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The Earth has 7 Lagrange points nearby since SEL1 and SEL2 (Sun-Earth Lagrange points 1 and 2, respectively) are only between 3 and 5 LD (lunar distances) away from the five EML (Earth-Moon Lagrange) points. Can it be useful for spacecraft to move in a sequence between SEL's and EML's in order to lower their missions' delta-v requirements? Has it been done already?

I would think that the monthly and substantial variation in their relative distances and angles should help lower delta-v for a wide range of maneuvers in this wider cislunar region. The SEL1-to-EML2 distance varies between 1,100,000 km and 1,900,000 km over a fortnight, while the EML points always have stationary locations relative to each other. But my impression is that SEL1-EML connections are rarely discussed.

enter image description here

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Can it be useful for spacecraft to move in a sequence between SEL's and EML's in order to lower their missions' delta-v requirements? Has it been done already?

Yes and yes. You are asking about weak stability boundary trajectories. These offer a significantly reduced delta V trajectory from the Earth to the Moon compared to a direct transfer. Two downsides compared to direct transfer are much longer transit times and rather narrow launch windows.

That is exactly how Japan's HITEN and NASA's GRAIL satellites went to the Moon. Those satellites used a weak stability boundary trajectory to significantly reduce delta V requirements. The GRAIL satellites went indirectly from the Earth to lunar orbit via the Sun-Earth L1 point.

Image of the trajectory of GRAIL A and B in the Sun Earth synodic frame. The vehicles were sent close to the Sun-Earth L1 point before traveling to the Moon, with each undergoing trajectory correction maneuvers prior to and after passing near the L1 point.

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    $\begingroup$ I've just found this excellent FISO telecon by David Folta about Artemis (with slides as PDF) where a transfer between EML points is described. It is clear that EML's are perturbed by the Sun. But I was really asking about transfers between an SEL and an EML. Or is there no point with going much farther than the EML's? $\endgroup$
    – LocalFluff
    Jul 23, 2015 at 22:26
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    $\begingroup$ Duh, sorry, now I see that you do write that Grail went by a SUN L-point to the Moon. Great! (I think it is esp interesting since Orion should be able to carry astronauts via SEL, but it is maybe less useful for fast spacecrafts?) $\endgroup$
    – LocalFluff
    Jul 23, 2015 at 22:34
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This isn't a direct answer to your question (which David H has done nicely), but it corrects a minor error in the question.

the EML points always have stationary locations relative to each other

That's not quite true. As I mentioned on Astronomy.SE

the size of the Moon's orbit varies over the course of the year. The maximum monthly variation occurs when the Earth-Moon system is near perihelion or aphelion.

That gives a ~7000 km (7%) variation in the distance between the Earth-Moon L1 and L2 points. Here are some plots (courtesy of JPL Horizons) which show the correlation between the distances. The points are plotted daily at midnight TDB (JPL's $T_{eph}$) over 13 (anomalistic) lunar months. I've included the corresponding plot for the SEL1-to-EML2 distance.

Earth-Moon distance EM L1 to L2 distance SEMB-L1 to EM-L2 distance

You can make your own plots using this live Sage / Python script.

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There is a great YT Channel from a NASA expert consultant on these concepts :

" https://www.youtube.com/c/RossDynamicsLab/search?query=sun%20earth%20moon%20Lagrang "

His LinkedIn : " https://www.linkedin.com/in/shaneross-professor/ "

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