3-Body Problem Equations Derived, Part 1: Inertial Frame and Non-dimensionalization | Topic 2

Equations of motion for a spacecraft in the Circular Restricted Three Body Problem (CR3BP) are derived. This model 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.
• The two primaries could be the Earth and Moon, the Sun and Earth, the Sun and Jupiter, etc.
• The equations of motion of derived in the inertial frame centered at the primaries' center of mass.
• We nondimensionalize the equations of motion, guided by the Buckingham Pi Theorem.
• The main parameter emerging is the mass parameter μ , which is the ratio of the mass of the smaller primary compared to their sum, for example for the Earth-Moon system, it is μ=0.01215
• The mass parameter μ is the only 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.
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We are Chapter 2.
► PDF Lecture Notes (Lecture 1 for this video)
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► 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.
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[@mahmudurrahman1696]

These lectures are really good. Thanks. Btw I few years ago two guys tried to solve the 3 body problem using the random walk approach.

[@BCarli1395]

The Buckingham Pi Theorem was a big stumbling block for me, and it was always at the beginning of the chapter or book so that I felt i needed to understand completely before moving on. I never understood the idea until i saw the algorothm you used. Like riding a bicycle-you dont undetstand what's going on until you do it, i guess. (I’ve found many such cases in math.)

[@ProfessorRoss]

Thanks. I have another video about non-dimensionalizing in physics, using a different mechanical system of the bead in a rotating hoop: czcams.com/video/hw8USek0euU/video.html

[@stefanonovelli9353]

It's amazing the way of conducting your class.It seems to me you talk about this first time!! That's amazing !! Isn't it in Italy(in general).I reserve as accurate comments when I hope I get the hang more about.So far so good.

[@ProfessorRoss]

Thank you! 😃

[@stefanonovelli9353]

The "Lagrange-Euler " equation solutions are stationary points,so at these points of the graph the function-derivative is zero."Lagrange- points" are equilibrium points where gravitational forces(of two massive bodies) and centrifugal force balance each other,but the third body (point,spacecraft) is moving.Isn't that the difference? Is it correct to say that? A thousand thanks.

[@simadavoodi]

really perfect

[@ThompPL1]

5:39. . . Here we have the mysterious "mu" ( but there's another one defined in Lecture #12 ).

[@momen8839]

Is there a relationship between Lagrange points and Lagrange euler equation ?

[@ProfessorRoss]

No, not really, except for the names of the discovers. Lagrange (and Euler) both worked on the 3-body problem and discovered the five famous equilibrium points. AND ALSO, but unrelatedly, they discovered the formulation of equations of motion of a system via the Euler-Lagrange equations (of both mechanics and variational calculus). It's important to note that Euler discovered 3 of the 5 points BEFORE Lagrange, but we typically drop Euler's name. I discuss Euler's paper where he funded L1, L2, and L3 here czcams.com/video/_cqC6q7OfRk/video.html

[@quantika72]

About the five body problem?

[@forheuristiclifeksh7836]

30:15