Hello!I might be totally wrong, but this is what I think is happening...If objects are "trapped" in a certain point in space in 3D, you can think of the potential (gravitational potential in this case) has a local minima in this point.So if you place an object at this exact location with no initial velocity, it will stay there forever...Now, if we think in a more realistic manner, when something is "put" into this L-point/local minima/trap, it will have some small velocity when it reaches the minimum point, and can slightly overshoot (jut like a pendulum).Mathematically, you can approximate most smooth and decent potentials as a quadratic function near the minima (Taylor series).So, its like this small overshoot causes the object to oscillate in this harmonic potential which results in an oscillating motion (again, like a pendulum).But remember, its a 3D space here, with the trapping potential being slightly different along 3 perpendicular directions. In addition, the initial velocity can be in any direction.So, you will have oscillations in all 3 directions with different amplitudes and frequencies, and when you project this onto one of the planes, it looks like Lissajous curves (which are oscillations in 2D).
Anyone, please correct me if I am wrong... (:Clear skies,SanthoshOn Tuesday, January 26, 2021 at 9:54:57 AM UTC+1 Sudhash Natarajan wrote:Hi All,Follow-up question from the session, that we couldn't cover this & more technical. Question was asked by Shakti."Why objects captured in lagrange points moves in Lissajous curves?"Experts, any opinions?Thanks
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Hey,I think for the case of a satellite, one would want a closed curve as the orbit for various reasons. So I guess the space agencies inject the satellite such that it gets into a closed curve.But in general, Im not sure if an open curve solution can sustain itself for a long time without finding some decay channels for a closed orbit...Given the source for the above thread, I can maybe get more info on this and try to do some simulations, as Im neither a theorist, nor into astro-dynamics to start from scratch...I am however quite curious, as an ion in a quadrupole trap (paul trap) has very similar micro motion. I want to know if there are some subtle difference (other than the standard ones due to AC field)...Clear skies,Santhosh
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BTW, the standard Classical Mechanics text by Goldstein discusses the three-body problem briefly in Section 3.12 and describes the Lagrange points. However, contrary to talking about Lissajous figures, the book says "Masses in the vicinity [of L₄ and L₅, the stable Lagrange points] experience a force of attraction towards [these points], and can find themselves in stable elliptical-shaped orbits around them."