Global Orbit Structure in the 3-Body Problem: Theorem and Examples | Topic 13

Write any sequence of letters representing the whereabouts of a spacecraft, and there exists a trajectory whose itinerary follows that sequence in the restricted 3-body problem, for example, a spacecraft in the gravity fields of the Earth and Moon.
This is another lecture in a series on the gravitational 3-body problem:
📚 3-Body Problem Orbital Dynamics Course
• Three-Body Problem Orb...
In this lecture, we talk about this theorem on global orbit structure, which Dr. Ross published in 2000, which holds near homoclinic-heteroclinic cycles in the circular restricted three-body problem. The proof is based on establishing Smale horseshoe-like dynamics occurring over 4 Poincaré sections, using generalized Conley-Moser conditions, and developing a system of symbolic dynamics.
We illustrate the theorem with examples.The theorem says we can construct an orbit with any *itinerary*, for example, for a spacecraft in the Earth-Moon system, an itinerary might contain,...
(...,M,X,M,E,M,E,...),
where E, M and X denote the different realms, as described in previous lectures (E = Earth realm, M = Moon realm, X = exterior realm). This corresponds to an orbit which goes from Moon realm to the exterior realm to the Moon realm, to the Earth realm, to the Moon realm, back to the Earth realm, etc.
Not only does there exist a trajectory which performs this itinerary, we can compute initial conditions for the trajectory, and it is *robust*, as there's an entire open set which performs this.
▶️ Next: 3-Body Problem Periodic Orbits & Stable Manifolds using Differential Correction, MATLAB | Topic 14
• 3-Body Problem Periodi...
▶️ Previous: Motion Near L4 and L5 Lagrange Points- Tadpole and Horseshoe Orbit, Trojan Asteroids and the Lucy Mission, Topic 12
• Motion Near L4, L5 Lag...
▶️ If you missed it: Introduction to the Three-Body Problem
• Three Body Problem Int...
▶️ Related: Applications to Dynamical Astronomy
• Interplanetary Transpo...
► Dr. Shane Ross is an Aerospace Engineering Professor at Virginia Tech. He has degrees in physics and engineering mathematics from Caltech, worked at NASA/JPL and Boeing on interplanetary trajectories, and is a world renowned expert on the 3-body problem.
► Reference: Chapter 3, "Heteroclinic Connection and Global Orbit Structure" of my free PDF book:
Dynamical Systems, the Three-Body Problem and Space Mission Design
Koon, Lo, Marsden, Ross (2022)
shaneross.com/b...
► PDF Lecture Notes (*Lecture 9* for this video)
is.gd/3BodyNotes
► TWITTER / X
/ rossdynamicslab
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, equilibria in the rotating frame.
The circular restricted 3-body problem (CR3BP) describes the motion of a body moving in the gravitational field of two primaries orbiting in a circle about their barycenter, with trajectories such as halo orbits, Lyapunov planar orbits, quasi-periodic orbits, quasi-halos, low-energy trajectories, etc.
• The two primaries could be the Earth & Moon, the Sun & Earth, the Sun & 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.
► Related Courses and Series Playlists by Dr. Ross
📚3-Body Problem Orbital Dynamics
is.gd/3BodyPro...
📚Space Manifolds
is.gd/SpaceMan...
📚Space Vehicle Dynamics
is.gd/SpaceVeh...
📚Lagrangian & 3D Rigid Body Dynamics
is.gd/Analytic...
📚Nonlinear Dynamics & Chaos
is.gd/Nonlinea...
📚Hamiltonian Dynamics
is.gd/Advanced...
📚Center Manifolds, Normal Forms, & Bifurcations
is.gd/CenterMa...
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[@chingizakniyazov1693]

Thanks Dr. Ross for doing these lectures.

[@ProfessorRoss]

So nice of you

[@kupo871]

Love all the videos Dr. Ross, ty! I'm going for my master's and these videos are an amazing resource

[@ProfessorRoss]

Glad you like them!