Motion Near L1 and L2: Linearized Equations of Motion in the 3-Body Problem | Topic 9

We study of the behavior of particle trajectories near the two Lagrange points L1 and L2, which are on either side of m2 along the x-axis. We are particularly interested in particles which have an 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.
▶️ Next: Geometry of Motion Near L1 and L2
• Geometry of Motion nea...
▶️ Previous: Calculating Collinear Lagrange Point Positions: L1, L2, L3
• Calculating Collinear ...
▶️ In Case you Missed It: Three-Body Problem Introduction
• Three Body Problem Int...
▶️ Related: Applications to Dynamical Astronomy
• Interplanetary Transpo...
► Reference: Section 2.6, "Linearization near the Collinear Equilibria" of my open-access PDF book:
Dynamical Systems, the Three-Body Problem and Space Mission Design.
Koon, Lo, Marsden, Ross (2011)
shaneross.com/b...
► PDF Lecture Notes (Lecture 5 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 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
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📚Center Manifolds, Normal Forms, and Bifurcations
is.gd/CenterMa...
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[@vinbo2232]

Thank you so much for the course. Waiting for the heteroclinic connection and invariant manifold part...

[@matteovissani1071]

In this case the linearized centre is a non-linear centre because the system has a conserved quantity? Thanks for your lectures. They are truly interesting.

[@ProfessorRoss]

Technically, we justify it by a theorem of Moser described in section 2.6 of the book: www.dept.aoe.vt.edu/~sdross/books/Ross_3BodyProblem_Book_2022.pdf
But I think of the non-linear center manifold as arising as a consequence of Liouville's theorem regarding volume preservation in phase space. I have a lecture on center manifolds in Hamiltonian systems czcams.com/video/Fz-yj_qEvuA/video.html
One could have a non-linear centre without Hamiltonian structure, as shown in an example here czcams.com/video/9rVscJwDpBo/video.html

[@matteovissani1071]

@@ProfessorRoss thanks!