A number of people (including Newton himself) have noticed that Newton's elliptical solution for orbits in an inverse square central force field (F ~ 1/r^2) is related to elliptical solutions for orbits in a linear central force field (F ~ r) -- i.e., a simple harmonic oscillator following Hooke's Law. This connection is most elegantly demonstrated when the orbit is in the complex plane, and the harmonic orbit is the simple _square_ of the inverse square law orbit [Needham1993] (equivalently, the inverse square orbit is (one of) the "square root(s)" of the harmonic orbit). This "trick" was also used by Levi-Civita [1920] to "regularize" the solutions to the 2-body problem to gracefully handle the case where the bodies would collide. There was apparently no 3-dimensional version of the Levi-Civita "trick", but Kustaanheimo and Stiefel [KS1965] (also [Waldvogel1972]) found a 4-dimensional version, which has been reinterpreted using quaternions by a number of people, including Vrbik [Vrbik2003] and Saha [Saha2009]. My problem: isn't the extension to the quaternion case completely trivial? After all, the 2-body system is inherently planar, so regardless of the orientation of the plane in 3-space, a simple rigid coordinate rotation should bring the complex plane into alignment with the plane of the 2-body problem. Thus, as Hamilton himself said many times (but not in these words!) in his Elements of Quaternions, the plane spanned by 1 and a "pure" unit quaternion q (pure <=> scalar part=0) is isomorphic to the complex plane, and hence any two quaternions lying in this (1,q) plane commute with one another, so the Levi-Civita algebra all carries over without having to worry about non-commutativity. I realize that the quaternion mapping used by both Vrbik and Saha may not work this way, but the trivial mapping suggested above should work just fine. What am I missing here? (Aarseth's NBODYx codes utilize a KS mapping locally for 2 bodies who are perilously close, so these kinds of mappings are part of the inner workings of some of the best n-body simulation codes in existence.) ----------------- (I was able to find all of these papers on the web w/o having to log in, except for the Levi-Civita and KS papers.) Saha, Prasenjit. "Interpreting the Kustaanheimo-Stiefel transform in gravitational dynamics. Arxiv 0803.4441v3 [astro-ph] 21 Aug 2009, pp1-5. Vrbik, Jan. "A novel solution to Kepler's problem". Eur. J. Phys 24 (2003), 575-583. Aarseth, Sverre J. "From NBODY1 to NBODY6: The Growth of an Industry". Astron. Soc. of the Pacific 111:1333-1346, Nov. 1999. Needham, Tristan. "Newton and the Transmutation of Force". Amer. Math. Monthly 100, 2 (Feb. 1993), 119-137. Waldvogel, Joerg. "A New Regularization of the Planar Problem of Three Bodies". Celestial Mechanics 6 (1972), 221-231. Kustaanheimo, P., Stiefel, E. J. Reine Angew. Math., 218, 204, 1965. Levi-Civita, T. Acta Math., 42, 99, 1920.