analytic...
📚Nonlinear Dynamics and Chaos
is.gd/Nonlinea...
📚Hamiltonian Dynamics
is.gd/Advanced...
📚Center Manifolds, Normal Forms, and Bifurcations
is.gd/CenterMa...
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[@majormelon8855]

Really appreciate these videos. Currently going through this textbook at the moment and these lectures are making it much easier to understand.

[@ProfessorRoss]

Excellent. I’ll post more as I find the time.

[@L0kias1]

@@ProfessorRoss post all you can

[@user-il9wp5fb1e]

Thanks a lot for the lecture. I really appreciate this.

[@RevelationOfDoom777]

Greetings Dr. Ross. Thanks a lot for publishing these lectures. I am really interested on the corellation between the eigenspace of monodromy matrix of the periodic lyapunov orbit and the eigenspace of fixed points. Is this subject discussed somewhere in this series of lectures eventually?

[@ProfessorRoss]

No, I don't discuss this. It would make an interesting analytical study. Basically, consider the Lyapunov orbit in the zero amplitude case and see how it relates to the eigenspace of the fixed point itself. That's what you mean, correct?

[@RevelationOfDoom777]

@@ProfessorRoss Later on, for the calculation of the unstable/stable manifolds you used the eigenvalues of the monodromy matrix. I have a hard time comprehending how do these manifolds relate to the tube-like structures predicted by the linearization around L2/L3, and how does one justify that. I hope that my question makes sense.

[@Astroponicist]

So, you'r saying that space is a 4 demential skate board park where the various common forms move in relation to each other according to the relationships of the major mass bodies? The fourth dimension being time. Thank you for this detailed introduction to multiplanetary navigation potentials.

[@ProfessorRoss]

Yes, that's what I'm saying. And the skate park is rotating. Some more about the potentials here, czcams.com/video/geDtmxtQFzM/video.html

[@L0kias1]

I love this