There are two Kepler problems: given the inverse-square law, find the trajectories; or, given Kepler’s laws, find the inverse-square law. Traditionally these problems are solved in the classroom via calculus, but the amount of calculus needed may be prohibitively high for a first-year course. Alternative solutions to the Kepler problems have been discovered, forgotten, and rediscovered for centuries. Many of these employ Hamilton’s hodograph, a graphical representation of an object’s velocity. This article demonstrates hodographic solutions to the Kepler problems, including an algorithm for the construction of parabolic trajectories.
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See Ref. 6, pp. 154–156. See also Richard P. Feynman, The Character of Physical Law (MIT, Cambridge, MA, 1967), Chap. 2, “The Relation of Mathematics to Physics,” pp. 35–37, for a geometric demonstration that a radial force guarantees Kepler’s second law, “equal areas are swept out in equal times.” Feynman credits his demonstration to Newton;
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