Sunday, 8 September 2019

newtonian gravity - Lagrange points (regions) for n-body systems


The title may be a bit misleading. They may not be points (small areas) at all in this case, but extended regions. Mathematically speaking:


Consider the origin for measurements at O. Let R(t) = {ri;i[1,n]} denote the time varying set of radius vectors of the n bodies, and let M = {mi;i[1,n]} denote their masses. Is it theoretically possible to determine the set of Lagrange points (or regions) L(t)=Lk such that a satellite placed at one of these Lk is fixed w.r.t. some subset of three bodies among the n bodies? I can attempt a brute force programming computation for simulation purposes, but a mathematical approach is appreciated. Even proving the existence of such points will be considered a great help. I honestly have no idea how to proceed with a generalised n-body system.



Answer



1) Regarding determination. No certitude, but I have the feeling that if one could theoretically determine the set of Lagrange points for n bodies, one wouldn't be very far of determining the n bodies trajectories, which we know to be impossible in the general case.



2) Regarding existence. for fixed t, the potential Ω(r) issued from gravity + centrifugal force reads Ω(r)=ni=1Gmi|rri|12|r|2ω2

in a suitable frame rotating with the n bodies with a non zero angular velocity ω.


To prove existence, it is enough to prove that Ω has a maximum somewhere. Take an arbitrary point ra different from the ri's, and let Ωa=Ω(ra). Let K={r; Ω(r)Ωa1}.

Because the function Ω(r) tends to when |r| or rri, K is compact and Ω is continuous on K, hence there exists an rLK where Ω reaches its maximum. Moreover, as Ω(r)=Ωa1 on the boundary of K, and Ω(rL)Ω(ra)>Ωa1, rL belongs to the interior of K. Hence rL is a local maximum of the function Ω on the whole space minus the ri's. Hence it is a Lagrange point (of type L4 - L5). Using the regularity of Ω and the implicit functions theorem, you can follow this point with the time for a certain time if it is not degenerate.


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