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The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity of solving the effect of three bodies which all pull on each other while moving, a total of six interactions. Mathematician Richard Arenstorf while at NASA solved a special case of this problem, by simplifying the interactions to four, because, the effect of the spacecraft's gravity upon the motion of the vastly more massive Earth and Moon is practically non-existent. Arenstorf found a stable orbit for a spacecraft orbiting between the Earth and Moon, shaped like an '8'

http://en.wikipedia.org/wiki/Richard_Arenstorf

Arenstorf's technical report is here

http://hdl.handle.net/2060/19630005545

Was Arenstorf's solution purely analytical, or did he use numerical mechanisms? Is the '8' shape an optimal path, meaning the route on which the spacecraft would expand the least amount of energy? If yes, how was this requirement included in the derivation in mathematical form?

If anyone has a clean derivation for this problem, that would be great, or any links to books, other papers, etc.

Thanks

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Crossposted to math.SE: math.stackexchange.com/questions/54735 –  Zev Chonoles Jul 31 '11 at 15:41
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5 Answers

Suggest going to Wolfram.com, demonstrations, search for chaos or n-body or 3 body to see solutions to these problems, including Montgomary's figure 8 and Simo's variety of multibody paths.

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The problem of 'optimal path' for going to moon has been studied under the topic of "Circular Restricted three-body problem" and "Planar circular three-body problem (PCR3BP)". Poincare' made major contributions in our understanding of complexity of such class of problems, with this (See his "New methods of celestial mechanics").

The major analytical breakthroughs were made in 60s by C. Conley and R. McGehee (See McGeee's thesis: "Some homoclinic orbits for the restricted three-body problem"), who described the geometry of the PCR3BP near the fixed points L1 and L2. The existence of homo-clinic orbits are proved for certain cases of the parameter $\mu$, which describes the ratio of mass of smaller of the two bodies with the mass of two bodies.

In 90s, the work of Marsden,Lo,Koon,Ross at JPL and Caltech (Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos, 2000) finally computed these homoclinic and heteroclinic intersections and showed that 'low-energy' travel between Earth and Moon can be understood in terms of these manifolds.

For an introduction to the mathematics and computation of 'low-energy' orbits in Restricted three body problem, I would recommend the following (free) e-book:

http://www2.esm.vt.edu/~sdross/books/space_book.html

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2-Body Problems also exist which have no specific solution such that there is a range of solutions for a given physical condition. This means solvability is not based on the number of bodies but the state and representation of space.

Indian Journal of Science and Technology published a physical proof called, “Binary Precession Solutions based on Synchronized Field Couplings”

http://www.indjst.org/index.php/indjst/article/view/30008/25962

In this research, a generalized wave function with classical characteristics was isolated within the motion of binary stars. The wave function provided the first tool for cracking the complex motion of DI Herculis and other binary stars that had several measured precession solutions.

http://xxx.lanl.gov/pdf/1111.3328v2.pdf

In this research, published about a year after the Indian Journal of Science and Technology publication, mathematicians from Imperial College London produced a proof for the physical existence of wave functions. The research was published in Nature Magazine.

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I can only give you a partial answer to the first question, after which I will add a comment on the TBP.

In a sense, Arenstorf's solution cannot be found by purely analytical methods since we only have approximate observations of the positions and interactions of celestial bodies. To give an example, the "official" model for the motion of the Moon is based on the two 900-page volumes of Ch. Delaunay from the 1860's. He computed three power series representing the position of the Moon, taking into account the perturbation of major bodies and the best current observations of the time.

But "computing the series" is not accurate. What he really did was find all the (hundreds of) terms up to the $7^{\rm th}$ degree. With so many intervening variables, this means writing three expressions that use 126 pages. In the end, the most accurate representation available is not a true series but a polynomial expression with approximate coefficients!

It is possible that a stable solution is found using interval arithmetic. This would amount to giving a formal proof that such an orbit exists for a set of parameters that include those that describe the positions of Earth and Mars. I have not looked at Arenstorf's report yet, so I do not know if this is what he did.


Now to the comment. It is technically false to say that "a general analytical solution for TBP is not known". The full expression of the solution in power series was given by Sundman in 1912 (see Wikipedia), and the solution for the $n$-body problem was given in 1991 by Q. Wang (there is a nice report of this in Math. Intelligencer 1996,18,p. 66–70). The reference is "The global solution of the n-body problem". Celestial Mechanics and Dynamical Astronomy 50 (1): 73–88.

Unfortunately, these solutions are purely theoretical as they converge too slowly for practical computation.

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Maybe this 2001 Notices of the AMS article (PDF link) by Richard Montgomery is useful: it describes a figure-8 solution to the 3-body problem as well as some other "coreographies". Some of the latter ones were found numerically by Carles Simó.

EDIT: It is probably not the same figure-8 solution mentioned in the question, since the masses of the three bodies are equal in the case of the Chenciner--Montgomery solution.

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The coreographies, of course, are for the $N$-body problem with $N$ often much larger than 3. –  José Figueroa-O'Farrill Jul 31 '11 at 15:23
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