Lagrange Points L4, L5 in 3-Body Problem: Derivation of Equilateral Point Location | Topic 7

The location of the equilibrium points in the Circular Restricted 3-Body Problem, starting with L4 and L5, the triangular points or equilateral points. L4 is 60 degrees ahead of the secondary (smaller) mass in its orbit and L5 is 60 degrees behind.
▶️ Next: Calculating Collinear Lagrange Point Positions: L1, L2, L3 in Restricted 3-Body Problem
• Calculating Collinear ...
▶️ Previous: 3-Body Problem Jacobi Constant, Zero Velocity Surfaces, Hill's Regions of Energetically Possible Motion
• 3-Body Problem Jacobi ...
▶️ In Case you Missed It: Three-Body Problem Introduction
• Three Body Problem Int...
▶️ Related: Applications to Dynamical Astronomy
• Interplanetary Transpo...
► Reference: Section 2.5, "Location of the Equilibrium Points" of my FREE PDF book:
Dynamical Systems, the Three-Body Problem and Space Mission Design.
Koon, Lo, Marsden, Ross (2011)
shaneross.com/b...
► PDF Lecture Notes (Lecture 4 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 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.
► 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...
#orbitalmechanics #threebodyproblem #LagrangePoint #L5society #space #oneill #oneillcolony #CR3BP #3body #3bodyproblem #mathematics #SpaceManifolds #SpaceManifoldDynamics #JamesWebb #NonlinearDynamics #gravity #SpaceTravel #SpaceManifold #DynamicalSystems #JamesWebbSpaceTelescope #space #solarSystem #NASA #dynamics #celestial #SpaceSuperhighway #InterplanetarySuperhighway #spaceHighway #spaceHighway #gravitational #mathematics #dynamicalAstronomy #astronomy #wormhole #physics #chaos #unstable #PeriodicOrbits #SaddlePoint #CriticalPoint #Halo #HaloOrbit #LibrationPoint #LagrangianPoint #LowEnergy #VirginiaTech #Caltech #JPL #LyapunovOrbit #CelestialMechanics #HamiltonianDynamics #planets #moons #multibody #GatewayStation #LunarGateway #L1gateway #ArchesOfChaos #cislunar #cislunarspace #orbitalDynamics #orbitalMechanics #celestialChaos #Chaotician #Boeing #JetPropulsionLab #Centaurs #Asteroids #Comets #TrojanAsteroid #Jupiter #JupiterFamily #JupiterFamilyComets #Hildas #quasiHildas #KuiperBelt

[@mdm_sgh]

thank you for this! really helpful.

[@ProfessorRoss]

Glad it was helpful!

[@vinbo2232]

The claim which singles x^2+y^2 out is really surprising.