Question from Open House Session - 26th Jan 2021

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Sudhash Natarajan

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Jan 26, 2021, 3:54:57 AM1/26/21
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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

Santhosh S

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Jan 27, 2021, 5:08:22 AM1/27/21
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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,
Santhosh

Akarsh Simha

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Jan 27, 2021, 6:08:10 AM1/27/21
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On Wed, Jan 27, 2021 at 02:08 Santhosh S <santho...@gmail.com> wrote:
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).

I was going for the exact same explanation, that you can expand the potential near the minimum as a quadratic potential and therefore a 3D harmonic oscillator; But for it to be a closed Lissajous orbit, doesn’t one need to show that the ratios of frequencies of the oscillators along the three dimensions are rational? I am not sure if this is the case or not. Is there anything indicating that the ratios of the curvatures of the potential in the three dimensions is relatively simple ratios like 1:2 and 3:4? I guess I don’t exactly know what the definition of a Lissajous curve is; I imagine things like a lemniscate etc which you get when the frequencies are locked in small rational ratios.

I tried writing down the equation for Lagrange points and stared it for a while, and quickly gave up... just finding the minima is challenging. Perhaps expanding the force around one of the minima is simpler if one has Mathematica, which I no longer do :-P

Perhaps if the person who asked the question could link the original source that made this statement, we can try digging in...



Anyone, please correct me if I am wrong... (:

Clear skies,
Santhosh



On 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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Akarsh Simha

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Jan 27, 2021, 6:12:01 AM1/27/21
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One more follow-up thought:

In the two body problem, conservation of angular momentum tells us that the orbit must lie in a plane. However, it’s not obvious to me that this is valid for the three body problem, since angular momentum can be absorbed by the sun or the planet... When we talk about a Lissajous curve, we’re usually referring to a planar curve...

Santhosh S

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Jan 27, 2021, 7:51:42 AM1/27/21
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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

Akarsh Simha

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Jan 27, 2021, 9:22:42 PM1/27/21
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On Wed, Jan 27, 2021 at 4:51 AM Santhosh S <santho...@gmail.com> wrote:
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

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."


Santhosh S

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Jan 28, 2021, 4:59:08 AM1/28/21
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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."

Exactly. Ellipse/circle is one specific Lissajous figure right..? I would expect similar behavior in central force problems... 
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