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About 2.5 million light years from the Milky Way is the Andromeda Galaxy.

We have about 4 billion years before the two collide as they approach each other things are getting to get interesting. But for the next 2 or 3 billion years it should be fairly calm.

Is there a Lagrangian point with these two galaxies? If so how long would it be stable/present for?

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Nope!

Lagrange points, also called libration points, are (in practice) extended locations that are well defined for a pair of bodies that are in a roughly circular, and periodic orbit.

The Andromeda and Milky way galaxies are moving almost straight at each other, on a collision course. Since their path is nothing like a circular or modestly elliptical orbit, classical Lagrange points are not defined. Once they collide, they will strongly interact, and cease being independent self-contained galaxies, so there is no periodicity to their motion either.

Further discussion:

Mathematically, if the two galaxies were treated as point masses, then one might ask if they could be considered bound in a highly elliptical orbit about their center of mass, or not. If so, and if gravity from other nearby galaxies in the local group can be ignored, then one might further ask if there are libration points or orbits defined. The elliptical restricted 3 body problem can have something similar to Lagrange points, but they would be small orbits themselves, not points. Since one or more of these caveats are probably not valid in the galactic collision example, it doesn't make sense to explore "mathematical curiosities" further here.

Also, see this helpful answer.

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  • $\begingroup$ Great answer, upvoted. For the OP, note that pretty much any randomly selected point in either galaxy is a "Lagrange point" in the sense you would experience virtually zero gravity, since stars are small and far apart. The "collision" you describe would be more the two galaxies passing through each other. The only real difference for planet-bound observers would be that you might see more stars in the sky. $\endgroup$ – barrycarter Jan 14 '17 at 17:19

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